| /* |
| * Double-precision x^y function. |
| * |
| * Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. |
| * See https://llvm.org/LICENSE.txt for license information. |
| * SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception |
| */ |
| |
| #include <float.h> |
| #include <math.h> |
| #include <stdint.h> |
| #include "math_config.h" |
| |
| /* |
| Worst-case error: 0.54 ULP (~= ulperr_exp + 1024*Ln2*relerr_log*2^53) |
| relerr_log: 1.3 * 2^-68 (Relative error of log, 1.5 * 2^-68 without fma) |
| ulperr_exp: 0.509 ULP (ULP error of exp, 0.511 ULP without fma) |
| */ |
| |
| #define T __pow_log_data.tab |
| #define A __pow_log_data.poly |
| #define Ln2hi __pow_log_data.ln2hi |
| #define Ln2lo __pow_log_data.ln2lo |
| #define N (1 << POW_LOG_TABLE_BITS) |
| #define OFF 0x3fe6955500000000 |
| |
| /* Top 12 bits of a double (sign and exponent bits). */ |
| static inline uint32_t |
| top12 (double x) |
| { |
| return asuint64 (x) >> 52; |
| } |
| |
| /* Compute y+TAIL = log(x) where the rounded result is y and TAIL has about |
| additional 15 bits precision. IX is the bit representation of x, but |
| normalized in the subnormal range using the sign bit for the exponent. */ |
| static inline double_t |
| log_inline (uint64_t ix, double_t *tail) |
| { |
| /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ |
| double_t z, r, y, invc, logc, logctail, kd, hi, t1, t2, lo, lo1, lo2, p; |
| uint64_t iz, tmp; |
| int k, i; |
| |
| /* x = 2^k z; where z is in range [OFF,2*OFF) and exact. |
| The range is split into N subintervals. |
| The ith subinterval contains z and c is near its center. */ |
| tmp = ix - OFF; |
| i = (tmp >> (52 - POW_LOG_TABLE_BITS)) % N; |
| k = (int64_t) tmp >> 52; /* arithmetic shift */ |
| iz = ix - (tmp & 0xfffULL << 52); |
| z = asdouble (iz); |
| kd = (double_t) k; |
| |
| /* log(x) = k*Ln2 + log(c) + log1p(z/c-1). */ |
| invc = T[i].invc; |
| logc = T[i].logc; |
| logctail = T[i].logctail; |
| |
| /* Note: 1/c is j/N or j/N/2 where j is an integer in [N,2N) and |
| |z/c - 1| < 1/N, so r = z/c - 1 is exactly representable. */ |
| #if HAVE_FAST_FMA |
| r = fma (z, invc, -1.0); |
| #else |
| /* Split z such that rhi, rlo and rhi*rhi are exact and |rlo| <= |r|. */ |
| double_t zhi = asdouble ((iz + (1ULL << 31)) & (-1ULL << 32)); |
| double_t zlo = z - zhi; |
| double_t rhi = zhi * invc - 1.0; |
| double_t rlo = zlo * invc; |
| r = rhi + rlo; |
| #endif |
| |
| /* k*Ln2 + log(c) + r. */ |
| t1 = kd * Ln2hi + logc; |
| t2 = t1 + r; |
| lo1 = kd * Ln2lo + logctail; |
| lo2 = t1 - t2 + r; |
| |
| /* Evaluation is optimized assuming superscalar pipelined execution. */ |
| double_t ar, ar2, ar3, lo3, lo4; |
| ar = A[0] * r; /* A[0] = -0.5. */ |
| ar2 = r * ar; |
| ar3 = r * ar2; |
| /* k*Ln2 + log(c) + r + A[0]*r*r. */ |
| #if HAVE_FAST_FMA |
| hi = t2 + ar2; |
| lo3 = fma (ar, r, -ar2); |
| lo4 = t2 - hi + ar2; |
| #else |
| double_t arhi = A[0] * rhi; |
| double_t arhi2 = rhi * arhi; |
| hi = t2 + arhi2; |
| lo3 = rlo * (ar + arhi); |
| lo4 = t2 - hi + arhi2; |
| #endif |
| /* p = log1p(r) - r - A[0]*r*r. */ |
| #if POW_LOG_POLY_ORDER == 8 |
| p = (ar3 |
| * (A[1] + r * A[2] + ar2 * (A[3] + r * A[4] + ar2 * (A[5] + r * A[6])))); |
| #endif |
| lo = lo1 + lo2 + lo3 + lo4 + p; |
| y = hi + lo; |
| *tail = hi - y + lo; |
| return y; |
| } |
| |
| #undef N |
| #undef T |
| #define N (1 << EXP_TABLE_BITS) |
| #define InvLn2N __exp_data.invln2N |
| #define NegLn2hiN __exp_data.negln2hiN |
| #define NegLn2loN __exp_data.negln2loN |
| #define Shift __exp_data.shift |
| #define T __exp_data.tab |
| #define C2 __exp_data.poly[5 - EXP_POLY_ORDER] |
| #define C3 __exp_data.poly[6 - EXP_POLY_ORDER] |
| #define C4 __exp_data.poly[7 - EXP_POLY_ORDER] |
| #define C5 __exp_data.poly[8 - EXP_POLY_ORDER] |
| #define C6 __exp_data.poly[9 - EXP_POLY_ORDER] |
| |
| /* Handle cases that may overflow or underflow when computing the result that |
| is scale*(1+TMP) without intermediate rounding. The bit representation of |
| scale is in SBITS, however it has a computed exponent that may have |
| overflown into the sign bit so that needs to be adjusted before using it as |
| a double. (int32_t)KI is the k used in the argument reduction and exponent |
| adjustment of scale, positive k here means the result may overflow and |
| negative k means the result may underflow. */ |
| static inline double |
| specialcase (double_t tmp, uint64_t sbits, uint64_t ki) |
| { |
| double_t scale, y; |
| |
| if ((ki & 0x80000000) == 0) |
| { |
| /* k > 0, the exponent of scale might have overflowed by <= 460. */ |
| sbits -= 1009ull << 52; |
| scale = asdouble (sbits); |
| y = 0x1p1009 * (scale + scale * tmp); |
| return check_oflow (eval_as_double (y)); |
| } |
| /* k < 0, need special care in the subnormal range. */ |
| sbits += 1022ull << 52; |
| /* Note: sbits is signed scale. */ |
| scale = asdouble (sbits); |
| y = scale + scale * tmp; |
| if (fabs (y) < 1.0) |
| { |
| /* Round y to the right precision before scaling it into the subnormal |
| range to avoid double rounding that can cause 0.5+E/2 ulp error where |
| E is the worst-case ulp error outside the subnormal range. So this |
| is only useful if the goal is better than 1 ulp worst-case error. */ |
| double_t hi, lo, one = 1.0; |
| if (y < 0.0) |
| one = -1.0; |
| lo = scale - y + scale * tmp; |
| hi = one + y; |
| lo = one - hi + y + lo; |
| y = eval_as_double (hi + lo) - one; |
| /* Fix the sign of 0. */ |
| if (y == 0.0) |
| y = asdouble (sbits & 0x8000000000000000); |
| /* The underflow exception needs to be signaled explicitly. */ |
| force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022); |
| } |
| y = 0x1p-1022 * y; |
| return check_uflow (eval_as_double (y)); |
| } |
| |
| #define SIGN_BIAS (0x800 << EXP_TABLE_BITS) |
| |
| /* Computes sign*exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|. |
| The sign_bias argument is SIGN_BIAS or 0 and sets the sign to -1 or 1. */ |
| static inline double |
| exp_inline (double_t x, double_t xtail, uint32_t sign_bias) |
| { |
| uint32_t abstop; |
| uint64_t ki, idx, top, sbits; |
| /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ |
| double_t kd, z, r, r2, scale, tail, tmp; |
| |
| abstop = top12 (x) & 0x7ff; |
| if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54))) |
| { |
| if (abstop - top12 (0x1p-54) >= 0x80000000) |
| { |
| /* Avoid spurious underflow for tiny x. */ |
| /* Note: 0 is common input. */ |
| double_t one = WANT_ROUNDING ? 1.0 + x : 1.0; |
| return sign_bias ? -one : one; |
| } |
| if (abstop >= top12 (1024.0)) |
| { |
| /* Note: inf and nan are already handled. */ |
| if (asuint64 (x) >> 63) |
| return __math_uflow (sign_bias); |
| else |
| return __math_oflow (sign_bias); |
| } |
| /* Large x is special cased below. */ |
| abstop = 0; |
| } |
| |
| /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */ |
| /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */ |
| z = InvLn2N * x; |
| #if TOINT_INTRINSICS |
| kd = roundtoint (z); |
| ki = converttoint (z); |
| #elif EXP_USE_TOINT_NARROW |
| /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */ |
| kd = eval_as_double (z + Shift); |
| ki = asuint64 (kd) >> 16; |
| kd = (double_t) (int32_t) ki; |
| #else |
| /* z - kd is in [-1, 1] in non-nearest rounding modes. */ |
| kd = eval_as_double (z + Shift); |
| ki = asuint64 (kd); |
| kd -= Shift; |
| #endif |
| r = x + kd * NegLn2hiN + kd * NegLn2loN; |
| /* The code assumes 2^-200 < |xtail| < 2^-8/N. */ |
| r += xtail; |
| /* 2^(k/N) ~= scale * (1 + tail). */ |
| idx = 2 * (ki % N); |
| top = (ki + sign_bias) << (52 - EXP_TABLE_BITS); |
| tail = asdouble (T[idx]); |
| /* This is only a valid scale when -1023*N < k < 1024*N. */ |
| sbits = T[idx + 1] + top; |
| /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */ |
| /* Evaluation is optimized assuming superscalar pipelined execution. */ |
| r2 = r * r; |
| /* Without fma the worst case error is 0.25/N ulp larger. */ |
| /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */ |
| #if EXP_POLY_ORDER == 4 |
| tmp = tail + r + r2 * C2 + r * r2 * (C3 + r * C4); |
| #elif EXP_POLY_ORDER == 5 |
| tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5); |
| #elif EXP_POLY_ORDER == 6 |
| tmp = tail + r + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6); |
| #endif |
| if (unlikely (abstop == 0)) |
| return specialcase (tmp, sbits, ki); |
| scale = asdouble (sbits); |
| /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there |
| is no spurious underflow here even without fma. */ |
| return eval_as_double (scale + scale * tmp); |
| } |
| |
| /* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is |
| the bit representation of a non-zero finite floating-point value. */ |
| static inline int |
| checkint (uint64_t iy) |
| { |
| int e = iy >> 52 & 0x7ff; |
| if (e < 0x3ff) |
| return 0; |
| if (e > 0x3ff + 52) |
| return 2; |
| if (iy & ((1ULL << (0x3ff + 52 - e)) - 1)) |
| return 0; |
| if (iy & (1ULL << (0x3ff + 52 - e))) |
| return 1; |
| return 2; |
| } |
| |
| /* Returns 1 if input is the bit representation of 0, infinity or nan. */ |
| static inline int |
| zeroinfnan (uint64_t i) |
| { |
| return 2 * i - 1 >= 2 * asuint64 (INFINITY) - 1; |
| } |
| |
| double |
| pow (double x, double y) |
| { |
| uint32_t sign_bias = 0; |
| uint64_t ix, iy; |
| uint32_t topx, topy; |
| |
| ix = asuint64 (x); |
| iy = asuint64 (y); |
| topx = top12 (x); |
| topy = top12 (y); |
| if (unlikely (topx - 0x001 >= 0x7ff - 0x001 |
| || (topy & 0x7ff) - 0x3be >= 0x43e - 0x3be)) |
| { |
| /* Note: if |y| > 1075 * ln2 * 2^53 ~= 0x1.749p62 then pow(x,y) = inf/0 |
| and if |y| < 2^-54 / 1075 ~= 0x1.e7b6p-65 then pow(x,y) = +-1. */ |
| /* Special cases: (x < 0x1p-126 or inf or nan) or |
| (|y| < 0x1p-65 or |y| >= 0x1p63 or nan). */ |
| if (unlikely (zeroinfnan (iy))) |
| { |
| if (2 * iy == 0) |
| return issignaling_inline (x) ? x + y : 1.0; |
| if (ix == asuint64 (1.0)) |
| return issignaling_inline (y) ? x + y : 1.0; |
| if (2 * ix > 2 * asuint64 (INFINITY) |
| || 2 * iy > 2 * asuint64 (INFINITY)) |
| return x + y; |
| if (2 * ix == 2 * asuint64 (1.0)) |
| return 1.0; |
| if ((2 * ix < 2 * asuint64 (1.0)) == !(iy >> 63)) |
| return 0.0; /* |x|<1 && y==inf or |x|>1 && y==-inf. */ |
| return y * y; |
| } |
| if (unlikely (zeroinfnan (ix))) |
| { |
| double_t x2 = x * x; |
| if (ix >> 63 && checkint (iy) == 1) |
| { |
| x2 = -x2; |
| sign_bias = 1; |
| } |
| if (WANT_ERRNO && 2 * ix == 0 && iy >> 63) |
| return __math_divzero (sign_bias); |
| /* Without the barrier some versions of clang hoist the 1/x2 and |
| thus division by zero exception can be signaled spuriously. */ |
| return iy >> 63 ? opt_barrier_double (1 / x2) : x2; |
| } |
| /* Here x and y are non-zero finite. */ |
| if (ix >> 63) |
| { |
| /* Finite x < 0. */ |
| int yint = checkint (iy); |
| if (yint == 0) |
| return __math_invalid (x); |
| if (yint == 1) |
| sign_bias = SIGN_BIAS; |
| ix &= 0x7fffffffffffffff; |
| topx &= 0x7ff; |
| } |
| if ((topy & 0x7ff) - 0x3be >= 0x43e - 0x3be) |
| { |
| /* Note: sign_bias == 0 here because y is not odd. */ |
| if (ix == asuint64 (1.0)) |
| return 1.0; |
| if ((topy & 0x7ff) < 0x3be) |
| { |
| /* |y| < 2^-65, x^y ~= 1 + y*log(x). */ |
| if (WANT_ROUNDING) |
| return ix > asuint64 (1.0) ? 1.0 + y : 1.0 - y; |
| else |
| return 1.0; |
| } |
| return (ix > asuint64 (1.0)) == (topy < 0x800) ? __math_oflow (0) |
| : __math_uflow (0); |
| } |
| if (topx == 0) |
| { |
| /* Normalize subnormal x so exponent becomes negative. */ |
| /* Without the barrier some versions of clang evaluate the mul |
| unconditionally causing spurious overflow exceptions. */ |
| ix = asuint64 (opt_barrier_double (x) * 0x1p52); |
| ix &= 0x7fffffffffffffff; |
| ix -= 52ULL << 52; |
| } |
| } |
| |
| double_t lo; |
| double_t hi = log_inline (ix, &lo); |
| double_t ehi, elo; |
| #if HAVE_FAST_FMA |
| ehi = y * hi; |
| elo = y * lo + fma (y, hi, -ehi); |
| #else |
| double_t yhi = asdouble (iy & -1ULL << 27); |
| double_t ylo = y - yhi; |
| double_t lhi = asdouble (asuint64 (hi) & -1ULL << 27); |
| double_t llo = hi - lhi + lo; |
| ehi = yhi * lhi; |
| elo = ylo * lhi + y * llo; /* |elo| < |ehi| * 2^-25. */ |
| #endif |
| return exp_inline (ehi, elo, sign_bias); |
| } |
| #if USE_GLIBC_ABI |
| strong_alias (pow, __pow_finite) |
| hidden_alias (pow, __ieee754_pow) |
| # if LDBL_MANT_DIG == 53 |
| long double powl (long double x, long double y) { return pow (x, y); } |
| # endif |
| #endif |