[444] | 1 | /* Single-precision pow function. |
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| 2 | Copyright (c) 2017 ARM Ltd. All rights reserved. |
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| 3 | |
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| 4 | Redistribution and use in source and binary forms, with or without |
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| 5 | modification, are permitted provided that the following conditions |
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| 6 | are met: |
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| 7 | 1. Redistributions of source code must retain the above copyright |
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| 8 | notice, this list of conditions and the following disclaimer. |
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| 9 | 2. Redistributions in binary form must reproduce the above copyright |
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| 10 | notice, this list of conditions and the following disclaimer in the |
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| 11 | documentation and/or other materials provided with the distribution. |
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| 12 | 3. The name of the company may not be used to endorse or promote |
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| 13 | products derived from this software without specific prior written |
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| 14 | permission. |
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| 15 | |
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| 16 | THIS SOFTWARE IS PROVIDED BY ARM LTD ``AS IS AND ANY EXPRESS OR IMPLIED |
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| 17 | WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF |
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| 18 | MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. |
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| 19 | IN NO EVENT SHALL ARM LTD BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, |
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| 20 | SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED |
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| 21 | TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR |
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| 22 | PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF |
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| 23 | LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING |
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| 24 | NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS |
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| 25 | SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */ |
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| 26 | |
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| 27 | #include "fdlibm.h" |
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| 28 | #if !__OBSOLETE_MATH |
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| 29 | |
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| 30 | #include <math.h> |
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| 31 | #include <stdint.h> |
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| 32 | #include "math_config.h" |
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| 33 | |
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| 34 | /* |
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| 35 | POWF_LOG2_POLY_ORDER = 5 |
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| 36 | EXP2F_TABLE_BITS = 5 |
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| 37 | |
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| 38 | ULP error: 0.82 (~ 0.5 + relerr*2^24) |
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| 39 | relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2) |
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| 40 | relerr_log2: 1.83 * 2^-33 (Relative error of logx.) |
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| 41 | relerr_exp2: 1.69 * 2^-34 (Relative error of exp2(ylogx).) |
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| 42 | */ |
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| 43 | |
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| 44 | #define N (1 << POWF_LOG2_TABLE_BITS) |
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| 45 | #define T __powf_log2_data.tab |
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| 46 | #define A __powf_log2_data.poly |
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| 47 | #define OFF 0x3f330000 |
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| 48 | |
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| 49 | /* Subnormal input is normalized so ix has negative biased exponent. |
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| 50 | Output is multiplied by N (POWF_SCALE) if TOINT_INTRINICS is set. */ |
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| 51 | static inline double_t |
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| 52 | log2_inline (uint32_t ix) |
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| 53 | { |
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| 54 | /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ |
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| 55 | double_t z, r, r2, r4, p, q, y, y0, invc, logc; |
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| 56 | uint32_t iz, top, tmp; |
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| 57 | int k, i; |
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| 58 | |
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| 59 | /* x = 2^k z; where z is in range [OFF,2*OFF] and exact. |
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| 60 | The range is split into N subintervals. |
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| 61 | The ith subinterval contains z and c is near its center. */ |
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| 62 | tmp = ix - OFF; |
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| 63 | i = (tmp >> (23 - POWF_LOG2_TABLE_BITS)) % N; |
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| 64 | top = tmp & 0xff800000; |
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| 65 | iz = ix - top; |
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| 66 | k = (int32_t) top >> (23 - POWF_SCALE_BITS); /* arithmetic shift */ |
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| 67 | invc = T[i].invc; |
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| 68 | logc = T[i].logc; |
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| 69 | z = (double_t) asfloat (iz); |
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| 70 | |
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| 71 | /* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */ |
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| 72 | r = z * invc - 1; |
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| 73 | y0 = logc + (double_t) k; |
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| 74 | |
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| 75 | /* Pipelined polynomial evaluation to approximate log1p(r)/ln2. */ |
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| 76 | r2 = r * r; |
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| 77 | y = A[0] * r + A[1]; |
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| 78 | p = A[2] * r + A[3]; |
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| 79 | r4 = r2 * r2; |
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| 80 | q = A[4] * r + y0; |
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| 81 | q = p * r2 + q; |
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| 82 | y = y * r4 + q; |
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| 83 | return y; |
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| 84 | } |
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| 85 | |
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| 86 | #undef N |
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| 87 | #undef T |
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| 88 | #define N (1 << EXP2F_TABLE_BITS) |
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| 89 | #define T __exp2f_data.tab |
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| 90 | #define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11)) |
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| 91 | |
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| 92 | /* The output of log2 and thus the input of exp2 is either scaled by N |
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| 93 | (in case of fast toint intrinsics) or not. The unscaled xd must be |
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| 94 | in [-1021,1023], sign_bias sets the sign of the result. */ |
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| 95 | static inline double_t |
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| 96 | exp2_inline (double_t xd, unsigned long sign_bias) |
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| 97 | { |
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| 98 | uint64_t ki, ski, t; |
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| 99 | /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ |
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| 100 | double_t kd, z, r, r2, y, s; |
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| 101 | |
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| 102 | #if TOINT_INTRINSICS |
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| 103 | # define C __exp2f_data.poly_scaled |
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| 104 | /* N*x = k + r with r in [-1/2, 1/2] */ |
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| 105 | kd = roundtoint (xd); /* k */ |
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| 106 | ki = converttoint (xd); |
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| 107 | #else |
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| 108 | # define C __exp2f_data.poly |
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| 109 | # define SHIFT __exp2f_data.shift_scaled |
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| 110 | /* x = k/N + r with r in [-1/(2N), 1/(2N)] */ |
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| 111 | kd = (double) (xd + SHIFT); /* Rounding to double precision is required. */ |
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| 112 | ki = asuint64 (kd); |
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| 113 | kd -= SHIFT; /* k/N */ |
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| 114 | #endif |
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| 115 | r = xd - kd; |
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| 116 | |
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| 117 | /* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */ |
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| 118 | t = T[ki % N]; |
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| 119 | ski = ki + sign_bias; |
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| 120 | t += ski << (52 - EXP2F_TABLE_BITS); |
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| 121 | s = asdouble (t); |
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| 122 | z = C[0] * r + C[1]; |
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| 123 | r2 = r * r; |
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| 124 | y = C[2] * r + 1; |
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| 125 | y = z * r2 + y; |
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| 126 | y = y * s; |
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| 127 | return y; |
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| 128 | } |
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| 129 | |
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| 130 | /* Returns 0 if not int, 1 if odd int, 2 if even int. */ |
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| 131 | static inline int |
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| 132 | checkint (uint32_t iy) |
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| 133 | { |
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| 134 | int e = iy >> 23 & 0xff; |
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| 135 | if (e < 0x7f) |
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| 136 | return 0; |
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| 137 | if (e > 0x7f + 23) |
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| 138 | return 2; |
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| 139 | if (iy & ((1 << (0x7f + 23 - e)) - 1)) |
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| 140 | return 0; |
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| 141 | if (iy & (1 << (0x7f + 23 - e))) |
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| 142 | return 1; |
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| 143 | return 2; |
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| 144 | } |
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| 145 | |
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| 146 | static inline int |
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| 147 | zeroinfnan (uint32_t ix) |
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| 148 | { |
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| 149 | return 2 * ix - 1 >= 2u * 0x7f800000 - 1; |
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| 150 | } |
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| 151 | |
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| 152 | float |
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| 153 | powf (float x, float y) |
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| 154 | { |
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| 155 | unsigned long sign_bias = 0; |
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| 156 | uint32_t ix, iy; |
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| 157 | |
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| 158 | ix = asuint (x); |
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| 159 | iy = asuint (y); |
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| 160 | if (__builtin_expect (ix - 0x00800000 >= 0x7f800000 - 0x00800000 |
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| 161 | || zeroinfnan (iy), |
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| 162 | 0)) |
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| 163 | { |
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| 164 | /* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan). */ |
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| 165 | if (__builtin_expect (zeroinfnan (iy), 0)) |
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| 166 | { |
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| 167 | if (2 * iy == 0) |
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| 168 | return issignalingf_inline (x) ? x + y : 1.0f; |
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| 169 | if (ix == 0x3f800000) |
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| 170 | return issignalingf_inline (y) ? x + y : 1.0f; |
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| 171 | if (2 * ix > 2u * 0x7f800000 || 2 * iy > 2u * 0x7f800000) |
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| 172 | return x + y; |
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| 173 | if (2 * ix == 2 * 0x3f800000) |
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| 174 | return 1.0f; |
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| 175 | if ((2 * ix < 2 * 0x3f800000) == !(iy & 0x80000000)) |
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| 176 | return 0.0f; /* |x|<1 && y==inf or |x|>1 && y==-inf. */ |
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| 177 | return y * y; |
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| 178 | } |
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| 179 | if (__builtin_expect (zeroinfnan (ix), 0)) |
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| 180 | { |
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| 181 | float_t x2 = x * x; |
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| 182 | if (ix & 0x80000000 && checkint (iy) == 1) |
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| 183 | { |
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| 184 | x2 = -x2; |
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| 185 | sign_bias = 1; |
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| 186 | } |
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| 187 | #if WANT_ERRNO |
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| 188 | if (2 * ix == 0 && iy & 0x80000000) |
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| 189 | return __math_divzerof (sign_bias); |
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| 190 | #endif |
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| 191 | return iy & 0x80000000 ? 1 / x2 : x2; |
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| 192 | } |
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| 193 | /* x and y are non-zero finite. */ |
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| 194 | if (ix & 0x80000000) |
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| 195 | { |
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| 196 | /* Finite x < 0. */ |
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| 197 | int yint = checkint (iy); |
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| 198 | if (yint == 0) |
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| 199 | return __math_invalidf (x); |
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| 200 | if (yint == 1) |
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| 201 | sign_bias = SIGN_BIAS; |
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| 202 | ix &= 0x7fffffff; |
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| 203 | } |
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| 204 | if (ix < 0x00800000) |
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| 205 | { |
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| 206 | /* Normalize subnormal x so exponent becomes negative. */ |
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| 207 | ix = asuint (x * 0x1p23f); |
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| 208 | ix &= 0x7fffffff; |
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| 209 | ix -= 23 << 23; |
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| 210 | } |
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| 211 | } |
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| 212 | double_t logx = log2_inline (ix); |
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| 213 | double_t ylogx = y * logx; /* Note: cannot overflow, y is single prec. */ |
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| 214 | if (__builtin_expect ((asuint64 (ylogx) >> 47 & 0xffff) |
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| 215 | >= asuint64 (126.0 * POWF_SCALE) >> 47, |
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| 216 | 0)) |
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| 217 | { |
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| 218 | /* |y*log(x)| >= 126. */ |
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| 219 | if (ylogx > 0x1.fffffffd1d571p+6 * POWF_SCALE) |
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| 220 | return __math_oflowf (sign_bias); |
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| 221 | if (ylogx <= -150.0 * POWF_SCALE) |
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| 222 | return __math_uflowf (sign_bias); |
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| 223 | #if WANT_ERRNO_UFLOW |
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| 224 | if (ylogx < -149.0 * POWF_SCALE) |
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| 225 | return __math_may_uflowf (sign_bias); |
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| 226 | #endif |
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| 227 | } |
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| 228 | return (float) exp2_inline (ylogx, sign_bias); |
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| 229 | } |
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| 230 | #endif /* !__OBSOLETE_MATH */ |
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