Index: head/lib/msun/src/s_erf.c =================================================================== --- head/lib/msun/src/s_erf.c (revision 268587) +++ head/lib/msun/src/s_erf.c (revision 268588) @@ -1,294 +1,301 @@ /* @(#)s_erf.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("$FreeBSD$"); /* double erf(double x) * double erfc(double x) * x * 2 |\ * erf(x) = --------- | exp(-t*t)dt * sqrt(pi) \| * 0 * * erfc(x) = 1-erf(x) * Note that * erf(-x) = -erf(x) * erfc(-x) = 2 - erfc(x) * * Method: * 1. For |x| in [0, 0.84375] * erf(x) = x + x*R(x^2) * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] * where R = P/Q where P is an odd poly of degree 8 and * Q is an odd poly of degree 10. * -57.90 * | R - (erf(x)-x)/x | <= 2 * * * Remark. The formula is derived by noting * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) * and that * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 * is close to one. The interval is chosen because the fix * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is * near 0.6174), and by some experiment, 0.84375 is chosen to * guarantee the error is less than one ulp for erf. * * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and * c = 0.84506291151 rounded to single (24 bits) * erf(x) = sign(x) * (c + P1(s)/Q1(s)) * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 * 1+(c+P1(s)/Q1(s)) if x < 0 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 * Remark: here we use the taylor series expansion at x=1. * erf(1+s) = erf(1) + s*Poly(s) * = 0.845.. + P1(s)/Q1(s) * That is, we use rational approximation to approximate * erf(1+s) - (c = (single)0.84506291151) * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] * where * P1(s) = degree 6 poly in s * Q1(s) = degree 6 poly in s * * 3. For x in [1.25,1/0.35(~2.857143)], * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) * erf(x) = 1 - erfc(x) * where * R1(z) = degree 7 poly in z, (z=1/x^2) * S1(z) = degree 8 poly in z * * 4. For x in [1/0.35,28] * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6 x >= 28 * erf(x) = sign(x) *(1 - tiny) (raise inexact) * erfc(x) = tiny*tiny (raise underflow) if x > 0 * = 2 - tiny if x<0 * * 7. Special case: * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, * erfc/erf(NaN) is NaN */ #include "math.h" #include "math_private.h" +/* XXX Prevent compilers from erroneously constant folding: */ +static const volatile double tiny= 1e-300; + static const double -tiny = 1e-300, -half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ -one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ -two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ +half= 0.5, +one = 1, +two = 2, /* c = (float)0.84506291151 */ erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ +/* + * In the domain [0, 2**-28], only the first term in the power series + * expansion of erf(x) is used. The magnitude of the first neglected + * terms is less than 2**-84. + */ efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ /* * Coefficients for approximation to erf on [0,0.84375] */ pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */ qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */ /* * Coefficients for approximation to erf in [0.84375,1.25] */ pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */ qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */ /* * Coefficients for approximation to erfc in [1.25,1/0.35] */ ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */ sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */ /* * Coefficients for approximation to erfc in [1/.35,28] */ rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */ sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ double erf(double x) { int32_t hx,ix,i; double R,S,P,Q,s,y,z,r; GET_HIGH_WORD(hx,x); ix = hx&0x7fffffff; if(ix>=0x7ff00000) { /* erf(nan)=nan */ i = ((u_int32_t)hx>>31)<<1; return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */ } if(ix < 0x3feb0000) { /* |x|<0.84375 */ if(ix < 0x3e300000) { /* |x|<2**-28 */ if (ix < 0x00800000) return (8*x+efx8*x)/8; /* avoid spurious underflow */ return x + efx*x; } z = x*x; r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); y = r/s; return x + x*y; } if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ s = fabs(x)-one; P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); if(hx>=0) return erx + P/Q; else return -erx - P/Q; } if (ix >= 0x40180000) { /* inf>|x|>=6 */ if(hx>=0) return one-tiny; else return tiny-one; } x = fabs(x); s = one/(x*x); if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */ R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7)))))); S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ s*sa8))))))); } else { /* |x| >= 1/0.35 */ R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6))))); S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))); } z = x; SET_LOW_WORD(z,0); r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S); if(hx>=0) return one-r/x; else return r/x-one; } double erfc(double x) { int32_t hx,ix; double R,S,P,Q,s,y,z,r; GET_HIGH_WORD(hx,x); ix = hx&0x7fffffff; if(ix>=0x7ff00000) { /* erfc(nan)=nan */ /* erfc(+-inf)=0,2 */ return (double)(((u_int32_t)hx>>31)<<1)+one/x; } if(ix < 0x3feb0000) { /* |x|<0.84375 */ if(ix < 0x3c700000) /* |x|<2**-56 */ return one-x; z = x*x; r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); y = r/s; if(hx < 0x3fd00000) { /* x<1/4 */ return one-(x+x*y); } else { r = x*y; r += (x-half); return half - r ; } } if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ s = fabs(x)-one; P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); if(hx>=0) { z = one-erx; return z - P/Q; } else { z = erx+P/Q; return one+z; } } if (ix < 0x403c0000) { /* |x|<28 */ x = fabs(x); s = one/(x*x); if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/ R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7)))))); S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ s*sa8))))))); } else { /* |x| >= 1/.35 ~ 2.857143 */ if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */ R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6))))); S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))); } z = x; SET_LOW_WORD(z,0); r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S); if(hx>0) return r/x; else return two-r/x; } else { if(hx>0) return tiny*tiny; else return two-tiny; } } Index: head/lib/msun/src/s_erff.c =================================================================== --- head/lib/msun/src/s_erff.c (revision 268587) +++ head/lib/msun/src/s_erff.c (revision 268588) @@ -1,182 +1,186 @@ /* s_erff.c -- float version of s_erf.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" +/* XXX Prevent compilers from erroneously constant folding: */ +static const volatile float tiny = 1e-30; + static const float -tiny = 1e-30, -half= 5.0000000000e-01, /* 0x3F000000 */ -one = 1.0000000000e+00, /* 0x3F800000 */ -two = 2.0000000000e+00, /* 0x40000000 */ +half= 0.5, +one = 1, +two = 2, +erx = 8.42697144e-01, /* 0x3f57bb00 */ /* - * Coefficients for approximation to erf on [0,0.84375] + * In the domain [0, 2**-14], only the first term in the power series + * expansion of erf(x) is used. The magnitude of the first neglected + * terms is less than 2**-42. */ efx = 1.2837916613e-01, /* 0x3e0375d4 */ efx8= 1.0270333290e+00, /* 0x3f8375d4 */ /* * Domain [0, 0.84375], range ~[-5.4446e-10,5.5197e-10]: * |(erf(x) - x)/x - p(x)/q(x)| < 2**-31. */ pp0 = 1.28379166e-01F, /* 0x1.06eba8p-3 */ pp1 = -3.36030394e-01F, /* -0x1.58185ap-2 */ pp2 = -1.86260219e-03F, /* -0x1.e8451ep-10 */ qq1 = 3.12324286e-01F, /* 0x1.3fd1f0p-2 */ qq2 = 2.16070302e-02F, /* 0x1.620274p-6 */ qq3 = -1.98859419e-03F, /* -0x1.04a626p-9 */ /* * Domain [0.84375, 1.25], range ~[-1.953e-11,1.940e-11]: * |(erf(x) - erx) - p(x)/q(x)| < 2**-36. */ -erx = 8.42697144e-01F, /* 0x1.af7600p-1. erf(1) rounded to 16 bits. */ pa0 = 3.64939137e-06F, /* 0x1.e9d022p-19 */ pa1 = 4.15109694e-01F, /* 0x1.a91284p-2 */ pa2 = -1.65179938e-01F, /* -0x1.5249dcp-3 */ pa3 = 1.10914491e-01F, /* 0x1.c64e46p-4 */ qa1 = 6.02074385e-01F, /* 0x1.344318p-1 */ qa2 = 5.35934687e-01F, /* 0x1.126608p-1 */ qa3 = 1.68576106e-01F, /* 0x1.593e6ep-3 */ qa4 = 5.62181212e-02F, /* 0x1.cc89f2p-5 */ /* * Domain [1.25,1/0.35], range ~[-7.043e-10,7.457e-10]: * |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-30 */ ra0 = -9.87132732e-03F, /* -0x1.4376b2p-7 */ ra1 = -5.53605914e-01F, /* -0x1.1b723cp-1 */ ra2 = -2.17589188e+00F, /* -0x1.1683a0p+1 */ ra3 = -1.43268085e+00F, /* -0x1.6ec42cp+0 */ sa1 = 5.45995426e+00F, /* 0x1.5d6fe4p+2 */ sa2 = 6.69798088e+00F, /* 0x1.acabb8p+2 */ sa3 = 1.43113089e+00F, /* 0x1.6e5e98p+0 */ sa4 = -5.77397496e-02F, /* -0x1.d90108p-5 */ /* * Domain [1/0.35, 11], range ~[-2.264e-13,2.336e-13]: * |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-42 */ rb0 = -9.86494310e-03F, /* -0x1.434124p-7 */ rb1 = -6.25171244e-01F, /* -0x1.401672p-1 */ rb2 = -6.16498327e+00F, /* -0x1.8a8f16p+2 */ rb3 = -1.66696873e+01F, /* -0x1.0ab70ap+4 */ rb4 = -9.53764343e+00F, /* -0x1.313460p+3 */ sb1 = 1.26884899e+01F, /* 0x1.96081cp+3 */ sb2 = 4.51839523e+01F, /* 0x1.6978bcp+5 */ sb3 = 4.72810211e+01F, /* 0x1.7a3f88p+5 */ sb4 = 8.93033314e+00F; /* 0x1.1dc54ap+3 */ float erff(float x) { int32_t hx,ix,i; float R,S,P,Q,s,y,z,r; GET_FLOAT_WORD(hx,x); ix = hx&0x7fffffff; if(ix>=0x7f800000) { /* erf(nan)=nan */ i = ((u_int32_t)hx>>31)<<1; return (float)(1-i)+one/x; /* erf(+-inf)=+-1 */ } if(ix < 0x3f580000) { /* |x|<0.84375 */ if(ix < 0x38800000) { /* |x|<2**-14 */ if (ix < 0x04000000) /* |x|<0x1p-119 */ return (8*x+efx8*x)/8; /* avoid spurious underflow */ return x + efx*x; } z = x*x; r = pp0+z*(pp1+z*pp2); s = one+z*(qq1+z*(qq2+z*qq3)); y = r/s; return x + x*y; } if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */ s = fabsf(x)-one; P = pa0+s*(pa1+s*(pa2+s*pa3)); Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4))); if(hx>=0) return erx + P/Q; else return -erx - P/Q; } if (ix >= 0x40800000) { /* inf>|x|>=4 */ if(hx>=0) return one-tiny; else return tiny-one; } x = fabsf(x); s = one/(x*x); if(ix< 0x4036DB6E) { /* |x| < 1/0.35 */ R=ra0+s*(ra1+s*(ra2+s*ra3)); S=one+s*(sa1+s*(sa2+s*(sa3+s*sa4))); } else { /* |x| >= 1/0.35 */ R=rb0+s*(rb1+s*(rb2+s*(rb3+s*rb4))); S=one+s*(sb1+s*(sb2+s*(sb3+s*sb4))); } SET_FLOAT_WORD(z,hx&0xffffe000); r = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S); if(hx>=0) return one-r/x; else return r/x-one; } float erfcf(float x) { int32_t hx,ix; float R,S,P,Q,s,y,z,r; GET_FLOAT_WORD(hx,x); ix = hx&0x7fffffff; if(ix>=0x7f800000) { /* erfc(nan)=nan */ /* erfc(+-inf)=0,2 */ return (float)(((u_int32_t)hx>>31)<<1)+one/x; } if(ix < 0x3f580000) { /* |x|<0.84375 */ if(ix < 0x33800000) /* |x|<2**-24 */ return one-x; z = x*x; r = pp0+z*(pp1+z*pp2); s = one+z*(qq1+z*(qq2+z*qq3)); y = r/s; if(hx < 0x3e800000) { /* x<1/4 */ return one-(x+x*y); } else { r = x*y; r += (x-half); return half - r ; } } if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */ s = fabsf(x)-one; P = pa0+s*(pa1+s*(pa2+s*pa3)); Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4))); if(hx>=0) { z = one-erx; return z - P/Q; } else { z = erx+P/Q; return one+z; } } if (ix < 0x41300000) { /* |x|<11 */ x = fabsf(x); s = one/(x*x); if(ix< 0x4036DB6D) { /* |x| < 1/.35 ~ 2.857143*/ R=ra0+s*(ra1+s*(ra2+s*ra3)); S=one+s*(sa1+s*(sa2+s*(sa3+s*sa4))); } else { /* |x| >= 1/.35 ~ 2.857143 */ if(hx<0&&ix>=0x40a00000) return two-tiny;/* x < -5 */ R=rb0+s*(rb1+s*(rb2+s*(rb3+s*rb4))); S=one+s*(sb1+s*(sb2+s*(sb3+s*sb4))); } SET_FLOAT_WORD(z,hx&0xffffe000); r = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S); if(hx>0) return r/x; else return two-r/x; } else { if(hx>0) return tiny*tiny; else return two-tiny; } }