/* * Double-precision x^y function. * * Copyright (c) 2018, Arm Limited. * SPDX-License-Identifier: MIT */ #include #include #include "libm.h" #include "exp_data.h" #include "pow_data.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 representible. */ #if __FP_FAST_FMA r = __builtin_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 __FP_FAST_FMA hi = t2 + ar2; lo3 = __builtin_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. */ p = (ar3 * (A[1] + r * A[2] + ar2 * (A[3] + r * A[4] + ar2 * (A[5] + r * A[6])))); 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 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. */ fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022); } y = 0x1p-1022 * y; return 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 (predict_false(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. */ tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5); if (predict_false(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 (predict_false(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 (predict_false(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 (predict_false(zeroinfnan(ix))) { double_t x2 = x * x; if (ix >> 63 && checkint(iy) == 1) x2 = -x2; /* Without the barrier some versions of clang hoist the 1/x2 and thus division by zero exception can be signaled spuriously. */ return iy >> 63 ? fp_barrier(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. */ ix = asuint64(x * 0x1p52); ix &= 0x7fffffffffffffff; ix -= 52ULL << 52; } } double_t lo; double_t hi = log_inline(ix, &lo); double_t ehi, elo; #if __FP_FAST_FMA ehi = y * hi; elo = y * lo + __builtin_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); }