Index: stable/10/lib/msun/src/e_exp.c =================================================================== --- stable/10/lib/msun/src/e_exp.c (revision 352834) +++ stable/10/lib/msun/src/e_exp.c (revision 352835) @@ -1,164 +1,164 @@ /* @(#)e_exp.c 1.6 04/04/22 */ /* * ==================================================== * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. * * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); /* __ieee754_exp(x) * Returns the exponential of x. * * Method * 1. Argument reduction: * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. * Given x, find r and integer k such that * * x = k*ln2 + r, |r| <= 0.5*ln2. * * Here r will be represented as r = hi-lo for better * accuracy. * * 2. Approximation of exp(r) by a special rational function on * the interval [0,0.34658]: * Write * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... * We use a special Remes algorithm on [0,0.34658] to generate * a polynomial of degree 5 to approximate R. The maximum error * of this polynomial approximation is bounded by 2**-59. In * other words, * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 * (where z=r*r, and the values of P1 to P5 are listed below) * and * | 5 | -59 * | 2.0+P1*z+...+P5*z - R(z) | <= 2 * | | * The computation of exp(r) thus becomes * 2*r * exp(r) = 1 + ------- * R - r * r*R1(r) * = 1 + r + ----------- (for better accuracy) * 2 - R1(r) * where * 2 4 10 * R1(r) = r - (P1*r + P2*r + ... + P5*r ). * * 3. Scale back to obtain exp(x): * From step 1, we have * exp(x) = 2^k * exp(r) * * Special cases: * exp(INF) is INF, exp(NaN) is NaN; * exp(-INF) is 0, and * for finite argument, only exp(0)=1 is exact. * * Accuracy: * according to an error analysis, the error is always less than * 1 ulp (unit in the last place). * * Misc. info. * For IEEE double * if x > 7.09782712893383973096e+02 then exp(x) overflow * if x < -7.45133219101941108420e+02 then exp(x) underflow * * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. */ #include #include "math.h" #include "math_private.h" static const double one = 1.0, halF[2] = {0.5,-0.5,}, o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ static volatile double huge = 1.0e+300, twom1000= 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0*/ double __ieee754_exp(double x) /* default IEEE double exp */ { double y,hi=0.0,lo=0.0,c,t,twopk; int32_t k=0,xsb; u_int32_t hx; GET_HIGH_WORD(hx,x); xsb = (hx>>31)&1; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out non-finite argument */ if(hx >= 0x40862E42) { /* if |x|>=709.78... */ if(hx>=0x7ff00000) { u_int32_t lx; GET_LOW_WORD(lx,x); if(((hx&0xfffff)|lx)!=0) return x+x; /* NaN */ else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ } if(x > o_threshold) return huge*huge; /* overflow */ if(x < u_threshold) return twom1000*twom1000; /* underflow */ } /* argument reduction */ if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; } else { k = (int)(invln2*x+halF[xsb]); t = k; hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ lo = t*ln2LO[0]; } STRICT_ASSIGN(double, x, hi - lo); } else if(hx < 0x3e300000) { /* when |x|<2**-28 */ if(huge+x>one) return one+x;/* trigger inexact */ } else k = 0; /* x is now in primary range */ t = x*x; if(k >= -1021) - INSERT_WORDS(twopk,0x3ff00000+(k<<20), 0); + INSERT_WORDS(twopk,((u_int32_t)(0x3ff+k))<<20, 0); else - INSERT_WORDS(twopk,0x3ff00000+((k+1000)<<20), 0); + INSERT_WORDS(twopk,((u_int32_t)(0x3ff+(k+1000)))<<20, 0); c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); if(k==0) return one-((x*c)/(c-2.0)-x); else y = one-((lo-(x*c)/(2.0-c))-hi); if(k >= -1021) { if (k==1024) return y*2.0*0x1p1023; return y*twopk; } else { return y*twopk*twom1000; } } #if (LDBL_MANT_DIG == 53) __weak_reference(exp, expl); #endif Index: stable/10/lib/msun/src/e_expf.c =================================================================== --- stable/10/lib/msun/src/e_expf.c (revision 352834) +++ stable/10/lib/msun/src/e_expf.c (revision 352835) @@ -1,98 +1,98 @@ /* e_expf.c -- float version of e_exp.c. * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); #include #include "math.h" #include "math_private.h" static const float one = 1.0, halF[2] = {0.5,-0.5,}, o_threshold= 8.8721679688e+01, /* 0x42b17180 */ u_threshold= -1.0397208405e+02, /* 0xc2cff1b5 */ ln2HI[2] ={ 6.9314575195e-01, /* 0x3f317200 */ -6.9314575195e-01,}, /* 0xbf317200 */ ln2LO[2] ={ 1.4286067653e-06, /* 0x35bfbe8e */ -1.4286067653e-06,}, /* 0xb5bfbe8e */ invln2 = 1.4426950216e+00, /* 0x3fb8aa3b */ /* * Domain [-0.34568, 0.34568], range ~[-4.278e-9, 4.447e-9]: * |x*(exp(x)+1)/(exp(x)-1) - p(x)| < 2**-27.74 */ P1 = 1.6666625440e-1, /* 0xaaaa8f.0p-26 */ P2 = -2.7667332906e-3; /* -0xb55215.0p-32 */ static volatile float huge = 1.0e+30, twom100 = 7.8886090522e-31; /* 2**-100=0x0d800000 */ float __ieee754_expf(float x) { float y,hi=0.0,lo=0.0,c,t,twopk; int32_t k=0,xsb; u_int32_t hx; GET_FLOAT_WORD(hx,x); xsb = (hx>>31)&1; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out non-finite argument */ if(hx >= 0x42b17218) { /* if |x|>=88.721... */ if(hx>0x7f800000) return x+x; /* NaN */ if(hx==0x7f800000) return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ if(x > o_threshold) return huge*huge; /* overflow */ if(x < u_threshold) return twom100*twom100; /* underflow */ } /* argument reduction */ if(hx > 0x3eb17218) { /* if |x| > 0.5 ln2 */ if(hx < 0x3F851592) { /* and |x| < 1.5 ln2 */ hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; } else { k = invln2*x+halF[xsb]; t = k; hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ lo = t*ln2LO[0]; } STRICT_ASSIGN(float, x, hi - lo); } else if(hx < 0x39000000) { /* when |x|<2**-14 */ if(huge+x>one) return one+x;/* trigger inexact */ } else k = 0; /* x is now in primary range */ t = x*x; if(k >= -125) - SET_FLOAT_WORD(twopk,0x3f800000+(k<<23)); + SET_FLOAT_WORD(twopk,((u_int32_t)(0x7f+k))<<23); else - SET_FLOAT_WORD(twopk,0x3f800000+((k+100)<<23)); + SET_FLOAT_WORD(twopk,((u_int32_t)(0x7f+(k+100)))<<23); c = x - t*(P1+t*P2); if(k==0) return one-((x*c)/(c-(float)2.0)-x); else y = one-((lo-(x*c)/((float)2.0-c))-hi); if(k >= -125) { if(k==128) return y*2.0F*0x1p127F; return y*twopk; } else { return y*twopk*twom100; } } Index: stable/10/lib/msun/src/s_expm1.c =================================================================== --- stable/10/lib/msun/src/s_expm1.c (revision 352834) +++ stable/10/lib/msun/src/s_expm1.c (revision 352835) @@ -1,222 +1,222 @@ /* @(#)s_expm1.c 5.1 93/09/24 */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); /* expm1(x) * Returns exp(x)-1, the exponential of x minus 1. * * Method * 1. Argument reduction: * Given x, find r and integer k such that * * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 * * Here a correction term c will be computed to compensate * the error in r when rounded to a floating-point number. * * 2. Approximating expm1(r) by a special rational function on * the interval [0,0.34658]: * Since * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... * we define R1(r*r) by * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) * That is, * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... * We use a special Reme algorithm on [0,0.347] to generate * a polynomial of degree 5 in r*r to approximate R1. The * maximum error of this polynomial approximation is bounded * by 2**-61. In other words, * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 * where Q1 = -1.6666666666666567384E-2, * Q2 = 3.9682539681370365873E-4, * Q3 = -9.9206344733435987357E-6, * Q4 = 2.5051361420808517002E-7, * Q5 = -6.2843505682382617102E-9; * z = r*r, * with error bounded by * | 5 | -61 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 * | | * * expm1(r) = exp(r)-1 is then computed by the following * specific way which minimize the accumulation rounding error: * 2 3 * r r [ 3 - (R1 + R1*r/2) ] * expm1(r) = r + --- + --- * [--------------------] * 2 2 [ 6 - r*(3 - R1*r/2) ] * * To compensate the error in the argument reduction, we use * expm1(r+c) = expm1(r) + c + expm1(r)*c * ~ expm1(r) + c + r*c * Thus c+r*c will be added in as the correction terms for * expm1(r+c). Now rearrange the term to avoid optimization * screw up: * ( 2 2 ) * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) * ( ) * * = r - E * 3. Scale back to obtain expm1(x): * From step 1, we have * expm1(x) = either 2^k*[expm1(r)+1] - 1 * = or 2^k*[expm1(r) + (1-2^-k)] * 4. Implementation notes: * (A). To save one multiplication, we scale the coefficient Qi * to Qi*2^i, and replace z by (x^2)/2. * (B). To achieve maximum accuracy, we compute expm1(x) by * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) * (ii) if k=0, return r-E * (iii) if k=-1, return 0.5*(r-E)-0.5 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) * else return 1.0+2.0*(r-E); * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else * (vii) return 2^k(1-((E+2^-k)-r)) * * Special cases: * expm1(INF) is INF, expm1(NaN) is NaN; * expm1(-INF) is -1, and * for finite argument, only expm1(0)=0 is exact. * * Accuracy: * according to an error analysis, the error is always less than * 1 ulp (unit in the last place). * * Misc. info. * For IEEE double * if x > 7.09782712893383973096e+02 then expm1(x) overflow * * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. */ #include #include "math.h" #include "math_private.h" static const double one = 1.0, tiny = 1.0e-300, o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */ Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ static volatile double huge = 1.0e+300; double expm1(double x) { double y,hi,lo,c,t,e,hxs,hfx,r1,twopk; int32_t k,xsb; u_int32_t hx; GET_HIGH_WORD(hx,x); xsb = hx&0x80000000; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out huge and non-finite argument */ if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ if(hx >= 0x40862E42) { /* if |x|>=709.78... */ if(hx>=0x7ff00000) { u_int32_t low; GET_LOW_WORD(low,x); if(((hx&0xfffff)|low)!=0) return x+x; /* NaN */ else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ } if(x > o_threshold) return huge*huge; /* overflow */ } if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ if(x+tiny<0.0) /* raise inexact */ return tiny-one; /* return -1 */ } } /* argument reduction */ if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ if(xsb==0) {hi = x - ln2_hi; lo = ln2_lo; k = 1;} else {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} } else { k = invln2*x+((xsb==0)?0.5:-0.5); t = k; hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ lo = t*ln2_lo; } STRICT_ASSIGN(double, x, hi - lo); c = (hi-x)-lo; } else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ t = huge+x; /* return x with inexact flags when x!=0 */ return x - (t-(huge+x)); } else k = 0; /* x is now in primary range */ hfx = 0.5*x; hxs = x*hfx; r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); t = 3.0-r1*hfx; e = hxs*((r1-t)/(6.0 - x*t)); if(k==0) return x - (x*e-hxs); /* c is 0 */ else { - INSERT_WORDS(twopk,0x3ff00000+(k<<20),0); /* 2^k */ + INSERT_WORDS(twopk,((u_int32_t)(0x3ff+k))<<20,0); /* 2^k */ e = (x*(e-c)-c); e -= hxs; if(k== -1) return 0.5*(x-e)-0.5; if(k==1) { if(x < -0.25) return -2.0*(e-(x+0.5)); else return one+2.0*(x-e); } if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ y = one-(e-x); if (k == 1024) y = y*2.0*0x1p1023; else y = y*twopk; return y-one; } t = one; if(k<20) { SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */ y = t-(e-x); y = y*twopk; } else { SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */ y = x-(e+t); y += one; y = y*twopk; } } return y; } #if (LDBL_MANT_DIG == 53) __weak_reference(expm1, expm1l); #endif Index: stable/10/lib/msun/src/s_expm1f.c =================================================================== --- stable/10/lib/msun/src/s_expm1f.c (revision 352834) +++ stable/10/lib/msun/src/s_expm1f.c (revision 352835) @@ -1,124 +1,124 @@ /* s_expm1f.c -- float version of s_expm1.c. * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); #include #include "math.h" #include "math_private.h" static const float one = 1.0, tiny = 1.0e-30, o_threshold = 8.8721679688e+01,/* 0x42b17180 */ ln2_hi = 6.9313812256e-01,/* 0x3f317180 */ ln2_lo = 9.0580006145e-06,/* 0x3717f7d1 */ invln2 = 1.4426950216e+00,/* 0x3fb8aa3b */ /* * Domain [-0.34568, 0.34568], range ~[-6.694e-10, 6.696e-10]: * |6 / x * (1 + 2 * (1 / (exp(x) - 1) - 1 / x)) - q(x)| < 2**-30.04 * Scaled coefficients: Qn_here = 2**n * Qn_for_q (see s_expm1.c): */ Q1 = -3.3333212137e-2, /* -0x888868.0p-28 */ Q2 = 1.5807170421e-3; /* 0xcf3010.0p-33 */ static volatile float huge = 1.0e+30; float expm1f(float x) { float y,hi,lo,c,t,e,hxs,hfx,r1,twopk; int32_t k,xsb; u_int32_t hx; GET_FLOAT_WORD(hx,x); xsb = hx&0x80000000; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out huge and non-finite argument */ if(hx >= 0x4195b844) { /* if |x|>=27*ln2 */ if(hx >= 0x42b17218) { /* if |x|>=88.721... */ if(hx>0x7f800000) return x+x; /* NaN */ if(hx==0x7f800000) return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ if(x > o_threshold) return huge*huge; /* overflow */ } if(xsb!=0) { /* x < -27*ln2, return -1.0 with inexact */ if(x+tiny<(float)0.0) /* raise inexact */ return tiny-one; /* return -1 */ } } /* argument reduction */ if(hx > 0x3eb17218) { /* if |x| > 0.5 ln2 */ if(hx < 0x3F851592) { /* and |x| < 1.5 ln2 */ if(xsb==0) {hi = x - ln2_hi; lo = ln2_lo; k = 1;} else {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} } else { k = invln2*x+((xsb==0)?(float)0.5:(float)-0.5); t = k; hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ lo = t*ln2_lo; } STRICT_ASSIGN(float, x, hi - lo); c = (hi-x)-lo; } else if(hx < 0x33000000) { /* when |x|<2**-25, return x */ t = huge+x; /* return x with inexact flags when x!=0 */ return x - (t-(huge+x)); } else k = 0; /* x is now in primary range */ hfx = (float)0.5*x; hxs = x*hfx; r1 = one+hxs*(Q1+hxs*Q2); t = (float)3.0-r1*hfx; e = hxs*((r1-t)/((float)6.0 - x*t)); if(k==0) return x - (x*e-hxs); /* c is 0 */ else { - SET_FLOAT_WORD(twopk,0x3f800000+(k<<23)); /* 2^k */ + SET_FLOAT_WORD(twopk,((u_int32_t)(0x7f+k))<<23); /* 2^k */ e = (x*(e-c)-c); e -= hxs; if(k== -1) return (float)0.5*(x-e)-(float)0.5; if(k==1) { if(x < (float)-0.25) return -(float)2.0*(e-(x+(float)0.5)); else return one+(float)2.0*(x-e); } if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ y = one-(e-x); if (k == 128) y = y*2.0F*0x1p127F; else y = y*twopk; return y-one; } t = one; if(k<23) { SET_FLOAT_WORD(t,0x3f800000 - (0x1000000>>k)); /* t=1-2^-k */ y = t-(e-x); y = y*twopk; } else { SET_FLOAT_WORD(t,((0x7f-k)<<23)); /* 2^-k */ y = x-(e+t); y += one; y = y*twopk; } } return y; } Index: stable/10 =================================================================== --- stable/10 (revision 352834) +++ stable/10 (revision 352835) Property changes on: stable/10 ___________________________________________________________________ Modified: svn:mergeinfo ## -0,0 +0,1 ## Merged /head:r352710 Index: stable/11/lib/msun/src/e_exp.c =================================================================== --- stable/11/lib/msun/src/e_exp.c (revision 352834) +++ stable/11/lib/msun/src/e_exp.c (revision 352835) @@ -1,164 +1,164 @@ /* @(#)e_exp.c 1.6 04/04/22 */ /* * ==================================================== * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. * * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); /* __ieee754_exp(x) * Returns the exponential of x. * * Method * 1. Argument reduction: * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. * Given x, find r and integer k such that * * x = k*ln2 + r, |r| <= 0.5*ln2. * * Here r will be represented as r = hi-lo for better * accuracy. * * 2. Approximation of exp(r) by a special rational function on * the interval [0,0.34658]: * Write * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... * We use a special Remes algorithm on [0,0.34658] to generate * a polynomial of degree 5 to approximate R. The maximum error * of this polynomial approximation is bounded by 2**-59. In * other words, * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 * (where z=r*r, and the values of P1 to P5 are listed below) * and * | 5 | -59 * | 2.0+P1*z+...+P5*z - R(z) | <= 2 * | | * The computation of exp(r) thus becomes * 2*r * exp(r) = 1 + ------- * R - r * r*R1(r) * = 1 + r + ----------- (for better accuracy) * 2 - R1(r) * where * 2 4 10 * R1(r) = r - (P1*r + P2*r + ... + P5*r ). * * 3. Scale back to obtain exp(x): * From step 1, we have * exp(x) = 2^k * exp(r) * * Special cases: * exp(INF) is INF, exp(NaN) is NaN; * exp(-INF) is 0, and * for finite argument, only exp(0)=1 is exact. * * Accuracy: * according to an error analysis, the error is always less than * 1 ulp (unit in the last place). * * Misc. info. * For IEEE double * if x > 7.09782712893383973096e+02 then exp(x) overflow * if x < -7.45133219101941108420e+02 then exp(x) underflow * * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. */ #include #include "math.h" #include "math_private.h" static const double one = 1.0, halF[2] = {0.5,-0.5,}, o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ static volatile double huge = 1.0e+300, twom1000= 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0*/ double __ieee754_exp(double x) /* default IEEE double exp */ { double y,hi=0.0,lo=0.0,c,t,twopk; int32_t k=0,xsb; u_int32_t hx; GET_HIGH_WORD(hx,x); xsb = (hx>>31)&1; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out non-finite argument */ if(hx >= 0x40862E42) { /* if |x|>=709.78... */ if(hx>=0x7ff00000) { u_int32_t lx; GET_LOW_WORD(lx,x); if(((hx&0xfffff)|lx)!=0) return x+x; /* NaN */ else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ } if(x > o_threshold) return huge*huge; /* overflow */ if(x < u_threshold) return twom1000*twom1000; /* underflow */ } /* argument reduction */ if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; } else { k = (int)(invln2*x+halF[xsb]); t = k; hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ lo = t*ln2LO[0]; } STRICT_ASSIGN(double, x, hi - lo); } else if(hx < 0x3e300000) { /* when |x|<2**-28 */ if(huge+x>one) return one+x;/* trigger inexact */ } else k = 0; /* x is now in primary range */ t = x*x; if(k >= -1021) - INSERT_WORDS(twopk,0x3ff00000+(k<<20), 0); + INSERT_WORDS(twopk,((u_int32_t)(0x3ff+k))<<20, 0); else - INSERT_WORDS(twopk,0x3ff00000+((k+1000)<<20), 0); + INSERT_WORDS(twopk,((u_int32_t)(0x3ff+(k+1000)))<<20, 0); c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); if(k==0) return one-((x*c)/(c-2.0)-x); else y = one-((lo-(x*c)/(2.0-c))-hi); if(k >= -1021) { if (k==1024) return y*2.0*0x1p1023; return y*twopk; } else { return y*twopk*twom1000; } } #if (LDBL_MANT_DIG == 53) __weak_reference(exp, expl); #endif Index: stable/11/lib/msun/src/e_expf.c =================================================================== --- stable/11/lib/msun/src/e_expf.c (revision 352834) +++ stable/11/lib/msun/src/e_expf.c (revision 352835) @@ -1,98 +1,98 @@ /* e_expf.c -- float version of e_exp.c. * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); #include #include "math.h" #include "math_private.h" static const float one = 1.0, halF[2] = {0.5,-0.5,}, o_threshold= 8.8721679688e+01, /* 0x42b17180 */ u_threshold= -1.0397208405e+02, /* 0xc2cff1b5 */ ln2HI[2] ={ 6.9314575195e-01, /* 0x3f317200 */ -6.9314575195e-01,}, /* 0xbf317200 */ ln2LO[2] ={ 1.4286067653e-06, /* 0x35bfbe8e */ -1.4286067653e-06,}, /* 0xb5bfbe8e */ invln2 = 1.4426950216e+00, /* 0x3fb8aa3b */ /* * Domain [-0.34568, 0.34568], range ~[-4.278e-9, 4.447e-9]: * |x*(exp(x)+1)/(exp(x)-1) - p(x)| < 2**-27.74 */ P1 = 1.6666625440e-1, /* 0xaaaa8f.0p-26 */ P2 = -2.7667332906e-3; /* -0xb55215.0p-32 */ static volatile float huge = 1.0e+30, twom100 = 7.8886090522e-31; /* 2**-100=0x0d800000 */ float __ieee754_expf(float x) { float y,hi=0.0,lo=0.0,c,t,twopk; int32_t k=0,xsb; u_int32_t hx; GET_FLOAT_WORD(hx,x); xsb = (hx>>31)&1; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out non-finite argument */ if(hx >= 0x42b17218) { /* if |x|>=88.721... */ if(hx>0x7f800000) return x+x; /* NaN */ if(hx==0x7f800000) return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ if(x > o_threshold) return huge*huge; /* overflow */ if(x < u_threshold) return twom100*twom100; /* underflow */ } /* argument reduction */ if(hx > 0x3eb17218) { /* if |x| > 0.5 ln2 */ if(hx < 0x3F851592) { /* and |x| < 1.5 ln2 */ hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; } else { k = invln2*x+halF[xsb]; t = k; hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ lo = t*ln2LO[0]; } STRICT_ASSIGN(float, x, hi - lo); } else if(hx < 0x39000000) { /* when |x|<2**-14 */ if(huge+x>one) return one+x;/* trigger inexact */ } else k = 0; /* x is now in primary range */ t = x*x; if(k >= -125) - SET_FLOAT_WORD(twopk,0x3f800000+(k<<23)); + SET_FLOAT_WORD(twopk,((u_int32_t)(0x7f+k))<<23); else - SET_FLOAT_WORD(twopk,0x3f800000+((k+100)<<23)); + SET_FLOAT_WORD(twopk,((u_int32_t)(0x7f+(k+100)))<<23); c = x - t*(P1+t*P2); if(k==0) return one-((x*c)/(c-(float)2.0)-x); else y = one-((lo-(x*c)/((float)2.0-c))-hi); if(k >= -125) { if(k==128) return y*2.0F*0x1p127F; return y*twopk; } else { return y*twopk*twom100; } } Index: stable/11/lib/msun/src/s_expm1.c =================================================================== --- stable/11/lib/msun/src/s_expm1.c (revision 352834) +++ stable/11/lib/msun/src/s_expm1.c (revision 352835) @@ -1,222 +1,222 @@ /* @(#)s_expm1.c 5.1 93/09/24 */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); /* expm1(x) * Returns exp(x)-1, the exponential of x minus 1. * * Method * 1. Argument reduction: * Given x, find r and integer k such that * * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 * * Here a correction term c will be computed to compensate * the error in r when rounded to a floating-point number. * * 2. Approximating expm1(r) by a special rational function on * the interval [0,0.34658]: * Since * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... * we define R1(r*r) by * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) * That is, * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... * We use a special Reme algorithm on [0,0.347] to generate * a polynomial of degree 5 in r*r to approximate R1. The * maximum error of this polynomial approximation is bounded * by 2**-61. In other words, * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 * where Q1 = -1.6666666666666567384E-2, * Q2 = 3.9682539681370365873E-4, * Q3 = -9.9206344733435987357E-6, * Q4 = 2.5051361420808517002E-7, * Q5 = -6.2843505682382617102E-9; * z = r*r, * with error bounded by * | 5 | -61 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 * | | * * expm1(r) = exp(r)-1 is then computed by the following * specific way which minimize the accumulation rounding error: * 2 3 * r r [ 3 - (R1 + R1*r/2) ] * expm1(r) = r + --- + --- * [--------------------] * 2 2 [ 6 - r*(3 - R1*r/2) ] * * To compensate the error in the argument reduction, we use * expm1(r+c) = expm1(r) + c + expm1(r)*c * ~ expm1(r) + c + r*c * Thus c+r*c will be added in as the correction terms for * expm1(r+c). Now rearrange the term to avoid optimization * screw up: * ( 2 2 ) * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) * ( ) * * = r - E * 3. Scale back to obtain expm1(x): * From step 1, we have * expm1(x) = either 2^k*[expm1(r)+1] - 1 * = or 2^k*[expm1(r) + (1-2^-k)] * 4. Implementation notes: * (A). To save one multiplication, we scale the coefficient Qi * to Qi*2^i, and replace z by (x^2)/2. * (B). To achieve maximum accuracy, we compute expm1(x) by * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) * (ii) if k=0, return r-E * (iii) if k=-1, return 0.5*(r-E)-0.5 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) * else return 1.0+2.0*(r-E); * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else * (vii) return 2^k(1-((E+2^-k)-r)) * * Special cases: * expm1(INF) is INF, expm1(NaN) is NaN; * expm1(-INF) is -1, and * for finite argument, only expm1(0)=0 is exact. * * Accuracy: * according to an error analysis, the error is always less than * 1 ulp (unit in the last place). * * Misc. info. * For IEEE double * if x > 7.09782712893383973096e+02 then expm1(x) overflow * * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. */ #include #include "math.h" #include "math_private.h" static const double one = 1.0, tiny = 1.0e-300, o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */ Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ static volatile double huge = 1.0e+300; double expm1(double x) { double y,hi,lo,c,t,e,hxs,hfx,r1,twopk; int32_t k,xsb; u_int32_t hx; GET_HIGH_WORD(hx,x); xsb = hx&0x80000000; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out huge and non-finite argument */ if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ if(hx >= 0x40862E42) { /* if |x|>=709.78... */ if(hx>=0x7ff00000) { u_int32_t low; GET_LOW_WORD(low,x); if(((hx&0xfffff)|low)!=0) return x+x; /* NaN */ else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ } if(x > o_threshold) return huge*huge; /* overflow */ } if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ if(x+tiny<0.0) /* raise inexact */ return tiny-one; /* return -1 */ } } /* argument reduction */ if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ if(xsb==0) {hi = x - ln2_hi; lo = ln2_lo; k = 1;} else {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} } else { k = invln2*x+((xsb==0)?0.5:-0.5); t = k; hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ lo = t*ln2_lo; } STRICT_ASSIGN(double, x, hi - lo); c = (hi-x)-lo; } else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ t = huge+x; /* return x with inexact flags when x!=0 */ return x - (t-(huge+x)); } else k = 0; /* x is now in primary range */ hfx = 0.5*x; hxs = x*hfx; r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); t = 3.0-r1*hfx; e = hxs*((r1-t)/(6.0 - x*t)); if(k==0) return x - (x*e-hxs); /* c is 0 */ else { - INSERT_WORDS(twopk,0x3ff00000+(k<<20),0); /* 2^k */ + INSERT_WORDS(twopk,((u_int32_t)(0x3ff+k))<<20,0); /* 2^k */ e = (x*(e-c)-c); e -= hxs; if(k== -1) return 0.5*(x-e)-0.5; if(k==1) { if(x < -0.25) return -2.0*(e-(x+0.5)); else return one+2.0*(x-e); } if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ y = one-(e-x); if (k == 1024) y = y*2.0*0x1p1023; else y = y*twopk; return y-one; } t = one; if(k<20) { SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */ y = t-(e-x); y = y*twopk; } else { SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */ y = x-(e+t); y += one; y = y*twopk; } } return y; } #if (LDBL_MANT_DIG == 53) __weak_reference(expm1, expm1l); #endif Index: stable/11/lib/msun/src/s_expm1f.c =================================================================== --- stable/11/lib/msun/src/s_expm1f.c (revision 352834) +++ stable/11/lib/msun/src/s_expm1f.c (revision 352835) @@ -1,124 +1,124 @@ /* s_expm1f.c -- float version of s_expm1.c. * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); #include #include "math.h" #include "math_private.h" static const float one = 1.0, tiny = 1.0e-30, o_threshold = 8.8721679688e+01,/* 0x42b17180 */ ln2_hi = 6.9313812256e-01,/* 0x3f317180 */ ln2_lo = 9.0580006145e-06,/* 0x3717f7d1 */ invln2 = 1.4426950216e+00,/* 0x3fb8aa3b */ /* * Domain [-0.34568, 0.34568], range ~[-6.694e-10, 6.696e-10]: * |6 / x * (1 + 2 * (1 / (exp(x) - 1) - 1 / x)) - q(x)| < 2**-30.04 * Scaled coefficients: Qn_here = 2**n * Qn_for_q (see s_expm1.c): */ Q1 = -3.3333212137e-2, /* -0x888868.0p-28 */ Q2 = 1.5807170421e-3; /* 0xcf3010.0p-33 */ static volatile float huge = 1.0e+30; float expm1f(float x) { float y,hi,lo,c,t,e,hxs,hfx,r1,twopk; int32_t k,xsb; u_int32_t hx; GET_FLOAT_WORD(hx,x); xsb = hx&0x80000000; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out huge and non-finite argument */ if(hx >= 0x4195b844) { /* if |x|>=27*ln2 */ if(hx >= 0x42b17218) { /* if |x|>=88.721... */ if(hx>0x7f800000) return x+x; /* NaN */ if(hx==0x7f800000) return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ if(x > o_threshold) return huge*huge; /* overflow */ } if(xsb!=0) { /* x < -27*ln2, return -1.0 with inexact */ if(x+tiny<(float)0.0) /* raise inexact */ return tiny-one; /* return -1 */ } } /* argument reduction */ if(hx > 0x3eb17218) { /* if |x| > 0.5 ln2 */ if(hx < 0x3F851592) { /* and |x| < 1.5 ln2 */ if(xsb==0) {hi = x - ln2_hi; lo = ln2_lo; k = 1;} else {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} } else { k = invln2*x+((xsb==0)?(float)0.5:(float)-0.5); t = k; hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ lo = t*ln2_lo; } STRICT_ASSIGN(float, x, hi - lo); c = (hi-x)-lo; } else if(hx < 0x33000000) { /* when |x|<2**-25, return x */ t = huge+x; /* return x with inexact flags when x!=0 */ return x - (t-(huge+x)); } else k = 0; /* x is now in primary range */ hfx = (float)0.5*x; hxs = x*hfx; r1 = one+hxs*(Q1+hxs*Q2); t = (float)3.0-r1*hfx; e = hxs*((r1-t)/((float)6.0 - x*t)); if(k==0) return x - (x*e-hxs); /* c is 0 */ else { - SET_FLOAT_WORD(twopk,0x3f800000+(k<<23)); /* 2^k */ + SET_FLOAT_WORD(twopk,((u_int32_t)(0x7f+k))<<23); /* 2^k */ e = (x*(e-c)-c); e -= hxs; if(k== -1) return (float)0.5*(x-e)-(float)0.5; if(k==1) { if(x < (float)-0.25) return -(float)2.0*(e-(x+(float)0.5)); else return one+(float)2.0*(x-e); } if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ y = one-(e-x); if (k == 128) y = y*2.0F*0x1p127F; else y = y*twopk; return y-one; } t = one; if(k<23) { SET_FLOAT_WORD(t,0x3f800000 - (0x1000000>>k)); /* t=1-2^-k */ y = t-(e-x); y = y*twopk; } else { SET_FLOAT_WORD(t,((0x7f-k)<<23)); /* 2^-k */ y = x-(e+t); y += one; y = y*twopk; } } return y; } Index: stable/11 =================================================================== --- stable/11 (revision 352834) +++ stable/11 (revision 352835) Property changes on: stable/11 ___________________________________________________________________ Modified: svn:mergeinfo ## -0,0 +0,1 ## Merged /head:r352710 Index: stable/12/lib/msun/src/e_exp.c =================================================================== --- stable/12/lib/msun/src/e_exp.c (revision 352834) +++ stable/12/lib/msun/src/e_exp.c (revision 352835) @@ -1,164 +1,164 @@ /* @(#)e_exp.c 1.6 04/04/22 */ /* * ==================================================== * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. * * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); /* __ieee754_exp(x) * Returns the exponential of x. * * Method * 1. Argument reduction: * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. * Given x, find r and integer k such that * * x = k*ln2 + r, |r| <= 0.5*ln2. * * Here r will be represented as r = hi-lo for better * accuracy. * * 2. Approximation of exp(r) by a special rational function on * the interval [0,0.34658]: * Write * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... * We use a special Remes algorithm on [0,0.34658] to generate * a polynomial of degree 5 to approximate R. The maximum error * of this polynomial approximation is bounded by 2**-59. In * other words, * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 * (where z=r*r, and the values of P1 to P5 are listed below) * and * | 5 | -59 * | 2.0+P1*z+...+P5*z - R(z) | <= 2 * | | * The computation of exp(r) thus becomes * 2*r * exp(r) = 1 + ------- * R - r * r*R1(r) * = 1 + r + ----------- (for better accuracy) * 2 - R1(r) * where * 2 4 10 * R1(r) = r - (P1*r + P2*r + ... + P5*r ). * * 3. Scale back to obtain exp(x): * From step 1, we have * exp(x) = 2^k * exp(r) * * Special cases: * exp(INF) is INF, exp(NaN) is NaN; * exp(-INF) is 0, and * for finite argument, only exp(0)=1 is exact. * * Accuracy: * according to an error analysis, the error is always less than * 1 ulp (unit in the last place). * * Misc. info. * For IEEE double * if x > 7.09782712893383973096e+02 then exp(x) overflow * if x < -7.45133219101941108420e+02 then exp(x) underflow * * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. */ #include #include "math.h" #include "math_private.h" static const double one = 1.0, halF[2] = {0.5,-0.5,}, o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ static volatile double huge = 1.0e+300, twom1000= 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0*/ double __ieee754_exp(double x) /* default IEEE double exp */ { double y,hi=0.0,lo=0.0,c,t,twopk; int32_t k=0,xsb; u_int32_t hx; GET_HIGH_WORD(hx,x); xsb = (hx>>31)&1; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out non-finite argument */ if(hx >= 0x40862E42) { /* if |x|>=709.78... */ if(hx>=0x7ff00000) { u_int32_t lx; GET_LOW_WORD(lx,x); if(((hx&0xfffff)|lx)!=0) return x+x; /* NaN */ else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ } if(x > o_threshold) return huge*huge; /* overflow */ if(x < u_threshold) return twom1000*twom1000; /* underflow */ } /* argument reduction */ if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; } else { k = (int)(invln2*x+halF[xsb]); t = k; hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ lo = t*ln2LO[0]; } STRICT_ASSIGN(double, x, hi - lo); } else if(hx < 0x3e300000) { /* when |x|<2**-28 */ if(huge+x>one) return one+x;/* trigger inexact */ } else k = 0; /* x is now in primary range */ t = x*x; if(k >= -1021) - INSERT_WORDS(twopk,0x3ff00000+(k<<20), 0); + INSERT_WORDS(twopk,((u_int32_t)(0x3ff+k))<<20, 0); else - INSERT_WORDS(twopk,0x3ff00000+((k+1000)<<20), 0); + INSERT_WORDS(twopk,((u_int32_t)(0x3ff+(k+1000)))<<20, 0); c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); if(k==0) return one-((x*c)/(c-2.0)-x); else y = one-((lo-(x*c)/(2.0-c))-hi); if(k >= -1021) { if (k==1024) return y*2.0*0x1p1023; return y*twopk; } else { return y*twopk*twom1000; } } #if (LDBL_MANT_DIG == 53) __weak_reference(exp, expl); #endif Index: stable/12/lib/msun/src/e_expf.c =================================================================== --- stable/12/lib/msun/src/e_expf.c (revision 352834) +++ stable/12/lib/msun/src/e_expf.c (revision 352835) @@ -1,98 +1,98 @@ /* e_expf.c -- float version of e_exp.c. * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); #include #include "math.h" #include "math_private.h" static const float one = 1.0, halF[2] = {0.5,-0.5,}, o_threshold= 8.8721679688e+01, /* 0x42b17180 */ u_threshold= -1.0397208405e+02, /* 0xc2cff1b5 */ ln2HI[2] ={ 6.9314575195e-01, /* 0x3f317200 */ -6.9314575195e-01,}, /* 0xbf317200 */ ln2LO[2] ={ 1.4286067653e-06, /* 0x35bfbe8e */ -1.4286067653e-06,}, /* 0xb5bfbe8e */ invln2 = 1.4426950216e+00, /* 0x3fb8aa3b */ /* * Domain [-0.34568, 0.34568], range ~[-4.278e-9, 4.447e-9]: * |x*(exp(x)+1)/(exp(x)-1) - p(x)| < 2**-27.74 */ P1 = 1.6666625440e-1, /* 0xaaaa8f.0p-26 */ P2 = -2.7667332906e-3; /* -0xb55215.0p-32 */ static volatile float huge = 1.0e+30, twom100 = 7.8886090522e-31; /* 2**-100=0x0d800000 */ float __ieee754_expf(float x) { float y,hi=0.0,lo=0.0,c,t,twopk; int32_t k=0,xsb; u_int32_t hx; GET_FLOAT_WORD(hx,x); xsb = (hx>>31)&1; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out non-finite argument */ if(hx >= 0x42b17218) { /* if |x|>=88.721... */ if(hx>0x7f800000) return x+x; /* NaN */ if(hx==0x7f800000) return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ if(x > o_threshold) return huge*huge; /* overflow */ if(x < u_threshold) return twom100*twom100; /* underflow */ } /* argument reduction */ if(hx > 0x3eb17218) { /* if |x| > 0.5 ln2 */ if(hx < 0x3F851592) { /* and |x| < 1.5 ln2 */ hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; } else { k = invln2*x+halF[xsb]; t = k; hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ lo = t*ln2LO[0]; } STRICT_ASSIGN(float, x, hi - lo); } else if(hx < 0x39000000) { /* when |x|<2**-14 */ if(huge+x>one) return one+x;/* trigger inexact */ } else k = 0; /* x is now in primary range */ t = x*x; if(k >= -125) - SET_FLOAT_WORD(twopk,0x3f800000+(k<<23)); + SET_FLOAT_WORD(twopk,((u_int32_t)(0x7f+k))<<23); else - SET_FLOAT_WORD(twopk,0x3f800000+((k+100)<<23)); + SET_FLOAT_WORD(twopk,((u_int32_t)(0x7f+(k+100)))<<23); c = x - t*(P1+t*P2); if(k==0) return one-((x*c)/(c-(float)2.0)-x); else y = one-((lo-(x*c)/((float)2.0-c))-hi); if(k >= -125) { if(k==128) return y*2.0F*0x1p127F; return y*twopk; } else { return y*twopk*twom100; } } Index: stable/12/lib/msun/src/s_expm1.c =================================================================== --- stable/12/lib/msun/src/s_expm1.c (revision 352834) +++ stable/12/lib/msun/src/s_expm1.c (revision 352835) @@ -1,222 +1,222 @@ /* @(#)s_expm1.c 5.1 93/09/24 */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); /* expm1(x) * Returns exp(x)-1, the exponential of x minus 1. * * Method * 1. Argument reduction: * Given x, find r and integer k such that * * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 * * Here a correction term c will be computed to compensate * the error in r when rounded to a floating-point number. * * 2. Approximating expm1(r) by a special rational function on * the interval [0,0.34658]: * Since * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... * we define R1(r*r) by * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) * That is, * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... * We use a special Reme algorithm on [0,0.347] to generate * a polynomial of degree 5 in r*r to approximate R1. The * maximum error of this polynomial approximation is bounded * by 2**-61. In other words, * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 * where Q1 = -1.6666666666666567384E-2, * Q2 = 3.9682539681370365873E-4, * Q3 = -9.9206344733435987357E-6, * Q4 = 2.5051361420808517002E-7, * Q5 = -6.2843505682382617102E-9; * z = r*r, * with error bounded by * | 5 | -61 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 * | | * * expm1(r) = exp(r)-1 is then computed by the following * specific way which minimize the accumulation rounding error: * 2 3 * r r [ 3 - (R1 + R1*r/2) ] * expm1(r) = r + --- + --- * [--------------------] * 2 2 [ 6 - r*(3 - R1*r/2) ] * * To compensate the error in the argument reduction, we use * expm1(r+c) = expm1(r) + c + expm1(r)*c * ~ expm1(r) + c + r*c * Thus c+r*c will be added in as the correction terms for * expm1(r+c). Now rearrange the term to avoid optimization * screw up: * ( 2 2 ) * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) * ( ) * * = r - E * 3. Scale back to obtain expm1(x): * From step 1, we have * expm1(x) = either 2^k*[expm1(r)+1] - 1 * = or 2^k*[expm1(r) + (1-2^-k)] * 4. Implementation notes: * (A). To save one multiplication, we scale the coefficient Qi * to Qi*2^i, and replace z by (x^2)/2. * (B). To achieve maximum accuracy, we compute expm1(x) by * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) * (ii) if k=0, return r-E * (iii) if k=-1, return 0.5*(r-E)-0.5 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) * else return 1.0+2.0*(r-E); * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else * (vii) return 2^k(1-((E+2^-k)-r)) * * Special cases: * expm1(INF) is INF, expm1(NaN) is NaN; * expm1(-INF) is -1, and * for finite argument, only expm1(0)=0 is exact. * * Accuracy: * according to an error analysis, the error is always less than * 1 ulp (unit in the last place). * * Misc. info. * For IEEE double * if x > 7.09782712893383973096e+02 then expm1(x) overflow * * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. */ #include #include "math.h" #include "math_private.h" static const double one = 1.0, tiny = 1.0e-300, o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */ Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ static volatile double huge = 1.0e+300; double expm1(double x) { double y,hi,lo,c,t,e,hxs,hfx,r1,twopk; int32_t k,xsb; u_int32_t hx; GET_HIGH_WORD(hx,x); xsb = hx&0x80000000; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out huge and non-finite argument */ if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ if(hx >= 0x40862E42) { /* if |x|>=709.78... */ if(hx>=0x7ff00000) { u_int32_t low; GET_LOW_WORD(low,x); if(((hx&0xfffff)|low)!=0) return x+x; /* NaN */ else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ } if(x > o_threshold) return huge*huge; /* overflow */ } if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ if(x+tiny<0.0) /* raise inexact */ return tiny-one; /* return -1 */ } } /* argument reduction */ if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ if(xsb==0) {hi = x - ln2_hi; lo = ln2_lo; k = 1;} else {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} } else { k = invln2*x+((xsb==0)?0.5:-0.5); t = k; hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ lo = t*ln2_lo; } STRICT_ASSIGN(double, x, hi - lo); c = (hi-x)-lo; } else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ t = huge+x; /* return x with inexact flags when x!=0 */ return x - (t-(huge+x)); } else k = 0; /* x is now in primary range */ hfx = 0.5*x; hxs = x*hfx; r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); t = 3.0-r1*hfx; e = hxs*((r1-t)/(6.0 - x*t)); if(k==0) return x - (x*e-hxs); /* c is 0 */ else { - INSERT_WORDS(twopk,0x3ff00000+(k<<20),0); /* 2^k */ + INSERT_WORDS(twopk,((u_int32_t)(0x3ff+k))<<20,0); /* 2^k */ e = (x*(e-c)-c); e -= hxs; if(k== -1) return 0.5*(x-e)-0.5; if(k==1) { if(x < -0.25) return -2.0*(e-(x+0.5)); else return one+2.0*(x-e); } if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ y = one-(e-x); if (k == 1024) y = y*2.0*0x1p1023; else y = y*twopk; return y-one; } t = one; if(k<20) { SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */ y = t-(e-x); y = y*twopk; } else { SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */ y = x-(e+t); y += one; y = y*twopk; } } return y; } #if (LDBL_MANT_DIG == 53) __weak_reference(expm1, expm1l); #endif Index: stable/12/lib/msun/src/s_expm1f.c =================================================================== --- stable/12/lib/msun/src/s_expm1f.c (revision 352834) +++ stable/12/lib/msun/src/s_expm1f.c (revision 352835) @@ -1,124 +1,124 @@ /* s_expm1f.c -- float version of s_expm1.c. * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); #include #include "math.h" #include "math_private.h" static const float one = 1.0, tiny = 1.0e-30, o_threshold = 8.8721679688e+01,/* 0x42b17180 */ ln2_hi = 6.9313812256e-01,/* 0x3f317180 */ ln2_lo = 9.0580006145e-06,/* 0x3717f7d1 */ invln2 = 1.4426950216e+00,/* 0x3fb8aa3b */ /* * Domain [-0.34568, 0.34568], range ~[-6.694e-10, 6.696e-10]: * |6 / x * (1 + 2 * (1 / (exp(x) - 1) - 1 / x)) - q(x)| < 2**-30.04 * Scaled coefficients: Qn_here = 2**n * Qn_for_q (see s_expm1.c): */ Q1 = -3.3333212137e-2, /* -0x888868.0p-28 */ Q2 = 1.5807170421e-3; /* 0xcf3010.0p-33 */ static volatile float huge = 1.0e+30; float expm1f(float x) { float y,hi,lo,c,t,e,hxs,hfx,r1,twopk; int32_t k,xsb; u_int32_t hx; GET_FLOAT_WORD(hx,x); xsb = hx&0x80000000; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out huge and non-finite argument */ if(hx >= 0x4195b844) { /* if |x|>=27*ln2 */ if(hx >= 0x42b17218) { /* if |x|>=88.721... */ if(hx>0x7f800000) return x+x; /* NaN */ if(hx==0x7f800000) return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ if(x > o_threshold) return huge*huge; /* overflow */ } if(xsb!=0) { /* x < -27*ln2, return -1.0 with inexact */ if(x+tiny<(float)0.0) /* raise inexact */ return tiny-one; /* return -1 */ } } /* argument reduction */ if(hx > 0x3eb17218) { /* if |x| > 0.5 ln2 */ if(hx < 0x3F851592) { /* and |x| < 1.5 ln2 */ if(xsb==0) {hi = x - ln2_hi; lo = ln2_lo; k = 1;} else {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} } else { k = invln2*x+((xsb==0)?(float)0.5:(float)-0.5); t = k; hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ lo = t*ln2_lo; } STRICT_ASSIGN(float, x, hi - lo); c = (hi-x)-lo; } else if(hx < 0x33000000) { /* when |x|<2**-25, return x */ t = huge+x; /* return x with inexact flags when x!=0 */ return x - (t-(huge+x)); } else k = 0; /* x is now in primary range */ hfx = (float)0.5*x; hxs = x*hfx; r1 = one+hxs*(Q1+hxs*Q2); t = (float)3.0-r1*hfx; e = hxs*((r1-t)/((float)6.0 - x*t)); if(k==0) return x - (x*e-hxs); /* c is 0 */ else { - SET_FLOAT_WORD(twopk,0x3f800000+(k<<23)); /* 2^k */ + SET_FLOAT_WORD(twopk,((u_int32_t)(0x7f+k))<<23); /* 2^k */ e = (x*(e-c)-c); e -= hxs; if(k== -1) return (float)0.5*(x-e)-(float)0.5; if(k==1) { if(x < (float)-0.25) return -(float)2.0*(e-(x+(float)0.5)); else return one+(float)2.0*(x-e); } if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ y = one-(e-x); if (k == 128) y = y*2.0F*0x1p127F; else y = y*twopk; return y-one; } t = one; if(k<23) { SET_FLOAT_WORD(t,0x3f800000 - (0x1000000>>k)); /* t=1-2^-k */ y = t-(e-x); y = y*twopk; } else { SET_FLOAT_WORD(t,((0x7f-k)<<23)); /* 2^-k */ y = x-(e+t); y += one; y = y*twopk; } } return y; } Index: stable/12 =================================================================== --- stable/12 (revision 352834) +++ stable/12 (revision 352835) Property changes on: stable/12 ___________________________________________________________________ Modified: svn:mergeinfo ## -0,0 +0,1 ## Merged /head:r352710 Index: stable/8/lib/msun/src/e_exp.c =================================================================== --- stable/8/lib/msun/src/e_exp.c (revision 352834) +++ stable/8/lib/msun/src/e_exp.c (revision 352835) @@ -1,158 +1,158 @@ /* @(#)e_exp.c 1.6 04/04/22 */ /* * ==================================================== * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. * * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); /* __ieee754_exp(x) * Returns the exponential of x. * * Method * 1. Argument reduction: * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. * Given x, find r and integer k such that * * x = k*ln2 + r, |r| <= 0.5*ln2. * * Here r will be represented as r = hi-lo for better * accuracy. * * 2. Approximation of exp(r) by a special rational function on * the interval [0,0.34658]: * Write * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... * We use a special Remes algorithm on [0,0.34658] to generate * a polynomial of degree 5 to approximate R. The maximum error * of this polynomial approximation is bounded by 2**-59. In * other words, * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 * (where z=r*r, and the values of P1 to P5 are listed below) * and * | 5 | -59 * | 2.0+P1*z+...+P5*z - R(z) | <= 2 * | | * The computation of exp(r) thus becomes * 2*r * exp(r) = 1 + ------- * R - r * r*R1(r) * = 1 + r + ----------- (for better accuracy) * 2 - R1(r) * where * 2 4 10 * R1(r) = r - (P1*r + P2*r + ... + P5*r ). * * 3. Scale back to obtain exp(x): * From step 1, we have * exp(x) = 2^k * exp(r) * * Special cases: * exp(INF) is INF, exp(NaN) is NaN; * exp(-INF) is 0, and * for finite argument, only exp(0)=1 is exact. * * Accuracy: * according to an error analysis, the error is always less than * 1 ulp (unit in the last place). * * Misc. info. * For IEEE double * if x > 7.09782712893383973096e+02 then exp(x) overflow * if x < -7.45133219101941108420e+02 then exp(x) underflow * * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. */ #include "math.h" #include "math_private.h" static const double one = 1.0, halF[2] = {0.5,-0.5,}, huge = 1.0e+300, o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ static volatile double twom1000= 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0*/ double __ieee754_exp(double x) /* default IEEE double exp */ { double y,hi=0.0,lo=0.0,c,t,twopk; int32_t k=0,xsb; u_int32_t hx; GET_HIGH_WORD(hx,x); xsb = (hx>>31)&1; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out non-finite argument */ if(hx >= 0x40862E42) { /* if |x|>=709.78... */ if(hx>=0x7ff00000) { u_int32_t lx; GET_LOW_WORD(lx,x); if(((hx&0xfffff)|lx)!=0) return x+x; /* NaN */ else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ } if(x > o_threshold) return huge*huge; /* overflow */ if(x < u_threshold) return twom1000*twom1000; /* underflow */ } /* argument reduction */ if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; } else { k = (int)(invln2*x+halF[xsb]); t = k; hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ lo = t*ln2LO[0]; } x = hi - lo; } else if(hx < 0x3e300000) { /* when |x|<2**-28 */ if(huge+x>one) return one+x;/* trigger inexact */ } else k = 0; /* x is now in primary range */ t = x*x; if(k >= -1021) - INSERT_WORDS(twopk,0x3ff00000+(k<<20), 0); + INSERT_WORDS(twopk,((u_int32_t)(0x3ff+k))<<20, 0); else - INSERT_WORDS(twopk,0x3ff00000+((k+1000)<<20), 0); + INSERT_WORDS(twopk,((u_int32_t)(0x3ff+(k+1000)))<<20, 0); c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); if(k==0) return one-((x*c)/(c-2.0)-x); else y = one-((lo-(x*c)/(2.0-c))-hi); if(k >= -1021) { if (k==1024) return y*2.0*0x1p1023; return y*twopk; } else { return y*twopk*twom1000; } } Index: stable/8/lib/msun/src/e_expf.c =================================================================== --- stable/8/lib/msun/src/e_expf.c (revision 352834) +++ stable/8/lib/msun/src/e_expf.c (revision 352835) @@ -1,95 +1,95 @@ /* e_expf.c -- float version of e_exp.c. * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); #include "math.h" #include "math_private.h" static const float one = 1.0, halF[2] = {0.5,-0.5,}, huge = 1.0e+30, o_threshold= 8.8721679688e+01, /* 0x42b17180 */ u_threshold= -1.0397208405e+02, /* 0xc2cff1b5 */ ln2HI[2] ={ 6.9314575195e-01, /* 0x3f317200 */ -6.9314575195e-01,}, /* 0xbf317200 */ ln2LO[2] ={ 1.4286067653e-06, /* 0x35bfbe8e */ -1.4286067653e-06,}, /* 0xb5bfbe8e */ invln2 = 1.4426950216e+00, /* 0x3fb8aa3b */ /* * Domain [-0.34568, 0.34568], range ~[-4.278e-9, 4.447e-9]: * |x*(exp(x)+1)/(exp(x)-1) - p(x)| < 2**-27.74 */ P1 = 1.6666625440e-1, /* 0xaaaa8f.0p-26 */ P2 = -2.7667332906e-3; /* -0xb55215.0p-32 */ static volatile float twom100 = 7.8886090522e-31; /* 2**-100=0x0d800000 */ float __ieee754_expf(float x) /* default IEEE double exp */ { float y,hi=0.0,lo=0.0,c,t,twopk; int32_t k=0,xsb; u_int32_t hx; GET_FLOAT_WORD(hx,x); xsb = (hx>>31)&1; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out non-finite argument */ if(hx >= 0x42b17218) { /* if |x|>=88.721... */ if(hx>0x7f800000) return x+x; /* NaN */ if(hx==0x7f800000) return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ if(x > o_threshold) return huge*huge; /* overflow */ if(x < u_threshold) return twom100*twom100; /* underflow */ } /* argument reduction */ if(hx > 0x3eb17218) { /* if |x| > 0.5 ln2 */ if(hx < 0x3F851592) { /* and |x| < 1.5 ln2 */ hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; } else { k = invln2*x+halF[xsb]; t = k; hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ lo = t*ln2LO[0]; } x = hi - lo; } else if(hx < 0x31800000) { /* when |x|<2**-28 */ if(huge+x>one) return one+x;/* trigger inexact */ } else k = 0; /* x is now in primary range */ t = x*x; if(k >= -125) - SET_FLOAT_WORD(twopk,0x3f800000+(k<<23)); + SET_FLOAT_WORD(twopk,((u_int32_t)(0x7f+k))<<23); else - SET_FLOAT_WORD(twopk,0x3f800000+((k+100)<<23)); + SET_FLOAT_WORD(twopk,((u_int32_t)(0x7f+(k+100)))<<23); c = x - t*(P1+t*P2); if(k==0) return one-((x*c)/(c-(float)2.0)-x); else y = one-((lo-(x*c)/((float)2.0-c))-hi); if(k >= -125) { if(k==128) return y*2.0F*0x1p127F; return y*twopk; } else { return y*twopk*twom100; } } Index: stable/8/lib/msun/src/s_expm1.c =================================================================== --- stable/8/lib/msun/src/s_expm1.c (revision 352834) +++ stable/8/lib/msun/src/s_expm1.c (revision 352835) @@ -1,216 +1,216 @@ /* @(#)s_expm1.c 5.1 93/09/24 */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); /* expm1(x) * Returns exp(x)-1, the exponential of x minus 1. * * Method * 1. Argument reduction: * Given x, find r and integer k such that * * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 * * Here a correction term c will be computed to compensate * the error in r when rounded to a floating-point number. * * 2. Approximating expm1(r) by a special rational function on * the interval [0,0.34658]: * Since * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... * we define R1(r*r) by * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) * That is, * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... * We use a special Reme algorithm on [0,0.347] to generate * a polynomial of degree 5 in r*r to approximate R1. The * maximum error of this polynomial approximation is bounded * by 2**-61. In other words, * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 * where Q1 = -1.6666666666666567384E-2, * Q2 = 3.9682539681370365873E-4, * Q3 = -9.9206344733435987357E-6, * Q4 = 2.5051361420808517002E-7, * Q5 = -6.2843505682382617102E-9; * z = r*r, * with error bounded by * | 5 | -61 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 * | | * * expm1(r) = exp(r)-1 is then computed by the following * specific way which minimize the accumulation rounding error: * 2 3 * r r [ 3 - (R1 + R1*r/2) ] * expm1(r) = r + --- + --- * [--------------------] * 2 2 [ 6 - r*(3 - R1*r/2) ] * * To compensate the error in the argument reduction, we use * expm1(r+c) = expm1(r) + c + expm1(r)*c * ~ expm1(r) + c + r*c * Thus c+r*c will be added in as the correction terms for * expm1(r+c). Now rearrange the term to avoid optimization * screw up: * ( 2 2 ) * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) * ( ) * * = r - E * 3. Scale back to obtain expm1(x): * From step 1, we have * expm1(x) = either 2^k*[expm1(r)+1] - 1 * = or 2^k*[expm1(r) + (1-2^-k)] * 4. Implementation notes: * (A). To save one multiplication, we scale the coefficient Qi * to Qi*2^i, and replace z by (x^2)/2. * (B). To achieve maximum accuracy, we compute expm1(x) by * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) * (ii) if k=0, return r-E * (iii) if k=-1, return 0.5*(r-E)-0.5 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) * else return 1.0+2.0*(r-E); * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else * (vii) return 2^k(1-((E+2^-k)-r)) * * Special cases: * expm1(INF) is INF, expm1(NaN) is NaN; * expm1(-INF) is -1, and * for finite argument, only expm1(0)=0 is exact. * * Accuracy: * according to an error analysis, the error is always less than * 1 ulp (unit in the last place). * * Misc. info. * For IEEE double * if x > 7.09782712893383973096e+02 then expm1(x) overflow * * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. */ #include "math.h" #include "math_private.h" static const double one = 1.0, huge = 1.0e+300, tiny = 1.0e-300, o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */ Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ double expm1(double x) { double y,hi,lo,c,t,e,hxs,hfx,r1,twopk; int32_t k,xsb; u_int32_t hx; GET_HIGH_WORD(hx,x); xsb = hx&0x80000000; /* sign bit of x */ if(xsb==0) y=x; else y= -x; /* y = |x| */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out huge and non-finite argument */ if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ if(hx >= 0x40862E42) { /* if |x|>=709.78... */ if(hx>=0x7ff00000) { u_int32_t low; GET_LOW_WORD(low,x); if(((hx&0xfffff)|low)!=0) return x+x; /* NaN */ else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ } if(x > o_threshold) return huge*huge; /* overflow */ } if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ if(x+tiny<0.0) /* raise inexact */ return tiny-one; /* return -1 */ } } /* argument reduction */ if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ if(xsb==0) {hi = x - ln2_hi; lo = ln2_lo; k = 1;} else {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} } else { k = invln2*x+((xsb==0)?0.5:-0.5); t = k; hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ lo = t*ln2_lo; } x = hi - lo; c = (hi-x)-lo; } else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ t = huge+x; /* return x with inexact flags when x!=0 */ return x - (t-(huge+x)); } else k = 0; /* x is now in primary range */ hfx = 0.5*x; hxs = x*hfx; r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); t = 3.0-r1*hfx; e = hxs*((r1-t)/(6.0 - x*t)); if(k==0) return x - (x*e-hxs); /* c is 0 */ else { - INSERT_WORDS(twopk,0x3ff00000+(k<<20),0); /* 2^k */ + INSERT_WORDS(twopk,((u_int32_t)(0x3ff+k))<<20,0); /* 2^k */ e = (x*(e-c)-c); e -= hxs; if(k== -1) return 0.5*(x-e)-0.5; if(k==1) { if(x < -0.25) return -2.0*(e-(x+0.5)); else return one+2.0*(x-e); } if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ y = one-(e-x); if (k == 1024) y = y*2.0*0x1p1023; else y = y*twopk; return y-one; } t = one; if(k<20) { SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */ y = t-(e-x); y = y*twopk; } else { SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */ y = x-(e+t); y += one; y = y*twopk; } } return y; } Index: stable/8/lib/msun/src/s_expm1f.c =================================================================== --- stable/8/lib/msun/src/s_expm1f.c (revision 352834) +++ stable/8/lib/msun/src/s_expm1f.c (revision 352835) @@ -1,122 +1,122 @@ /* s_expm1f.c -- float version of s_expm1.c. * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); #include "math.h" #include "math_private.h" static const float one = 1.0, huge = 1.0e+30, tiny = 1.0e-30, o_threshold = 8.8721679688e+01,/* 0x42b17180 */ ln2_hi = 6.9313812256e-01,/* 0x3f317180 */ ln2_lo = 9.0580006145e-06,/* 0x3717f7d1 */ invln2 = 1.4426950216e+00,/* 0x3fb8aa3b */ /* * Domain [-0.34568, 0.34568], range ~[-6.694e-10, 6.696e-10]: * |6 / x * (1 + 2 * (1 / (exp(x) - 1) - 1 / x)) - q(x)| < 2**-30.04 * Scaled coefficients: Qn_here = 2**n * Qn_for_q (see s_expm1.c): */ Q1 = -3.3333212137e-2, /* -0x888868.0p-28 */ Q2 = 1.5807170421e-3; /* 0xcf3010.0p-33 */ float expm1f(float x) { float y,hi,lo,c,t,e,hxs,hfx,r1,twopk; int32_t k,xsb; u_int32_t hx; GET_FLOAT_WORD(hx,x); xsb = hx&0x80000000; /* sign bit of x */ if(xsb==0) y=x; else y= -x; /* y = |x| */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out huge and non-finite argument */ if(hx >= 0x4195b844) { /* if |x|>=27*ln2 */ if(hx >= 0x42b17218) { /* if |x|>=88.721... */ if(hx>0x7f800000) return x+x; /* NaN */ if(hx==0x7f800000) return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ if(x > o_threshold) return huge*huge; /* overflow */ } if(xsb!=0) { /* x < -27*ln2, return -1.0 with inexact */ if(x+tiny<(float)0.0) /* raise inexact */ return tiny-one; /* return -1 */ } } /* argument reduction */ if(hx > 0x3eb17218) { /* if |x| > 0.5 ln2 */ if(hx < 0x3F851592) { /* and |x| < 1.5 ln2 */ if(xsb==0) {hi = x - ln2_hi; lo = ln2_lo; k = 1;} else {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} } else { k = invln2*x+((xsb==0)?(float)0.5:(float)-0.5); t = k; hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ lo = t*ln2_lo; } x = hi - lo; c = (hi-x)-lo; } else if(hx < 0x33000000) { /* when |x|<2**-25, return x */ t = huge+x; /* return x with inexact flags when x!=0 */ return x - (t-(huge+x)); } else k = 0; /* x is now in primary range */ hfx = (float)0.5*x; hxs = x*hfx; r1 = one+hxs*(Q1+hxs*Q2); t = (float)3.0-r1*hfx; e = hxs*((r1-t)/((float)6.0 - x*t)); if(k==0) return x - (x*e-hxs); /* c is 0 */ else { - SET_FLOAT_WORD(twopk,0x3f800000+(k<<23)); /* 2^k */ + SET_FLOAT_WORD(twopk,((u_int32_t)(0x7f+k))<<23); /* 2^k */ e = (x*(e-c)-c); e -= hxs; if(k== -1) return (float)0.5*(x-e)-(float)0.5; if(k==1) { if(x < (float)-0.25) return -(float)2.0*(e-(x+(float)0.5)); else return one+(float)2.0*(x-e); } if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ y = one-(e-x); if (k == 128) y = y*2.0F*0x1p127F; else y = y*twopk; return y-one; } t = one; if(k<23) { SET_FLOAT_WORD(t,0x3f800000 - (0x1000000>>k)); /* t=1-2^-k */ y = t-(e-x); y = y*twopk; } else { SET_FLOAT_WORD(t,((0x7f-k)<<23)); /* 2^-k */ y = x-(e+t); y += one; y = y*twopk; } } return y; } Index: stable/8/lib/msun =================================================================== --- stable/8/lib/msun (revision 352834) +++ stable/8/lib/msun (revision 352835) Property changes on: stable/8/lib/msun ___________________________________________________________________ Modified: svn:mergeinfo ## -0,0 +0,1 ## Merged /head/lib/msun:r352710 Index: stable/8/lib =================================================================== --- stable/8/lib (revision 352834) +++ stable/8/lib (revision 352835) Property changes on: stable/8/lib ___________________________________________________________________ Modified: svn:mergeinfo ## -0,0 +0,1 ## Merged /head/lib:r352710 Index: stable/8 =================================================================== --- stable/8 (revision 352834) +++ stable/8 (revision 352835) Property changes on: stable/8 ___________________________________________________________________ Modified: svn:mergeinfo ## -0,0 +0,1 ## Merged /head:r352710 Index: stable/9/lib/msun/src/e_exp.c =================================================================== --- stable/9/lib/msun/src/e_exp.c (revision 352834) +++ stable/9/lib/msun/src/e_exp.c (revision 352835) @@ -1,158 +1,158 @@ /* @(#)e_exp.c 1.6 04/04/22 */ /* * ==================================================== * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. * * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); /* __ieee754_exp(x) * Returns the exponential of x. * * Method * 1. Argument reduction: * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. * Given x, find r and integer k such that * * x = k*ln2 + r, |r| <= 0.5*ln2. * * Here r will be represented as r = hi-lo for better * accuracy. * * 2. Approximation of exp(r) by a special rational function on * the interval [0,0.34658]: * Write * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... * We use a special Remes algorithm on [0,0.34658] to generate * a polynomial of degree 5 to approximate R. The maximum error * of this polynomial approximation is bounded by 2**-59. In * other words, * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 * (where z=r*r, and the values of P1 to P5 are listed below) * and * | 5 | -59 * | 2.0+P1*z+...+P5*z - R(z) | <= 2 * | | * The computation of exp(r) thus becomes * 2*r * exp(r) = 1 + ------- * R - r * r*R1(r) * = 1 + r + ----------- (for better accuracy) * 2 - R1(r) * where * 2 4 10 * R1(r) = r - (P1*r + P2*r + ... + P5*r ). * * 3. Scale back to obtain exp(x): * From step 1, we have * exp(x) = 2^k * exp(r) * * Special cases: * exp(INF) is INF, exp(NaN) is NaN; * exp(-INF) is 0, and * for finite argument, only exp(0)=1 is exact. * * Accuracy: * according to an error analysis, the error is always less than * 1 ulp (unit in the last place). * * Misc. info. * For IEEE double * if x > 7.09782712893383973096e+02 then exp(x) overflow * if x < -7.45133219101941108420e+02 then exp(x) underflow * * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. */ #include "math.h" #include "math_private.h" static const double one = 1.0, halF[2] = {0.5,-0.5,}, huge = 1.0e+300, o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ static volatile double twom1000= 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0*/ double __ieee754_exp(double x) /* default IEEE double exp */ { double y,hi=0.0,lo=0.0,c,t,twopk; int32_t k=0,xsb; u_int32_t hx; GET_HIGH_WORD(hx,x); xsb = (hx>>31)&1; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out non-finite argument */ if(hx >= 0x40862E42) { /* if |x|>=709.78... */ if(hx>=0x7ff00000) { u_int32_t lx; GET_LOW_WORD(lx,x); if(((hx&0xfffff)|lx)!=0) return x+x; /* NaN */ else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ } if(x > o_threshold) return huge*huge; /* overflow */ if(x < u_threshold) return twom1000*twom1000; /* underflow */ } /* argument reduction */ if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; } else { k = (int)(invln2*x+halF[xsb]); t = k; hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ lo = t*ln2LO[0]; } x = hi - lo; } else if(hx < 0x3e300000) { /* when |x|<2**-28 */ if(huge+x>one) return one+x;/* trigger inexact */ } else k = 0; /* x is now in primary range */ t = x*x; if(k >= -1021) - INSERT_WORDS(twopk,0x3ff00000+(k<<20), 0); + INSERT_WORDS(twopk,((u_int32_t)(0x3ff+k))<<20, 0); else - INSERT_WORDS(twopk,0x3ff00000+((k+1000)<<20), 0); + INSERT_WORDS(twopk,((u_int32_t)(0x3ff+(k+1000)))<<20, 0); c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); if(k==0) return one-((x*c)/(c-2.0)-x); else y = one-((lo-(x*c)/(2.0-c))-hi); if(k >= -1021) { if (k==1024) return y*2.0*0x1p1023; return y*twopk; } else { return y*twopk*twom1000; } } Index: stable/9/lib/msun/src/e_expf.c =================================================================== --- stable/9/lib/msun/src/e_expf.c (revision 352834) +++ stable/9/lib/msun/src/e_expf.c (revision 352835) @@ -1,95 +1,95 @@ /* e_expf.c -- float version of e_exp.c. * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); #include "math.h" #include "math_private.h" static const float one = 1.0, halF[2] = {0.5,-0.5,}, huge = 1.0e+30, o_threshold= 8.8721679688e+01, /* 0x42b17180 */ u_threshold= -1.0397208405e+02, /* 0xc2cff1b5 */ ln2HI[2] ={ 6.9314575195e-01, /* 0x3f317200 */ -6.9314575195e-01,}, /* 0xbf317200 */ ln2LO[2] ={ 1.4286067653e-06, /* 0x35bfbe8e */ -1.4286067653e-06,}, /* 0xb5bfbe8e */ invln2 = 1.4426950216e+00, /* 0x3fb8aa3b */ /* * Domain [-0.34568, 0.34568], range ~[-4.278e-9, 4.447e-9]: * |x*(exp(x)+1)/(exp(x)-1) - p(x)| < 2**-27.74 */ P1 = 1.6666625440e-1, /* 0xaaaa8f.0p-26 */ P2 = -2.7667332906e-3; /* -0xb55215.0p-32 */ static volatile float twom100 = 7.8886090522e-31; /* 2**-100=0x0d800000 */ float __ieee754_expf(float x) /* default IEEE double exp */ { float y,hi=0.0,lo=0.0,c,t,twopk; int32_t k=0,xsb; u_int32_t hx; GET_FLOAT_WORD(hx,x); xsb = (hx>>31)&1; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out non-finite argument */ if(hx >= 0x42b17218) { /* if |x|>=88.721... */ if(hx>0x7f800000) return x+x; /* NaN */ if(hx==0x7f800000) return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ if(x > o_threshold) return huge*huge; /* overflow */ if(x < u_threshold) return twom100*twom100; /* underflow */ } /* argument reduction */ if(hx > 0x3eb17218) { /* if |x| > 0.5 ln2 */ if(hx < 0x3F851592) { /* and |x| < 1.5 ln2 */ hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; } else { k = invln2*x+halF[xsb]; t = k; hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ lo = t*ln2LO[0]; } x = hi - lo; } else if(hx < 0x39000000) { /* when |x|<2**-14 */ if(huge+x>one) return one+x;/* trigger inexact */ } else k = 0; /* x is now in primary range */ t = x*x; if(k >= -125) - SET_FLOAT_WORD(twopk,0x3f800000+(k<<23)); + SET_FLOAT_WORD(twopk,((u_int32_t)(0x7f+k))<<23); else - SET_FLOAT_WORD(twopk,0x3f800000+((k+100)<<23)); + SET_FLOAT_WORD(twopk,((u_int32_t)(0x7f+(k+100)))<<23); c = x - t*(P1+t*P2); if(k==0) return one-((x*c)/(c-(float)2.0)-x); else y = one-((lo-(x*c)/((float)2.0-c))-hi); if(k >= -125) { if(k==128) return y*2.0F*0x1p127F; return y*twopk; } else { return y*twopk*twom100; } } Index: stable/9/lib/msun/src/s_expm1.c =================================================================== --- stable/9/lib/msun/src/s_expm1.c (revision 352834) +++ stable/9/lib/msun/src/s_expm1.c (revision 352835) @@ -1,216 +1,216 @@ /* @(#)s_expm1.c 5.1 93/09/24 */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); /* expm1(x) * Returns exp(x)-1, the exponential of x minus 1. * * Method * 1. Argument reduction: * Given x, find r and integer k such that * * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 * * Here a correction term c will be computed to compensate * the error in r when rounded to a floating-point number. * * 2. Approximating expm1(r) by a special rational function on * the interval [0,0.34658]: * Since * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... * we define R1(r*r) by * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) * That is, * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... * We use a special Reme algorithm on [0,0.347] to generate * a polynomial of degree 5 in r*r to approximate R1. The * maximum error of this polynomial approximation is bounded * by 2**-61. In other words, * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 * where Q1 = -1.6666666666666567384E-2, * Q2 = 3.9682539681370365873E-4, * Q3 = -9.9206344733435987357E-6, * Q4 = 2.5051361420808517002E-7, * Q5 = -6.2843505682382617102E-9; * z = r*r, * with error bounded by * | 5 | -61 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 * | | * * expm1(r) = exp(r)-1 is then computed by the following * specific way which minimize the accumulation rounding error: * 2 3 * r r [ 3 - (R1 + R1*r/2) ] * expm1(r) = r + --- + --- * [--------------------] * 2 2 [ 6 - r*(3 - R1*r/2) ] * * To compensate the error in the argument reduction, we use * expm1(r+c) = expm1(r) + c + expm1(r)*c * ~ expm1(r) + c + r*c * Thus c+r*c will be added in as the correction terms for * expm1(r+c). Now rearrange the term to avoid optimization * screw up: * ( 2 2 ) * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) * ( ) * * = r - E * 3. Scale back to obtain expm1(x): * From step 1, we have * expm1(x) = either 2^k*[expm1(r)+1] - 1 * = or 2^k*[expm1(r) + (1-2^-k)] * 4. Implementation notes: * (A). To save one multiplication, we scale the coefficient Qi * to Qi*2^i, and replace z by (x^2)/2. * (B). To achieve maximum accuracy, we compute expm1(x) by * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) * (ii) if k=0, return r-E * (iii) if k=-1, return 0.5*(r-E)-0.5 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) * else return 1.0+2.0*(r-E); * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else * (vii) return 2^k(1-((E+2^-k)-r)) * * Special cases: * expm1(INF) is INF, expm1(NaN) is NaN; * expm1(-INF) is -1, and * for finite argument, only expm1(0)=0 is exact. * * Accuracy: * according to an error analysis, the error is always less than * 1 ulp (unit in the last place). * * Misc. info. * For IEEE double * if x > 7.09782712893383973096e+02 then expm1(x) overflow * * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. */ #include "math.h" #include "math_private.h" static const double one = 1.0, huge = 1.0e+300, tiny = 1.0e-300, o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */ Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ double expm1(double x) { double y,hi,lo,c,t,e,hxs,hfx,r1,twopk; int32_t k,xsb; u_int32_t hx; GET_HIGH_WORD(hx,x); xsb = hx&0x80000000; /* sign bit of x */ if(xsb==0) y=x; else y= -x; /* y = |x| */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out huge and non-finite argument */ if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ if(hx >= 0x40862E42) { /* if |x|>=709.78... */ if(hx>=0x7ff00000) { u_int32_t low; GET_LOW_WORD(low,x); if(((hx&0xfffff)|low)!=0) return x+x; /* NaN */ else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ } if(x > o_threshold) return huge*huge; /* overflow */ } if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ if(x+tiny<0.0) /* raise inexact */ return tiny-one; /* return -1 */ } } /* argument reduction */ if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ if(xsb==0) {hi = x - ln2_hi; lo = ln2_lo; k = 1;} else {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} } else { k = invln2*x+((xsb==0)?0.5:-0.5); t = k; hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ lo = t*ln2_lo; } x = hi - lo; c = (hi-x)-lo; } else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ t = huge+x; /* return x with inexact flags when x!=0 */ return x - (t-(huge+x)); } else k = 0; /* x is now in primary range */ hfx = 0.5*x; hxs = x*hfx; r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); t = 3.0-r1*hfx; e = hxs*((r1-t)/(6.0 - x*t)); if(k==0) return x - (x*e-hxs); /* c is 0 */ else { - INSERT_WORDS(twopk,0x3ff00000+(k<<20),0); /* 2^k */ + INSERT_WORDS(twopk,((u_int32_t)(0x3ff+k))<<20,0); /* 2^k */ e = (x*(e-c)-c); e -= hxs; if(k== -1) return 0.5*(x-e)-0.5; if(k==1) { if(x < -0.25) return -2.0*(e-(x+0.5)); else return one+2.0*(x-e); } if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ y = one-(e-x); if (k == 1024) y = y*2.0*0x1p1023; else y = y*twopk; return y-one; } t = one; if(k<20) { SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */ y = t-(e-x); y = y*twopk; } else { SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */ y = x-(e+t); y += one; y = y*twopk; } } return y; } Index: stable/9/lib/msun/src/s_expm1f.c =================================================================== --- stable/9/lib/msun/src/s_expm1f.c (revision 352834) +++ stable/9/lib/msun/src/s_expm1f.c (revision 352835) @@ -1,122 +1,122 @@ /* s_expm1f.c -- float version of s_expm1.c. * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include __FBSDID("$FreeBSD$"); #include "math.h" #include "math_private.h" static const float one = 1.0, huge = 1.0e+30, tiny = 1.0e-30, o_threshold = 8.8721679688e+01,/* 0x42b17180 */ ln2_hi = 6.9313812256e-01,/* 0x3f317180 */ ln2_lo = 9.0580006145e-06,/* 0x3717f7d1 */ invln2 = 1.4426950216e+00,/* 0x3fb8aa3b */ /* * Domain [-0.34568, 0.34568], range ~[-6.694e-10, 6.696e-10]: * |6 / x * (1 + 2 * (1 / (exp(x) - 1) - 1 / x)) - q(x)| < 2**-30.04 * Scaled coefficients: Qn_here = 2**n * Qn_for_q (see s_expm1.c): */ Q1 = -3.3333212137e-2, /* -0x888868.0p-28 */ Q2 = 1.5807170421e-3; /* 0xcf3010.0p-33 */ float expm1f(float x) { float y,hi,lo,c,t,e,hxs,hfx,r1,twopk; int32_t k,xsb; u_int32_t hx; GET_FLOAT_WORD(hx,x); xsb = hx&0x80000000; /* sign bit of x */ if(xsb==0) y=x; else y= -x; /* y = |x| */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out huge and non-finite argument */ if(hx >= 0x4195b844) { /* if |x|>=27*ln2 */ if(hx >= 0x42b17218) { /* if |x|>=88.721... */ if(hx>0x7f800000) return x+x; /* NaN */ if(hx==0x7f800000) return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ if(x > o_threshold) return huge*huge; /* overflow */ } if(xsb!=0) { /* x < -27*ln2, return -1.0 with inexact */ if(x+tiny<(float)0.0) /* raise inexact */ return tiny-one; /* return -1 */ } } /* argument reduction */ if(hx > 0x3eb17218) { /* if |x| > 0.5 ln2 */ if(hx < 0x3F851592) { /* and |x| < 1.5 ln2 */ if(xsb==0) {hi = x - ln2_hi; lo = ln2_lo; k = 1;} else {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} } else { k = invln2*x+((xsb==0)?(float)0.5:(float)-0.5); t = k; hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ lo = t*ln2_lo; } x = hi - lo; c = (hi-x)-lo; } else if(hx < 0x33000000) { /* when |x|<2**-25, return x */ t = huge+x; /* return x with inexact flags when x!=0 */ return x - (t-(huge+x)); } else k = 0; /* x is now in primary range */ hfx = (float)0.5*x; hxs = x*hfx; r1 = one+hxs*(Q1+hxs*Q2); t = (float)3.0-r1*hfx; e = hxs*((r1-t)/((float)6.0 - x*t)); if(k==0) return x - (x*e-hxs); /* c is 0 */ else { - SET_FLOAT_WORD(twopk,0x3f800000+(k<<23)); /* 2^k */ + SET_FLOAT_WORD(twopk,((u_int32_t)(0x7f+k))<<23); /* 2^k */ e = (x*(e-c)-c); e -= hxs; if(k== -1) return (float)0.5*(x-e)-(float)0.5; if(k==1) { if(x < (float)-0.25) return -(float)2.0*(e-(x+(float)0.5)); else return one+(float)2.0*(x-e); } if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ y = one-(e-x); if (k == 128) y = y*2.0F*0x1p127F; else y = y*twopk; return y-one; } t = one; if(k<23) { SET_FLOAT_WORD(t,0x3f800000 - (0x1000000>>k)); /* t=1-2^-k */ y = t-(e-x); y = y*twopk; } else { SET_FLOAT_WORD(t,((0x7f-k)<<23)); /* 2^-k */ y = x-(e+t); y += one; y = y*twopk; } } return y; } Index: stable/9/lib/msun =================================================================== --- stable/9/lib/msun (revision 352834) +++ stable/9/lib/msun (revision 352835) Property changes on: stable/9/lib/msun ___________________________________________________________________ Modified: svn:mergeinfo ## -0,0 +0,1 ## Merged /head/lib/msun:r352710 Index: stable/9/lib =================================================================== --- stable/9/lib (revision 352834) +++ stable/9/lib (revision 352835) Property changes on: stable/9/lib ___________________________________________________________________ Modified: svn:mergeinfo ## -0,0 +0,1 ## Merged /head/lib:r352710 Index: stable/9 =================================================================== --- stable/9 (revision 352834) +++ stable/9 (revision 352835) Property changes on: stable/9 ___________________________________________________________________ Modified: svn:mergeinfo ## -0,0 +0,1 ## Merged /head:r352710