| #include "libm.h" |
| #include <fenv.h> |
| |
| #if LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 |
| /* exact add, assumes exponent_x >= exponent_y */ |
| static void add(long double* hi, long double* lo, long double x, long double y) { |
| long double r; |
| |
| r = x + y; |
| *hi = r; |
| r -= x; |
| *lo = y - r; |
| } |
| |
| /* exact mul, assumes no over/underflow */ |
| static void mul(long double* hi, long double* lo, long double x, long double y) { |
| static const long double c = 1.0 + 0x1p32L; |
| long double cx, xh, xl, cy, yh, yl; |
| |
| cx = c * x; |
| xh = (x - cx) + cx; |
| xl = x - xh; |
| cy = c * y; |
| yh = (y - cy) + cy; |
| yl = y - yh; |
| *hi = x * y; |
| *lo = (xh * yh - *hi) + xh * yl + xl * yh + xl * yl; |
| } |
| |
| /* |
| assume (long double)(hi+lo) == hi |
| return an adjusted hi so that rounding it to double (or less) precision is correct |
| */ |
| static long double adjust(long double hi, long double lo) { |
| union ldshape uhi, ulo; |
| |
| if (lo == 0) |
| return hi; |
| uhi.f = hi; |
| if (uhi.i.m & 0x3ff) |
| return hi; |
| ulo.f = lo; |
| if ((uhi.i.se & 0x8000) == (ulo.i.se & 0x8000)) |
| uhi.i.m++; |
| else { |
| /* handle underflow and take care of ld80 implicit msb */ |
| if (uhi.i.m << 1 == 0) { |
| uhi.i.m = 0; |
| uhi.i.se--; |
| } |
| uhi.i.m--; |
| } |
| return uhi.f; |
| } |
| |
| /* adjusted add so the result is correct when rounded to double (or less) precision */ |
| static long double dadd(long double x, long double y) { |
| add(&x, &y, x, y); |
| return adjust(x, y); |
| } |
| |
| /* adjusted mul so the result is correct when rounded to double (or less) precision */ |
| static long double dmul(long double x, long double y) { |
| mul(&x, &y, x, y); |
| return adjust(x, y); |
| } |
| |
| static int getexp(long double x) { |
| union ldshape u; |
| u.f = x; |
| return u.i.se & 0x7fff; |
| } |
| |
| double fma(double x, double y, double z) { |
| PRAGMA_STDC_FENV_ACCESS_ON |
| long double hi, lo1, lo2, xy; |
| int round, ez, exy; |
| |
| /* handle +-inf,nan */ |
| if (!isfinite(x) || !isfinite(y)) |
| return x * y + z; |
| if (!isfinite(z)) |
| return z; |
| /* handle +-0 */ |
| if (x == 0.0 || y == 0.0) |
| return x * y + z; |
| round = fegetround(); |
| if (z == 0.0) { |
| if (round == FE_TONEAREST) |
| return dmul(x, y); |
| return x * y; |
| } |
| |
| /* exact mul and add require nearest rounding */ |
| /* spurious inexact exceptions may be raised */ |
| fesetround(FE_TONEAREST); |
| mul(&xy, &lo1, x, y); |
| exy = getexp(xy); |
| ez = getexp(z); |
| if (ez > exy) { |
| add(&hi, &lo2, z, xy); |
| } else if (ez > exy - 12) { |
| add(&hi, &lo2, xy, z); |
| if (hi == 0) { |
| /* |
| xy + z is 0, but it should be calculated with the |
| original rounding mode so the sign is correct, if the |
| compiler does not support FENV_ACCESS ON it does not |
| know about the changed rounding mode and eliminates |
| the xy + z below without the volatile memory access |
| */ |
| volatile double z_; |
| fesetround(round); |
| z_ = z; |
| return (xy + z_) + lo1; |
| } |
| } else { |
| /* |
| ez <= exy - 12 |
| the 12 extra bits (1guard, 11round+sticky) are needed so with |
| lo = dadd(lo1, lo2) |
| elo <= ehi - 11, and we use the last 10 bits in adjust so |
| dadd(hi, lo) |
| gives correct result when rounded to double |
| */ |
| hi = xy; |
| lo2 = z; |
| } |
| /* |
| the result is stored before return for correct precision and exceptions |
| |
| one corner case is when the underflow flag should be raised because |
| the precise result is an inexact subnormal double, but the calculated |
| long double result is an exact subnormal double |
| (so rounding to double does not raise exceptions) |
| |
| in nearest rounding mode dadd takes care of this: the last bit of the |
| result is adjusted so rounding sees an inexact value when it should |
| |
| in non-nearest rounding mode fenv is used for the workaround |
| */ |
| fesetround(round); |
| if (round == FE_TONEAREST) |
| z = dadd(hi, dadd(lo1, lo2)); |
| else { |
| #if defined(FE_INEXACT) && defined(FE_UNDERFLOW) |
| int e = fetestexcept(FE_INEXACT); |
| feclearexcept(FE_INEXACT); |
| #endif |
| z = hi + (lo1 + lo2); |
| #if defined(FE_INEXACT) && defined(FE_UNDERFLOW) |
| if (getexp(z) < 0x3fff - 1022 && fetestexcept(FE_INEXACT)) |
| feraiseexcept(FE_UNDERFLOW); |
| else if (e) |
| feraiseexcept(FE_INEXACT); |
| #endif |
| } |
| return z; |
| } |
| #else |
| /* origin: FreeBSD /usr/src/lib/msun/src/s_fma.c */ |
| /*- |
| * Copyright (c) 2005-2011 David Schultz <das@FreeBSD.ORG> |
| * 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. |
| */ |
| |
| /* |
| * 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, with a small tweak: The least significant bit of the |
| * result is adjusted into a sticky bit summarizing all the bits that |
| * were lost to rounding. This adjustment negates the effects of double |
| * rounding when the result is added to another number with a higher |
| * exponent. For an explanation of round and sticky bits, see any reference |
| * on FPU design, e.g., |
| * |
| * J. Coonen. An Implementation Guide to a Proposed Standard for |
| * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980. |
| */ |
| static inline double add_adjusted(double a, double b) { |
| struct dd sum; |
| union { |
| double f; |
| uint64_t i; |
| } uhi, ulo; |
| |
| sum = dd_add(a, b); |
| if (sum.lo != 0) { |
| uhi.f = sum.hi; |
| if ((uhi.i & 1) == 0) { |
| /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */ |
| ulo.f = sum.lo; |
| uhi.i += 1 - ((uhi.i ^ ulo.i) >> 62); |
| sum.hi = uhi.f; |
| } |
| } |
| return (sum.hi); |
| } |
| |
| /* |
| * Compute ldexp(a+b, scale) with a single rounding error. It is assumed |
| * that the result will be subnormal, and care is taken to ensure that |
| * double rounding does not occur. |
| */ |
| static inline double add_and_denormalize(double a, double b, int scale) { |
| struct dd sum; |
| union { |
| double f; |
| uint64_t i; |
| } uhi, ulo; |
| int bits_lost; |
| |
| sum = dd_add(a, b); |
| |
| /* |
| * If we are losing at least two bits of accuracy to denormalization, |
| * then the first lost bit becomes a round bit, and we adjust the |
| * lowest bit of sum.hi to make it a sticky bit summarizing all the |
| * bits in sum.lo. With the sticky bit adjusted, the hardware will |
| * break any ties in the correct direction. |
| * |
| * If we are losing only one bit to denormalization, however, we must |
| * break the ties manually. |
| */ |
| if (sum.lo != 0) { |
| uhi.f = sum.hi; |
| bits_lost = -((int)(uhi.i >> 52) & 0x7ff) - scale + 1; |
| if ((bits_lost != 1) ^ (int)(uhi.i & 1)) { |
| /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */ |
| ulo.f = sum.lo; |
| uhi.i += 1 - (((uhi.i ^ ulo.i) >> 62) & 2); |
| sum.hi = uhi.f; |
| } |
| } |
| return scalbn(sum.hi, scale); |
| } |
| |
| /* |
| * 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. |
| */ |
| double fma(double x, double y, double z) { |
| PRAGMA_STDC_FENV_ACCESS_ON |
| double xs, ys, zs, adj; |
| struct dd xy, r; |
| 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 (!isfinite(x) || !isfinite(y)) |
| return (x * y + z); |
| if (!isfinite(z)) |
| return (z); |
| if (x == 0.0 || y == 0.0) |
| return (x * y + z); |
| if (z == 0.0) |
| return (x * y); |
| |
| 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) { |
| #ifdef FE_INEXACT |
| feraiseexcept(FE_INEXACT); |
| #endif |
| #ifdef FE_UNDERFLOW |
| if (!isnormal(z)) |
| feraiseexcept(FE_UNDERFLOW); |
| #endif |
| switch (oround) { |
| default: /* FE_TONEAREST */ |
| return (z); |
| #ifdef FE_TOWARDZERO |
| case FE_TOWARDZERO: |
| if ((x > 0.0) ^ (y < 0.0) ^ (z < 0.0)) |
| return (z); |
| else |
| return (nextafter(z, 0)); |
| #endif |
| #ifdef FE_DOWNWARD |
| case FE_DOWNWARD: |
| if ((x > 0.0) ^ (y < 0.0)) |
| return (z); |
| else |
| return (nextafter(z, -INFINITY)); |
| #endif |
| #ifdef FE_UPWARD |
| case FE_UPWARD: |
| if ((x > 0.0) ^ (y < 0.0)) |
| return (nextafter(z, INFINITY)); |
| else |
| return (z); |
| #endif |
| } |
| } |
| if (spread <= DBL_MANT_DIG * 2) |
| zs = scalbn(zs, -spread); |
| else |
| zs = copysign(DBL_MIN, zs); |
| |
| fesetround(FE_TONEAREST); |
| |
| /* |
| * Basic approach for round-to-nearest: |
| * |
| * (xy.hi, xy.lo) = x * y (exact) |
| * (r.hi, r.lo) = xy.hi + z (exact) |
| * adj = xy.lo + r.lo (inexact; low bit is sticky) |
| * result = r.hi + adj (correctly rounded) |
| */ |
| xy = dd_mul(xs, ys); |
| r = dd_add(xy.hi, zs); |
| |
| spread = ex + ey; |
| |
| if (r.hi == 0.0) { |
| /* |
| * When the addends cancel to 0, ensure that the result has |
| * the correct sign. |
| */ |
| fesetround(oround); |
| volatile double vzs = zs; /* XXX gcc CSE bug workaround */ |
| return xy.hi + vzs + scalbn(xy.lo, spread); |
| } |
| |
| if (oround != FE_TONEAREST) { |
| /* |
| * There is no need to worry about double rounding in directed |
| * rounding modes. |
| * But underflow may not be raised properly, example in downward rounding: |
| * fma(0x1.000000001p-1000, 0x1.000000001p-30, -0x1p-1066) |
| */ |
| double ret; |
| #if defined(FE_INEXACT) && defined(FE_UNDERFLOW) |
| int e = fetestexcept(FE_INEXACT); |
| feclearexcept(FE_INEXACT); |
| #endif |
| fesetround(oround); |
| adj = r.lo + xy.lo; |
| ret = scalbn(r.hi + adj, spread); |
| #if defined(FE_INEXACT) && defined(FE_UNDERFLOW) |
| if (ilogb(ret) < -1022 && fetestexcept(FE_INEXACT)) |
| feraiseexcept(FE_UNDERFLOW); |
| else if (e) |
| feraiseexcept(FE_INEXACT); |
| #endif |
| return ret; |
| } |
| |
| adj = add_adjusted(r.lo, xy.lo); |
| if (spread + ilogb(r.hi) > -1023) |
| return scalbn(r.hi + adj, spread); |
| else |
| return add_and_denormalize(r.hi, adj, spread); |
| } |
| #endif |