1/* 2 * Double-precision x^y function. 3 * 4 * Copyright (c) 2018, Arm Limited. 5 * SPDX-License-Identifier: MIT 6 */ 7 8#include <math.h> 9#include <stdint.h> 10#include "libm.h" 11#include "exp_data.h" 12#include "pow_data.h" 13 14/* 15Worst-case error: 0.54 ULP (~= ulperr_exp + 1024*Ln2*relerr_log*2^53) 16relerr_log: 1.3 * 2^-68 (Relative error of log, 1.5 * 2^-68 without fma) 17ulperr_exp: 0.509 ULP (ULP error of exp, 0.511 ULP without fma) 18*/ 19 20#define T __pow_log_data.tab 21#define A __pow_log_data.poly 22#define Ln2hi __pow_log_data.ln2hi 23#define Ln2lo __pow_log_data.ln2lo 24#define N (1 << POW_LOG_TABLE_BITS) 25#define OFF 0x3fe6955500000000 26 27/* Top 12 bits of a double (sign and exponent bits). */ 28static inline uint32_t top12(double x) 29{ 30 return asuint64(x) >> 52; 31} 32 33/* Compute y+TAIL = log(x) where the rounded result is y and TAIL has about 34 additional 15 bits precision. IX is the bit representation of x, but 35 normalized in the subnormal range using the sign bit for the exponent. */ 36static inline double_t log_inline(uint64_t ix, double_t *tail) 37{ 38 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 39 double_t z, r, y, invc, logc, logctail, kd, hi, t1, t2, lo, lo1, lo2, p; 40 uint64_t iz, tmp; 41 int k, i; 42 43 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact. 44 The range is split into N subintervals. 45 The ith subinterval contains z and c is near its center. */ 46 tmp = ix - OFF; 47 i = (tmp >> (52 - POW_LOG_TABLE_BITS)) % N; 48 k = (int64_t)tmp >> 52; /* arithmetic shift */ 49 iz = ix - (tmp & 0xfffULL << 52); 50 z = asdouble(iz); 51 kd = (double_t)k; 52 53 /* log(x) = k*Ln2 + log(c) + log1p(z/c-1). */ 54 invc = T[i].invc; 55 logc = T[i].logc; 56 logctail = T[i].logctail; 57 58 /* Note: 1/c is j/N or j/N/2 where j is an integer in [N,2N) and 59 |z/c - 1| < 1/N, so r = z/c - 1 is exactly representible. */ 60#if __FP_FAST_FMA 61 r = __builtin_fma(z, invc, -1.0); 62#else 63 /* Split z such that rhi, rlo and rhi*rhi are exact and |rlo| <= |r|. */ 64 double_t zhi = asdouble((iz + (1ULL << 31)) & (-1ULL << 32)); 65 double_t zlo = z - zhi; 66 double_t rhi = zhi * invc - 1.0; 67 double_t rlo = zlo * invc; 68 r = rhi + rlo; 69#endif 70 71 /* k*Ln2 + log(c) + r. */ 72 t1 = kd * Ln2hi + logc; 73 t2 = t1 + r; 74 lo1 = kd * Ln2lo + logctail; 75 lo2 = t1 - t2 + r; 76 77 /* Evaluation is optimized assuming superscalar pipelined execution. */ 78 double_t ar, ar2, ar3, lo3, lo4; 79 ar = A[0] * r; /* A[0] = -0.5. */ 80 ar2 = r * ar; 81 ar3 = r * ar2; 82 /* k*Ln2 + log(c) + r + A[0]*r*r. */ 83#if __FP_FAST_FMA 84 hi = t2 + ar2; 85 lo3 = __builtin_fma(ar, r, -ar2); 86 lo4 = t2 - hi + ar2; 87#else 88 double_t arhi = A[0] * rhi; 89 double_t arhi2 = rhi * arhi; 90 hi = t2 + arhi2; 91 lo3 = rlo * (ar + arhi); 92 lo4 = t2 - hi + arhi2; 93#endif 94 /* p = log1p(r) - r - A[0]*r*r. */ 95 p = (ar3 * (A[1] + r * A[2] + 96 ar2 * (A[3] + r * A[4] + ar2 * (A[5] + r * A[6])))); 97 lo = lo1 + lo2 + lo3 + lo4 + p; 98 y = hi + lo; 99 *tail = hi - y + lo; 100 return y; 101} 102 103#undef N 104#undef T 105#define N (1 << EXP_TABLE_BITS) 106#define InvLn2N __exp_data.invln2N 107#define NegLn2hiN __exp_data.negln2hiN 108#define NegLn2loN __exp_data.negln2loN 109#define Shift __exp_data.shift 110#define T __exp_data.tab 111#define C2 __exp_data.poly[5 - EXP_POLY_ORDER] 112#define C3 __exp_data.poly[6 - EXP_POLY_ORDER] 113#define C4 __exp_data.poly[7 - EXP_POLY_ORDER] 114#define C5 __exp_data.poly[8 - EXP_POLY_ORDER] 115#define C6 __exp_data.poly[9 - EXP_POLY_ORDER] 116 117/* Handle cases that may overflow or underflow when computing the result that 118 is scale*(1+TMP) without intermediate rounding. The bit representation of 119 scale is in SBITS, however it has a computed exponent that may have 120 overflown into the sign bit so that needs to be adjusted before using it as 121 a double. (int32_t)KI is the k used in the argument reduction and exponent 122 adjustment of scale, positive k here means the result may overflow and 123 negative k means the result may underflow. */ 124static inline double specialcase(double_t tmp, uint64_t sbits, uint64_t ki) 125{ 126 double_t scale, y; 127 128 if ((ki & 0x80000000) == 0) { 129 /* k > 0, the exponent of scale might have overflowed by <= 460. */ 130 sbits -= 1009ull << 52; 131 scale = asdouble(sbits); 132 y = 0x1p1009 * (scale + scale * tmp); 133 return eval_as_double(y); 134 } 135 /* k < 0, need special care in the subnormal range. */ 136 sbits += 1022ull << 52; 137 /* Note: sbits is signed scale. */ 138 scale = asdouble(sbits); 139 y = scale + scale * tmp; 140 if (fabs(y) < 1.0) { 141 /* Round y to the right precision before scaling it into the subnormal 142 range to avoid double rounding that can cause 0.5+E/2 ulp error where 143 E is the worst-case ulp error outside the subnormal range. So this 144 is only useful if the goal is better than 1 ulp worst-case error. */ 145 double_t hi, lo, one = 1.0; 146 if (y < 0.0) 147 one = -1.0; 148 lo = scale - y + scale * tmp; 149 hi = one + y; 150 lo = one - hi + y + lo; 151 y = eval_as_double(hi + lo) - one; 152 /* Fix the sign of 0. */ 153 if (y == 0.0) 154 y = asdouble(sbits & 0x8000000000000000); 155 /* The underflow exception needs to be signaled explicitly. */ 156 fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022); 157 } 158 y = 0x1p-1022 * y; 159 return eval_as_double(y); 160} 161 162#define SIGN_BIAS (0x800 << EXP_TABLE_BITS) 163 164/* Computes sign*exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|. 165 The sign_bias argument is SIGN_BIAS or 0 and sets the sign to -1 or 1. */ 166static inline double exp_inline(double_t x, double_t xtail, uint32_t sign_bias) 167{ 168 uint32_t abstop; 169 uint64_t ki, idx, top, sbits; 170 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 171 double_t kd, z, r, r2, scale, tail, tmp; 172 173 abstop = top12(x) & 0x7ff; 174 if (predict_false(abstop - top12(0x1p-54) >= 175 top12(512.0) - top12(0x1p-54))) { 176 if (abstop - top12(0x1p-54) >= 0x80000000) { 177 /* Avoid spurious underflow for tiny x. */ 178 /* Note: 0 is common input. */ 179 double_t one = WANT_ROUNDING ? 1.0 + x : 1.0; 180 return sign_bias ? -one : one; 181 } 182 if (abstop >= top12(1024.0)) { 183 /* Note: inf and nan are already handled. */ 184 if (asuint64(x) >> 63) 185 return __math_uflow(sign_bias); 186 else 187 return __math_oflow(sign_bias); 188 } 189 /* Large x is special cased below. */ 190 abstop = 0; 191 } 192 193 /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */ 194 /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */ 195 z = InvLn2N * x; 196#if TOINT_INTRINSICS 197 kd = roundtoint(z); 198 ki = converttoint(z); 199#elif EXP_USE_TOINT_NARROW 200 /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */ 201 kd = eval_as_double(z + Shift); 202 ki = asuint64(kd) >> 16; 203 kd = (double_t)(int32_t)ki; 204#else 205 /* z - kd is in [-1, 1] in non-nearest rounding modes. */ 206 kd = eval_as_double(z + Shift); 207 ki = asuint64(kd); 208 kd -= Shift; 209#endif 210 r = x + kd * NegLn2hiN + kd * NegLn2loN; 211 /* The code assumes 2^-200 < |xtail| < 2^-8/N. */ 212 r += xtail; 213 /* 2^(k/N) ~= scale * (1 + tail). */ 214 idx = 2 * (ki % N); 215 top = (ki + sign_bias) << (52 - EXP_TABLE_BITS); 216 tail = asdouble(T[idx]); 217 /* This is only a valid scale when -1023*N < k < 1024*N. */ 218 sbits = T[idx + 1] + top; 219 /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */ 220 /* Evaluation is optimized assuming superscalar pipelined execution. */ 221 r2 = r * r; 222 /* Without fma the worst case error is 0.25/N ulp larger. */ 223 /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */ 224 tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5); 225 if (predict_false(abstop == 0)) 226 return specialcase(tmp, sbits, ki); 227 scale = asdouble(sbits); 228 /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there 229 is no spurious underflow here even without fma. */ 230 return eval_as_double(scale + scale * tmp); 231} 232 233/* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is 234 the bit representation of a non-zero finite floating-point value. */ 235static inline int checkint(uint64_t iy) 236{ 237 int e = iy >> 52 & 0x7ff; 238 if (e < 0x3ff) 239 return 0; 240 if (e > 0x3ff + 52) 241 return 2; 242 if (iy & ((1ULL << (0x3ff + 52 - e)) - 1)) 243 return 0; 244 if (iy & (1ULL << (0x3ff + 52 - e))) 245 return 1; 246 return 2; 247} 248 249/* Returns 1 if input is the bit representation of 0, infinity or nan. */ 250static inline int zeroinfnan(uint64_t i) 251{ 252 return 2 * i - 1 >= 2 * asuint64(INFINITY) - 1; 253} 254 255double pow(double x, double y) 256{ 257 uint32_t sign_bias = 0; 258 uint64_t ix, iy; 259 uint32_t topx, topy; 260 261 ix = asuint64(x); 262 iy = asuint64(y); 263 topx = top12(x); 264 topy = top12(y); 265 if (predict_false(topx - 0x001 >= 0x7ff - 0x001 || 266 (topy & 0x7ff) - 0x3be >= 0x43e - 0x3be)) { 267 /* Note: if |y| > 1075 * ln2 * 2^53 ~= 0x1.749p62 then pow(x,y) = inf/0 268 and if |y| < 2^-54 / 1075 ~= 0x1.e7b6p-65 then pow(x,y) = +-1. */ 269 /* Special cases: (x < 0x1p-126 or inf or nan) or 270 (|y| < 0x1p-65 or |y| >= 0x1p63 or nan). */ 271 if (predict_false(zeroinfnan(iy))) { 272 if (2 * iy == 0) 273 return issignaling_inline(x) ? x + y : 1.0; 274 if (ix == asuint64(1.0)) 275 return issignaling_inline(y) ? x + y : 1.0; 276 if (2 * ix > 2 * asuint64(INFINITY) || 277 2 * iy > 2 * asuint64(INFINITY)) 278 return x + y; 279 if (2 * ix == 2 * asuint64(1.0)) 280 return 1.0; 281 if ((2 * ix < 2 * asuint64(1.0)) == !(iy >> 63)) 282 return 0.0; /* |x|<1 && y==inf or |x|>1 && y==-inf. */ 283 return y * y; 284 } 285 if (predict_false(zeroinfnan(ix))) { 286 double_t x2 = x * x; 287 if (ix >> 63 && checkint(iy) == 1) 288 x2 = -x2; 289 /* Without the barrier some versions of clang hoist the 1/x2 and 290 thus division by zero exception can be signaled spuriously. */ 291 return iy >> 63 ? fp_barrier(1 / x2) : x2; 292 } 293 /* Here x and y are non-zero finite. */ 294 if (ix >> 63) { 295 /* Finite x < 0. */ 296 int yint = checkint(iy); 297 if (yint == 0) 298 return __math_invalid(x); 299 if (yint == 1) 300 sign_bias = SIGN_BIAS; 301 ix &= 0x7fffffffffffffff; 302 topx &= 0x7ff; 303 } 304 if ((topy & 0x7ff) - 0x3be >= 0x43e - 0x3be) { 305 /* Note: sign_bias == 0 here because y is not odd. */ 306 if (ix == asuint64(1.0)) 307 return 1.0; 308 if ((topy & 0x7ff) < 0x3be) { 309 /* |y| < 2^-65, x^y ~= 1 + y*log(x). */ 310 if (WANT_ROUNDING) 311 return ix > asuint64(1.0) ? 1.0 + y : 312 1.0 - y; 313 else 314 return 1.0; 315 } 316 return (ix > asuint64(1.0)) == (topy < 0x800) ? 317 __math_oflow(0) : 318 __math_uflow(0); 319 } 320 if (topx == 0) { 321 /* Normalize subnormal x so exponent becomes negative. */ 322 ix = asuint64(x * 0x1p52); 323 ix &= 0x7fffffffffffffff; 324 ix -= 52ULL << 52; 325 } 326 } 327 328 double_t lo; 329 double_t hi = log_inline(ix, &lo); 330 double_t ehi, elo; 331#if __FP_FAST_FMA 332 ehi = y * hi; 333 elo = y * lo + __builtin_fma(y, hi, -ehi); 334#else 335 double_t yhi = asdouble(iy & -1ULL << 27); 336 double_t ylo = y - yhi; 337 double_t lhi = asdouble(asuint64(hi) & -1ULL << 27); 338 double_t llo = hi - lhi + lo; 339 ehi = yhi * lhi; 340 elo = ylo * lhi + y * llo; /* |elo| < |ehi| * 2^-25. */ 341#endif 342 return exp_inline(ehi, elo, sign_bias); 343} 344