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1/*2 * Double-precision 2^x function.3 *4 * Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.5 * See https://llvm.org/LICENSE.txt for license information.6 * SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception7 */8 9#include <float.h>10#include <math.h>11#include <stdint.h>12#include "math_config.h"13 14#define N (1 << EXP_TABLE_BITS)15#define Shift __exp_data.exp2_shift16#define T __exp_data.tab17#define C1 __exp_data.exp2_poly[0]18#define C2 __exp_data.exp2_poly[1]19#define C3 __exp_data.exp2_poly[2]20#define C4 __exp_data.exp2_poly[3]21#define C5 __exp_data.exp2_poly[4]22#define C6 __exp_data.exp2_poly[5]23 24/* Handle cases that may overflow or underflow when computing the result that25 is scale*(1+TMP) without intermediate rounding. The bit representation of26 scale is in SBITS, however it has a computed exponent that may have27 overflown into the sign bit so that needs to be adjusted before using it as28 a double. (int32_t)KI is the k used in the argument reduction and exponent29 adjustment of scale, positive k here means the result may overflow and30 negative k means the result may underflow. */31static inline double32specialcase (double_t tmp, uint64_t sbits, uint64_t ki)33{34 double_t scale, y;35 36 if ((ki & 0x80000000) == 0)37 {38 /* k > 0, the exponent of scale might have overflowed by 1. */39 sbits -= 1ull << 52;40 scale = asdouble (sbits);41 y = 2 * (scale + scale * tmp);42 return check_oflow (eval_as_double (y));43 }44 /* k < 0, need special care in the subnormal range. */45 sbits += 1022ull << 52;46 scale = asdouble (sbits);47 y = scale + scale * tmp;48 if (y < 1.0)49 {50 /* Round y to the right precision before scaling it into the subnormal51 range to avoid double rounding that can cause 0.5+E/2 ulp error where52 E is the worst-case ulp error outside the subnormal range. So this53 is only useful if the goal is better than 1 ulp worst-case error. */54 double_t hi, lo;55 lo = scale - y + scale * tmp;56 hi = 1.0 + y;57 lo = 1.0 - hi + y + lo;58 y = eval_as_double (hi + lo) - 1.0;59 /* Avoid -0.0 with downward rounding. */60 if (WANT_ROUNDING && y == 0.0)61 y = 0.0;62 /* The underflow exception needs to be signaled explicitly. */63 force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);64 }65 y = 0x1p-1022 * y;66 return check_uflow (eval_as_double (y));67}68 69/* Top 12 bits of a double (sign and exponent bits). */70static inline uint32_t71top12 (double x)72{73 return asuint64 (x) >> 52;74}75 76double77exp2 (double x)78{79 uint32_t abstop;80 uint64_t ki, idx, top, sbits;81 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */82 double_t kd, r, r2, scale, tail, tmp;83 84 abstop = top12 (x) & 0x7ff;85 if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))86 {87 if (abstop - top12 (0x1p-54) >= 0x80000000)88 /* Avoid spurious underflow for tiny x. */89 /* Note: 0 is common input. */90 return WANT_ROUNDING ? 1.0 + x : 1.0;91 if (abstop >= top12 (1024.0))92 {93 if (asuint64 (x) == asuint64 (-INFINITY))94 return 0.0;95 if (abstop >= top12 (INFINITY))96 return 1.0 + x;97 if (!(asuint64 (x) >> 63))98 return __math_oflow (0);99 else if (asuint64 (x) >= asuint64 (-1075.0))100 return __math_uflow (0);101 }102 if (2 * asuint64 (x) > 2 * asuint64 (928.0))103 /* Large x is special cased below. */104 abstop = 0;105 }106 107 /* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)]. */108 /* x = k/N + r, with int k and r in [-1/2N, 1/2N]. */109 kd = eval_as_double (x + Shift);110 ki = asuint64 (kd); /* k. */111 kd -= Shift; /* k/N for int k. */112 r = x - kd;113 /* 2^(k/N) ~= scale * (1 + tail). */114 idx = 2 * (ki % N);115 top = ki << (52 - EXP_TABLE_BITS);116 tail = asdouble (T[idx]);117 /* This is only a valid scale when -1023*N < k < 1024*N. */118 sbits = T[idx + 1] + top;119 /* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1). */120 /* Evaluation is optimized assuming superscalar pipelined execution. */121 r2 = r * r;122 /* Without fma the worst case error is 0.5/N ulp larger. */123 /* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp. */124#if EXP2_POLY_ORDER == 4125 tmp = tail + r * C1 + r2 * C2 + r * r2 * (C3 + r * C4);126#elif EXP2_POLY_ORDER == 5127 tmp = tail + r * C1 + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);128#elif EXP2_POLY_ORDER == 6129 tmp = tail + r * C1 + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6);130#endif131 if (unlikely (abstop == 0))132 return specialcase (tmp, sbits, ki);133 scale = asdouble (sbits);134 /* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-928, so there135 is no spurious underflow here even without fma. */136 return eval_as_double (scale + scale * tmp);137}138#if USE_GLIBC_ABI139strong_alias (exp2, __exp2_finite)140hidden_alias (exp2, __ieee754_exp2)141# if LDBL_MANT_DIG == 53142long double exp2l (long double x) { return exp2 (x); }143# endif144#endif145