src/math/pow.c

   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 /*
  15 Worst-case error: 0.54 ULP (~= ulperr_exp + 1024*Ln2*relerr_log*2^53)
  16 relerr_log: 1.3 * 2^-68 (Relative error of log, 1.5 * 2^-68 without fma)
  17 ulperr_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).  */
  28 static 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.  */
  36 static 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.  */
 124 static 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.  */
 166 static 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.  */
 235 static 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.  */
 250 static inline int zeroinfnan(uint64_t i)
 251 {
 252         return 2 * i - 1 >= 2 * asuint64(INFINITY) - 1;
 253 }
 254
 255 double 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 }