diff options
Diffstat (limited to 'src')
| -rw-r--r-- | src/math/fma.c | 585 |
1 files changed, 154 insertions, 431 deletions
diff --git a/src/math/fma.c b/src/math/fma.c index 741ccd75..0c6f90c9 100644 --- a/src/math/fma.c +++ b/src/math/fma.c @@ -1,460 +1,183 @@ -#include <fenv.h> -#include "libm.h" +#include <stdint.h> +#include <float.h> +#include <math.h> +#include "atomic.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; +#define ASUINT64(x) ((union {double f; uint64_t i;}){x}).i +#define ZEROINFNAN (0x7ff-0x3ff-52-1) - 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; -} +struct num { uint64_t m; int e; int sign; }; -/* -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) +static struct num normalize(double x) { - 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--; + uint64_t ix = ASUINT64(x); + int e = ix>>52; + int sign = e & 0x800; + e &= 0x7ff; + if (!e) { + ix = ASUINT64(x*0x1p63); + e = ix>>52 & 0x7ff; + e = e ? e-63 : 0x800; } - return uhi.f; + ix &= (1ull<<52)-1; + ix |= 1ull<<52; + ix <<= 1; + e -= 0x3ff + 52 + 1; + return (struct num){ix,e,sign}; } -/* adjusted add so the result is correct when rounded to double (or less) precision */ -static long double dadd(long double x, long double y) +static void mul(uint64_t *hi, uint64_t *lo, uint64_t x, uint64_t 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; + uint64_t t1,t2,t3; + uint64_t xlo = (uint32_t)x, xhi = x>>32; + uint64_t ylo = (uint32_t)y, yhi = y>>32; + + t1 = xlo*ylo; + t2 = xlo*yhi + xhi*ylo; + t3 = xhi*yhi; + *lo = t1 + (t2<<32); + *hi = t3 + (t2>>32) + (t1 > *lo); } 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)) + /* normalize so top 10bits and last bit are 0 */ + struct num nx, ny, nz; + nx = normalize(x); + ny = normalize(y); + nz = normalize(z); + + if (nx.e >= ZEROINFNAN || ny.e >= ZEROINFNAN) return x*y + z; - if (!isfinite(z)) + if (nz.e >= ZEROINFNAN) { + if (nz.e > ZEROINFNAN) /* z==0 */ + return x*y + 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; + /* mul: r = x*y */ + uint64_t rhi, rlo, zhi, zlo; + mul(&rhi, &rlo, nx.m, ny.m); + /* either top 20 or 21 bits of rhi and last 2 bits of rlo are 0 */ + + /* align exponents */ + int e = nx.e + ny.e; + int d = nz.e - e; + /* shift bits z<<=kz, r>>=kr, so kz+kr == d, set e = e+kr (== ez-kz) */ + if (d > 0) { + if (d < 64) { + zlo = nz.m<<d; + zhi = nz.m>>64-d; + } else { + zlo = 0; + zhi = nz.m; + e = nz.e - 64; + d -= 64; + if (d == 0) { + } else if (d < 64) { + rlo = rhi<<64-d | rlo>>d | !!(rlo<<64-d); + rhi = rhi>>d; + } else { + rlo = 1; + rhi = 0; + } } } 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; + zhi = 0; + d = -d; + if (d == 0) { + zlo = nz.m; + } else if (d < 64) { + zlo = nz.m>>d | !!(nz.m<<64-d); + } else { + zlo = 1; } } - 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 + /* add */ + int sign = nx.sign^ny.sign; + int samesign = !(sign^nz.sign); + int nonzero = 1; + if (samesign) { + /* r += z */ + rlo += zlo; + rhi += zhi + (rlo < zlo); + } else { + /* r -= z */ + uint64_t t = rlo; + rlo -= zlo; + rhi = rhi - zhi - (t < rlo); + if (rhi>>63) { + rlo = -rlo; + rhi = -rhi-!!rlo; + sign = !sign; } + nonzero = !!rhi; } - 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); + /* set rhi to top 63bit of the result (last bit is sticky) */ + if (nonzero) { + e += 64; + d = a_clz_64(rhi)-1; + /* note: d > 0 */ + rhi = rhi<<d | rlo>>64-d | !!(rlo<<d); + } else if (rlo) { + d = a_clz_64(rlo)-1; + if (d < 0) + rhi = rlo>>1 | (rlo&1); + else + rhi = rlo<<d; + } else { + /* exact +-0 */ + return x*y + z; } - - 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; + e -= d; + + /* convert to double */ + int64_t i = rhi; /* i is in [1<<62,(1<<63)-1] */ + if (sign) + i = -i; + double r = i; /* |r| is in [0x1p62,0x1p63] */ + + if (e < -1022-62) { + /* result is subnormal before rounding */ + if (e == -1022-63) { + double c = 0x1p63; + if (sign) + c = -c; + if (r == c) { + /* min normal after rounding, underflow depends + on arch behaviour which can be imitated by + a double to float conversion */ + float fltmin = 0x0.ffffff8p-63*FLT_MIN * r; + return DBL_MIN/FLT_MIN * fltmin; + } + /* one bit is lost when scaled, add another top bit to + only round once at conversion if it is inexact */ + if (rhi << 53) { + i = rhi>>1 | (rhi&1) | 1ull<<62; + if (sign) + i = -i; + r = i; + r = 2*r - c; /* remove top bit */ + + /* raise underflow portably, such that it + cannot be optimized away */ + { + double_t tiny = DBL_MIN/FLT_MIN * r; + r += (double)(tiny*tiny) * (r-r); + } + } + } else { + /* only round once when scaled */ + d = 10; + i = ( rhi>>d | !!(rhi<<64-d) ) << d; + if (sign) + i = -i; + r = i; + } } - - 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); + return scalbn(r, e); } -#endif |