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 "math_config.h"
11
12 /*
13 Worst-case error: 0.54 ULP (~= ulperr_exp + 1024*Ln2*relerr_log*2^53)
14 relerr_log: 1.3 * 2^-68 (Relative error of log, 1.5 * 2^-68 without fma)
15 ulperr_exp: 0.509 ULP (ULP error of exp, 0.511 ULP without fma)
16 */
17
18 #define T __pow_log_data.tab
19 #define A __pow_log_data.poly
20 #define Ln2hi __pow_log_data.ln2hi
21 #define Ln2lo __pow_log_data.ln2lo
22 #define N (1 << POW_LOG_TABLE_BITS)
23 #define OFF 0x3fe6955500000000
24
25 /* Top 12 bits of a double (sign and exponent bits). */
26 static inline uint32_t
top12(double x)27 top12 (double x)
28 {
29 return asuint64 (x) >> 52;
30 }
31
32 /* Compute y+TAIL = log(x) where the rounded result is y and TAIL has about
33 additional 15 bits precision. IX is the bit representation of x, but
34 normalized in the subnormal range using the sign bit for the exponent. */
35 static inline double_t
log_inline(uint64_t ix,double_t * tail)36 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 HAVE_FAST_FMA
61 r = 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 HAVE_FAST_FMA
84 hi = t2 + ar2;
85 lo3 = 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 #if POW_LOG_POLY_ORDER == 8
96 p = (ar3
97 * (A[1] + r * A[2] + ar2 * (A[3] + r * A[4] + ar2 * (A[5] + r * A[6]))));
98 #endif
99 lo = lo1 + lo2 + lo3 + lo4 + p;
100 y = hi + lo;
101 *tail = hi - y + lo;
102 return y;
103 }
104
105 #undef N
106 #undef T
107 #define N (1 << EXP_TABLE_BITS)
108 #define InvLn2N __exp_data.invln2N
109 #define NegLn2hiN __exp_data.negln2hiN
110 #define NegLn2loN __exp_data.negln2loN
111 #define Shift __exp_data.shift
112 #define T __exp_data.tab
113 #define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
114 #define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
115 #define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
116 #define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
117 #define C6 __exp_data.poly[9 - EXP_POLY_ORDER]
118
119 /* Handle cases that may overflow or underflow when computing the result that
120 is scale*(1+TMP) without intermediate rounding. The bit representation of
121 scale is in SBITS, however it has a computed exponent that may have
122 overflown into the sign bit so that needs to be adjusted before using it as
123 a double. (int32_t)KI is the k used in the argument reduction and exponent
124 adjustment of scale, positive k here means the result may overflow and
125 negative k means the result may underflow. */
126 static inline double
specialcase(double_t tmp,uint64_t sbits,uint64_t ki)127 specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
128 {
129 double_t scale, y;
130
131 if ((ki & 0x80000000) == 0)
132 {
133 /* k > 0, the exponent of scale might have overflowed by <= 460. */
134 sbits -= 1009ull << 52;
135 scale = asdouble (sbits);
136 y = 0x1p1009 * (scale + scale * tmp);
137 return check_oflow (eval_as_double (y));
138 }
139 /* k < 0, need special care in the subnormal range. */
140 sbits += 1022ull << 52;
141 /* Note: sbits is signed scale. */
142 scale = asdouble (sbits);
143 y = scale + scale * tmp;
144 if (fabs (y) < 1.0)
145 {
146 /* Round y to the right precision before scaling it into the subnormal
147 range to avoid double rounding that can cause 0.5+E/2 ulp error where
148 E is the worst-case ulp error outside the subnormal range. So this
149 is only useful if the goal is better than 1 ulp worst-case error. */
150 double_t hi, lo, one = 1.0;
151 if (y < 0.0)
152 one = -1.0;
153 lo = scale - y + scale * tmp;
154 hi = one + y;
155 lo = one - hi + y + lo;
156 y = eval_as_double (hi + lo) - one;
157 /* Fix the sign of 0. */
158 if (y == 0.0)
159 y = asdouble (sbits & 0x8000000000000000);
160 /* The underflow exception needs to be signaled explicitly. */
161 force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
162 }
163 y = 0x1p-1022 * y;
164 return check_uflow (eval_as_double (y));
165 }
166
167 #define SIGN_BIAS (0x800 << EXP_TABLE_BITS)
168
169 /* Computes sign*exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|.
170 The sign_bias argument is SIGN_BIAS or 0 and sets the sign to -1 or 1. */
171 static inline double
exp_inline(double_t x,double_t xtail,uint32_t sign_bias)172 exp_inline (double_t x, double_t xtail, uint32_t sign_bias)
173 {
174 uint32_t abstop;
175 uint64_t ki, idx, top, sbits;
176 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
177 double_t kd, z, r, r2, scale, tail, tmp;
178
179 abstop = top12 (x) & 0x7ff;
180 if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))
181 {
182 if (abstop - top12 (0x1p-54) >= 0x80000000)
183 {
184 /* Avoid spurious underflow for tiny x. */
185 /* Note: 0 is common input. */
186 double_t one = WANT_ROUNDING ? 1.0 + x : 1.0;
187 return sign_bias ? -one : one;
188 }
189 if (abstop >= top12 (1024.0))
190 {
191 /* Note: inf and nan are already handled. */
192 if (asuint64 (x) >> 63)
193 return __math_uflow (sign_bias);
194 else
195 return __math_oflow (sign_bias);
196 }
197 /* Large x is special cased below. */
198 abstop = 0;
199 }
200
201 /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */
202 /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */
203 z = InvLn2N * x;
204 #if TOINT_INTRINSICS
205 kd = roundtoint (z);
206 ki = converttoint (z);
207 #elif EXP_USE_TOINT_NARROW
208 /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */
209 kd = eval_as_double (z + Shift);
210 ki = asuint64 (kd) >> 16;
211 kd = (double_t) (int32_t) ki;
212 #else
213 /* z - kd is in [-1, 1] in non-nearest rounding modes. */
214 kd = eval_as_double (z + Shift);
215 ki = asuint64 (kd);
216 kd -= Shift;
217 #endif
218 r = x + kd * NegLn2hiN + kd * NegLn2loN;
219 /* The code assumes 2^-200 < |xtail| < 2^-8/N. */
220 r += xtail;
221 /* 2^(k/N) ~= scale * (1 + tail). */
222 idx = 2 * (ki % N);
223 top = (ki + sign_bias) << (52 - EXP_TABLE_BITS);
224 tail = asdouble (T[idx]);
225 /* This is only a valid scale when -1023*N < k < 1024*N. */
226 sbits = T[idx + 1] + top;
227 /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */
228 /* Evaluation is optimized assuming superscalar pipelined execution. */
229 r2 = r * r;
230 /* Without fma the worst case error is 0.25/N ulp larger. */
231 /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */
232 #if EXP_POLY_ORDER == 4
233 tmp = tail + r + r2 * C2 + r * r2 * (C3 + r * C4);
234 #elif EXP_POLY_ORDER == 5
235 tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
236 #elif EXP_POLY_ORDER == 6
237 tmp = tail + r + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6);
238 #endif
239 if (unlikely (abstop == 0))
240 return specialcase (tmp, sbits, ki);
241 scale = asdouble (sbits);
242 /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
243 is no spurious underflow here even without fma. */
244 return eval_as_double (scale + scale * tmp);
245 }
246
247 /* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is
248 the bit representation of a non-zero finite floating-point value. */
249 static inline int
checkint(uint64_t iy)250 checkint (uint64_t iy)
251 {
252 int e = iy >> 52 & 0x7ff;
253 if (e < 0x3ff)
254 return 0;
255 if (e > 0x3ff + 52)
256 return 2;
257 if (iy & ((1ULL << (0x3ff + 52 - e)) - 1))
258 return 0;
259 if (iy & (1ULL << (0x3ff + 52 - e)))
260 return 1;
261 return 2;
262 }
263
264 /* Returns 1 if input is the bit representation of 0, infinity or nan. */
265 static inline int
zeroinfnan(uint64_t i)266 zeroinfnan (uint64_t i)
267 {
268 return 2 * i - 1 >= 2 * asuint64 (INFINITY) - 1;
269 }
270
271 double
pow(double x,double y)272 pow (double x, double y)
273 {
274 uint32_t sign_bias = 0;
275 uint64_t ix, iy;
276 uint32_t topx, topy;
277
278 ix = asuint64 (x);
279 iy = asuint64 (y);
280 topx = top12 (x);
281 topy = top12 (y);
282 if (unlikely (topx - 0x001 >= 0x7ff - 0x001
283 || (topy & 0x7ff) - 0x3be >= 0x43e - 0x3be))
284 {
285 /* Note: if |y| > 1075 * ln2 * 2^53 ~= 0x1.749p62 then pow(x,y) = inf/0
286 and if |y| < 2^-54 / 1075 ~= 0x1.e7b6p-65 then pow(x,y) = +-1. */
287 /* Special cases: (x < 0x1p-126 or inf or nan) or
288 (|y| < 0x1p-65 or |y| >= 0x1p63 or nan). */
289 if (unlikely (zeroinfnan (iy)))
290 {
291 if (2 * iy == 0)
292 return issignaling_inline (x) ? x + y : 1.0;
293 if (ix == asuint64 (1.0))
294 return issignaling_inline (y) ? x + y : 1.0;
295 if (2 * ix > 2 * asuint64 (INFINITY)
296 || 2 * iy > 2 * asuint64 (INFINITY))
297 return x + y;
298 if (2 * ix == 2 * asuint64 (1.0))
299 return 1.0;
300 if ((2 * ix < 2 * asuint64 (1.0)) == !(iy >> 63))
301 return 0.0; /* |x|<1 && y==inf or |x|>1 && y==-inf. */
302 return y * y;
303 }
304 if (unlikely (zeroinfnan (ix)))
305 {
306 double_t x2 = x * x;
307 if (ix >> 63 && checkint (iy) == 1)
308 {
309 x2 = -x2;
310 sign_bias = 1;
311 }
312 if (WANT_ERRNO && 2 * ix == 0 && iy >> 63)
313 return __math_divzero (sign_bias);
314 /* Without the barrier some versions of clang hoist the 1/x2 and
315 thus division by zero exception can be signaled spuriously. */
316 return iy >> 63 ? opt_barrier_double (1 / x2) : x2;
317 }
318 /* Here x and y are non-zero finite. */
319 if (ix >> 63)
320 {
321 /* Finite x < 0. */
322 int yint = checkint (iy);
323 if (yint == 0)
324 return __math_invalid (x);
325 if (yint == 1)
326 sign_bias = SIGN_BIAS;
327 ix &= 0x7fffffffffffffff;
328 topx &= 0x7ff;
329 }
330 if ((topy & 0x7ff) - 0x3be >= 0x43e - 0x3be)
331 {
332 /* Note: sign_bias == 0 here because y is not odd. */
333 if (ix == asuint64 (1.0))
334 return 1.0;
335 if ((topy & 0x7ff) < 0x3be)
336 {
337 /* |y| < 2^-65, x^y ~= 1 + y*log(x). */
338 if (WANT_ROUNDING)
339 return ix > asuint64 (1.0) ? 1.0 + y : 1.0 - y;
340 else
341 return 1.0;
342 }
343 return (ix > asuint64 (1.0)) == (topy < 0x800) ? __math_oflow (0)
344 : __math_uflow (0);
345 }
346 if (topx == 0)
347 {
348 /* Normalize subnormal x so exponent becomes negative. */
349 ix = asuint64 (x * 0x1p52);
350 ix &= 0x7fffffffffffffff;
351 ix -= 52ULL << 52;
352 }
353 }
354
355 double_t lo;
356 double_t hi = log_inline (ix, &lo);
357 double_t ehi, elo;
358 #if HAVE_FAST_FMA
359 ehi = y * hi;
360 elo = y * lo + fma (y, hi, -ehi);
361 #else
362 double_t yhi = asdouble (iy & -1ULL << 27);
363 double_t ylo = y - yhi;
364 double_t lhi = asdouble (asuint64 (hi) & -1ULL << 27);
365 double_t llo = hi - lhi + lo;
366 ehi = yhi * lhi;
367 elo = ylo * lhi + y * llo; /* |elo| < |ehi| * 2^-25. */
368 #endif
369 return exp_inline (ehi, elo, sign_bias);
370 }
371 #if USE_GLIBC_ABI
372 strong_alias (pow, __pow_finite)
373 hidden_alias (pow, __ieee754_pow)
374 #endif
375