diff --git a/lib/msun/src/s_fma.c b/lib/msun/src/s_fma.c index ad1fc4aef49d..aefbd8e72474 100644 --- a/lib/msun/src/s_fma.c +++ b/lib/msun/src/s_fma.c @@ -1,207 +1,243 @@ /*- - * Copyright (c) 2005 David Schultz + * Copyright (c) 2005-2011 David Schultz * All rights reserved. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * 1. Redistributions of source code must retain the above copyright * notice, this list of conditions and the following disclaimer. * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in the * documentation and/or other materials provided with the distribution. * * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF * SUCH DAMAGE. */ #include __FBSDID("$FreeBSD$"); #include #include #include +/* + * A struct dd represents a floating-point number with twice the precision + * of a double. We maintain the invariant that "hi" stores the 53 high-order + * bits of the result. + */ +struct dd { + double hi; + double lo; +}; + +/* + * Compute a+b exactly, returning the exact result in a struct dd. We assume + * that both a and b are finite, but make no assumptions about their relative + * magnitudes. + */ +static inline struct dd +dd_add(double a, double b) +{ + struct dd ret; + double s; + + ret.hi = a + b; + s = ret.hi - a; + ret.lo = (a - (ret.hi - s)) + (b - s); + return (ret); +} + +/* + * Compute a*b exactly, returning the exact result in a struct dd. We assume + * that both a and b are normalized, so no underflow or overflow will occur. + * The current rounding mode must be round-to-nearest. + */ +static inline struct dd +dd_mul(double a, double b) +{ + static const double split = 0x1p27 + 1.0; + struct dd ret; + double ha, hb, la, lb, p, q; + + p = a * split; + ha = a - p; + ha += p; + la = a - ha; + + p = b * split; + hb = b - p; + hb += p; + lb = b - hb; + + p = ha * hb; + q = ha * lb + la * hb; + + ret.hi = p + q; + ret.lo = p - ret.hi + q + la * lb; + return (ret); +} + /* * Fused multiply-add: Compute x * y + z with a single rounding error. * * We use scaling to avoid overflow/underflow, along with the * canonical precision-doubling technique adapted from: * * Dekker, T. A Floating-Point Technique for Extending the * Available Precision. Numer. Math. 18, 224-242 (1971). * * This algorithm is sensitive to the rounding precision. FPUs such * as the i387 must be set in double-precision mode if variables are * to be stored in FP registers in order to avoid incorrect results. * This is the default on FreeBSD, but not on many other systems. * * Hardware instructions should be used on architectures that support it, * since this implementation will likely be several times slower. */ #if LDBL_MANT_DIG != 113 double fma(double x, double y, double z) { - static const double split = 0x1p27 + 1.0; double xs, ys, zs; - double c, cc, hx, hy, p, q, tx, ty; - double r, rr, s; + struct dd xy, r, r2; + double p; + double s; int oround; int ex, ey, ez; int spread; /* * Handle special cases. The order of operations and the particular * return values here are crucial in handling special cases involving * infinities, NaNs, overflows, and signed zeroes correctly. */ if (x == 0.0 || y == 0.0) return (x * y + z); if (z == 0.0) return (x * y); if (!isfinite(x) || !isfinite(y)) return (x * y + z); if (!isfinite(z)) return (z); xs = frexp(x, &ex); ys = frexp(y, &ey); zs = frexp(z, &ez); oround = fegetround(); spread = ex + ey - ez; /* * If x * y and z are many orders of magnitude apart, the scaling * will overflow, so we handle these cases specially. Rounding * modes other than FE_TONEAREST are painful. */ if (spread > DBL_MANT_DIG * 2) { fenv_t env; feraiseexcept(FE_INEXACT); switch(oround) { case FE_TONEAREST: return (x * y); case FE_TOWARDZERO: if (x > 0.0 ^ y < 0.0 ^ z < 0.0) return (x * y); feholdexcept(&env); - r = x * y; + s = x * y; if (!fetestexcept(FE_INEXACT)) - r = nextafter(r, 0); + s = nextafter(s, 0); feupdateenv(&env); - return (r); + return (s); case FE_DOWNWARD: if (z > 0.0) return (x * y); feholdexcept(&env); - r = x * y; + s = x * y; if (!fetestexcept(FE_INEXACT)) - r = nextafter(r, -INFINITY); + s = nextafter(s, -INFINITY); feupdateenv(&env); - return (r); + return (s); default: /* FE_UPWARD */ if (z < 0.0) return (x * y); feholdexcept(&env); - r = x * y; + s = x * y; if (!fetestexcept(FE_INEXACT)) - r = nextafter(r, INFINITY); + s = nextafter(s, INFINITY); feupdateenv(&env); - return (r); + return (s); } } if (spread < -DBL_MANT_DIG) { feraiseexcept(FE_INEXACT); if (!isnormal(z)) feraiseexcept(FE_UNDERFLOW); switch (oround) { case FE_TONEAREST: return (z); case FE_TOWARDZERO: if (x > 0.0 ^ y < 0.0 ^ z < 0.0) return (z); else return (nextafter(z, 0)); case FE_DOWNWARD: if (x > 0.0 ^ y < 0.0) return (z); else return (nextafter(z, -INFINITY)); default: /* FE_UPWARD */ if (x > 0.0 ^ y < 0.0) return (nextafter(z, INFINITY)); else return (z); } } - /* - * Use Dekker's algorithm to perform the multiplication and - * subsequent addition in twice the machine precision. - * Arrange so that x * y = c + cc, and x * y + z = r + rr. - */ fesetround(FE_TONEAREST); - p = xs * split; - hx = xs - p; - hx += p; - tx = xs - hx; - - p = ys * split; - hy = ys - p; - hy += p; - ty = ys - hy; - - p = hx * hy; - q = hx * ty + tx * hy; - c = p + q; - cc = p - c + q + tx * ty; - + xy = dd_mul(xs, ys); zs = ldexp(zs, -spread); - r = c + zs; - s = r - c; - rr = (c - (r - s)) + (zs - s) + cc; + r = dd_add(xy.hi, zs); + r.lo += xy.lo; spread = ex + ey; - if (spread + ilogb(r) > -1023) { + if (spread + ilogb(r.hi) > -1023) { fesetround(oround); - r = r + rr; + r.hi = r.hi + r.lo; } else { /* * The result is subnormal, so we round before scaling to * avoid double rounding. */ - p = ldexp(copysign(0x1p-1022, r), -spread); - c = r + p; - s = c - r; - cc = (r - (c - s)) + (p - s) + rr; + p = ldexp(copysign(0x1p-1022, r.hi), -spread); + r2 = dd_add(r.hi, p); + r2.lo += r.lo; fesetround(oround); - r = (c + cc) - p; + r.hi = (r2.hi + r2.lo) - p; } - return (ldexp(r, spread)); + return (ldexp(r.hi, spread)); } #else /* LDBL_MANT_DIG == 113 */ /* * 113 bits of precision is more than twice the precision of a double, * so it is enough to represent the intermediate product exactly. */ double fma(double x, double y, double z) { return ((long double)x * y + z); } #endif /* LDBL_MANT_DIG != 113 */ #if (LDBL_MANT_DIG == 53) __weak_reference(fma, fmal); #endif diff --git a/lib/msun/src/s_fmal.c b/lib/msun/src/s_fmal.c index 4d5d1141b44b..464dcb5e86df 100644 --- a/lib/msun/src/s_fmal.c +++ b/lib/msun/src/s_fmal.c @@ -1,187 +1,223 @@ /*- - * Copyright (c) 2005 David Schultz + * Copyright (c) 2005-2011 David Schultz * All rights reserved. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * 1. Redistributions of source code must retain the above copyright * notice, this list of conditions and the following disclaimer. * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in the * documentation and/or other materials provided with the distribution. * * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF * SUCH DAMAGE. */ #include __FBSDID("$FreeBSD$"); #include #include #include +/* + * A struct dd represents a floating-point number with twice the precision + * of a long double. We maintain the invariant that "hi" stores the high-order + * bits of the result. + */ +struct dd { + long double hi; + long double lo; +}; + +/* + * Compute a+b exactly, returning the exact result in a struct dd. We assume + * that both a and b are finite, but make no assumptions about their relative + * magnitudes. + */ +static inline struct dd +dd_add(long double a, long double b) +{ + struct dd ret; + long double s; + + ret.hi = a + b; + s = ret.hi - a; + ret.lo = (a - (ret.hi - s)) + (b - s); + return (ret); +} + +/* + * Compute a*b exactly, returning the exact result in a struct dd. We assume + * that both a and b are normalized, so no underflow or overflow will occur. + * The current rounding mode must be round-to-nearest. + */ +static inline struct dd +dd_mul(long double a, long double b) +{ +#if LDBL_MANT_DIG == 64 + static const long double split = 0x1p32L + 1.0; +#elif LDBL_MANT_DIG == 113 + static const long double split = 0x1p57L + 1.0; +#endif + struct dd ret; + long double ha, hb, la, lb, p, q; + + p = a * split; + ha = a - p; + ha += p; + la = a - ha; + + p = b * split; + hb = b - p; + hb += p; + lb = b - hb; + + p = ha * hb; + q = ha * lb + la * hb; + + ret.hi = p + q; + ret.lo = p - ret.hi + q + la * lb; + return (ret); +} + /* * Fused multiply-add: Compute x * y + z with a single rounding error. * * We use scaling to avoid overflow/underflow, along with the * canonical precision-doubling technique adapted from: * * Dekker, T. A Floating-Point Technique for Extending the * Available Precision. Numer. Math. 18, 224-242 (1971). */ long double fmal(long double x, long double y, long double z) { -#if LDBL_MANT_DIG == 64 - static const long double split = 0x1p32L + 1.0; -#elif LDBL_MANT_DIG == 113 - static const long double split = 0x1p57L + 1.0; -#endif long double xs, ys, zs; - long double c, cc, hx, hy, p, q, tx, ty; - long double r, rr, s; + struct dd xy, r, r2; + long double p; + long double s; int oround; int ex, ey, ez; int spread; /* * Handle special cases. The order of operations and the particular * return values here are crucial in handling special cases involving * infinities, NaNs, overflows, and signed zeroes correctly. */ if (x == 0.0 || y == 0.0) return (x * y + z); if (z == 0.0) return (x * y); if (!isfinite(x) || !isfinite(y)) return (x * y + z); if (!isfinite(z)) return (z); xs = frexpl(x, &ex); ys = frexpl(y, &ey); zs = frexpl(z, &ez); oround = fegetround(); spread = ex + ey - ez; /* * If x * y and z are many orders of magnitude apart, the scaling * will overflow, so we handle these cases specially. Rounding * modes other than FE_TONEAREST are painful. */ if (spread > LDBL_MANT_DIG * 2) { fenv_t env; feraiseexcept(FE_INEXACT); switch(oround) { case FE_TONEAREST: return (x * y); case FE_TOWARDZERO: if (x > 0.0 ^ y < 0.0 ^ z < 0.0) return (x * y); feholdexcept(&env); - r = x * y; + s = x * y; if (!fetestexcept(FE_INEXACT)) - r = nextafterl(r, 0); + s = nextafterl(s, 0); feupdateenv(&env); - return (r); + return (s); case FE_DOWNWARD: if (z > 0.0) return (x * y); feholdexcept(&env); - r = x * y; + s = x * y; if (!fetestexcept(FE_INEXACT)) - r = nextafterl(r, -INFINITY); + s = nextafterl(s, -INFINITY); feupdateenv(&env); - return (r); + return (s); default: /* FE_UPWARD */ if (z < 0.0) return (x * y); feholdexcept(&env); - r = x * y; + s = x * y; if (!fetestexcept(FE_INEXACT)) - r = nextafterl(r, INFINITY); + s = nextafterl(s, INFINITY); feupdateenv(&env); - return (r); + return (s); } } if (spread < -LDBL_MANT_DIG) { feraiseexcept(FE_INEXACT); if (!isnormal(z)) feraiseexcept(FE_UNDERFLOW); switch (oround) { case FE_TONEAREST: return (z); case FE_TOWARDZERO: if (x > 0.0 ^ y < 0.0 ^ z < 0.0) return (z); else return (nextafterl(z, 0)); case FE_DOWNWARD: if (x > 0.0 ^ y < 0.0) return (z); else return (nextafterl(z, -INFINITY)); default: /* FE_UPWARD */ if (x > 0.0 ^ y < 0.0) return (nextafterl(z, INFINITY)); else return (z); } } - /* - * Use Dekker's algorithm to perform the multiplication and - * subsequent addition in twice the machine precision. - * Arrange so that x * y = c + cc, and x * y + z = r + rr. - */ fesetround(FE_TONEAREST); - p = xs * split; - hx = xs - p; - hx += p; - tx = xs - hx; - - p = ys * split; - hy = ys - p; - hy += p; - ty = ys - hy; - - p = hx * hy; - q = hx * ty + tx * hy; - c = p + q; - cc = p - c + q + tx * ty; - + xy = dd_mul(xs, ys); zs = ldexpl(zs, -spread); - r = c + zs; - s = r - c; - rr = (c - (r - s)) + (zs - s) + cc; + r = dd_add(xy.hi, zs); + r.lo += xy.lo; spread = ex + ey; - if (spread + ilogbl(r) > -16383) { + if (spread + ilogbl(r.hi) > -16383) { fesetround(oround); - r = r + rr; + r.hi = r.hi + r.lo; } else { /* * The result is subnormal, so we round before scaling to * avoid double rounding. */ - p = ldexpl(copysignl(0x1p-16382L, r), -spread); - c = r + p; - s = c - r; - cc = (r - (c - s)) + (p - s) + rr; + p = ldexpl(copysignl(0x1p-16382L, r.hi), -spread); + r2 = dd_add(r.hi, p); + r2.lo += r.lo; fesetround(oround); - r = (c + cc) - p; + r.hi = (r2.hi + r2.lo) - p; } - return (ldexpl(r, spread)); + return (ldexpl(r.hi, spread)); }