232 lines
6.8 KiB
C
232 lines
6.8 KiB
C
/* Single-precision pow function.
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Copyright (c) 2017-2018 ARM Ltd. All rights reserved.
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Redistribution and use in source and binary forms, with or without
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modification, are permitted provided that the following conditions
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are met:
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1. Redistributions of source code must retain the above copyright
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notice, this list of conditions and the following disclaimer.
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2. Redistributions in binary form must reproduce the above copyright
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notice, this list of conditions and the following disclaimer in the
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documentation and/or other materials provided with the distribution.
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3. The name of the company may not be used to endorse or promote
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products derived from this software without specific prior written
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permission.
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THIS SOFTWARE IS PROVIDED BY ARM LTD ``AS IS AND ANY EXPRESS OR IMPLIED
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WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
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MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
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IN NO EVENT SHALL ARM LTD BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
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TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
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PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
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LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
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NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
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SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */
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#include "fdlibm.h"
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#if !__OBSOLETE_MATH
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#include <math.h>
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#include <stdint.h>
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#include "math_config.h"
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/*
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POWF_LOG2_POLY_ORDER = 5
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EXP2F_TABLE_BITS = 5
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ULP error: 0.82 (~ 0.5 + relerr*2^24)
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relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2)
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relerr_log2: 1.83 * 2^-33 (Relative error of logx.)
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relerr_exp2: 1.69 * 2^-34 (Relative error of exp2(ylogx).)
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*/
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#define N (1 << POWF_LOG2_TABLE_BITS)
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#define T __powf_log2_data.tab
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#define A __powf_log2_data.poly
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#define OFF 0x3f330000
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/* Subnormal input is normalized so ix has negative biased exponent.
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Output is multiplied by N (POWF_SCALE) if TOINT_INTRINICS is set. */
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static inline double_t
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log2_inline (uint32_t ix)
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{
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/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
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double_t z, r, r2, r4, p, q, y, y0, invc, logc;
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uint32_t iz, top, tmp;
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int k, i;
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/* x = 2^k z; where z is in range [OFF,2*OFF] and exact.
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The range is split into N subintervals.
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The ith subinterval contains z and c is near its center. */
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tmp = ix - OFF;
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i = (tmp >> (23 - POWF_LOG2_TABLE_BITS)) % N;
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top = tmp & 0xff800000;
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iz = ix - top;
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k = (int32_t) top >> (23 - POWF_SCALE_BITS); /* arithmetic shift */
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invc = T[i].invc;
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logc = T[i].logc;
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z = (double_t) asfloat (iz);
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/* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */
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r = z * invc - 1;
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y0 = logc + (double_t) k;
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/* Pipelined polynomial evaluation to approximate log1p(r)/ln2. */
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r2 = r * r;
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y = A[0] * r + A[1];
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p = A[2] * r + A[3];
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r4 = r2 * r2;
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q = A[4] * r + y0;
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q = p * r2 + q;
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y = y * r4 + q;
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return y;
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}
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#undef N
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#undef T
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#define N (1 << EXP2F_TABLE_BITS)
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#define T __exp2f_data.tab
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#define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11))
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/* The output of log2 and thus the input of exp2 is either scaled by N
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(in case of fast toint intrinsics) or not. The unscaled xd must be
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in [-1021,1023], sign_bias sets the sign of the result. */
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static inline double_t
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exp2_inline (double_t xd, uint32_t sign_bias)
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{
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uint64_t ki, ski, t;
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/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
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double_t kd, z, r, r2, y, s;
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#if TOINT_INTRINSICS
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# define C __exp2f_data.poly_scaled
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/* N*x = k + r with r in [-1/2, 1/2] */
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kd = roundtoint (xd); /* k */
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ki = converttoint (xd);
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#else
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# define C __exp2f_data.poly
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# define SHIFT __exp2f_data.shift_scaled
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/* x = k/N + r with r in [-1/(2N), 1/(2N)] */
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kd = (double) (xd + SHIFT); /* Rounding to double precision is required. */
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ki = asuint64 (kd);
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kd -= SHIFT; /* k/N */
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#endif
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r = xd - kd;
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/* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
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t = T[ki % N];
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ski = ki + sign_bias;
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t += ski << (52 - EXP2F_TABLE_BITS);
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s = asdouble (t);
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z = C[0] * r + C[1];
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r2 = r * r;
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y = C[2] * r + 1;
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y = z * r2 + y;
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y = y * s;
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return y;
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}
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/* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is
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the bit representation of a non-zero finite floating-point value. */
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static inline int
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checkint (uint32_t iy)
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{
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int e = iy >> 23 & 0xff;
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if (e < 0x7f)
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return 0;
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if (e > 0x7f + 23)
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return 2;
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if (iy & ((1 << (0x7f + 23 - e)) - 1))
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return 0;
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if (iy & (1 << (0x7f + 23 - e)))
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return 1;
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return 2;
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}
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static inline int
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zeroinfnan (uint32_t ix)
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{
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return 2 * ix - 1 >= 2u * 0x7f800000 - 1;
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}
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float
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powf (float x, float y)
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{
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uint32_t sign_bias = 0;
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uint32_t ix, iy;
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ix = asuint (x);
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iy = asuint (y);
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if (__builtin_expect (ix - 0x00800000 >= 0x7f800000 - 0x00800000
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|| zeroinfnan (iy),
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0))
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{
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/* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan). */
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if (__builtin_expect (zeroinfnan (iy), 0))
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{
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if (2 * iy == 0)
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return issignalingf_inline (x) ? x + y : 1.0f;
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if (ix == 0x3f800000)
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return issignalingf_inline (y) ? x + y : 1.0f;
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if (2 * ix > 2u * 0x7f800000 || 2 * iy > 2u * 0x7f800000)
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return x + y;
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if (2 * ix == 2 * 0x3f800000)
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return 1.0f;
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if ((2 * ix < 2 * 0x3f800000) == !(iy & 0x80000000))
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return 0.0f; /* |x|<1 && y==inf or |x|>1 && y==-inf. */
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return y * y;
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}
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if (__builtin_expect (zeroinfnan (ix), 0))
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{
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float_t x2 = x * x;
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if (ix & 0x80000000 && checkint (iy) == 1)
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{
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x2 = -x2;
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sign_bias = 1;
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}
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#if WANT_ERRNO
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if (2 * ix == 0 && iy & 0x80000000)
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return __math_divzerof (sign_bias);
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#endif
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return iy & 0x80000000 ? 1 / x2 : x2;
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}
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/* x and y are non-zero finite. */
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if (ix & 0x80000000)
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{
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/* Finite x < 0. */
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int yint = checkint (iy);
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if (yint == 0)
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return __math_invalidf (x);
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if (yint == 1)
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sign_bias = SIGN_BIAS;
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ix &= 0x7fffffff;
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}
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if (ix < 0x00800000)
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{
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/* Normalize subnormal x so exponent becomes negative. */
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ix = asuint (x * 0x1p23f);
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ix &= 0x7fffffff;
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ix -= 23 << 23;
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}
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}
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double_t logx = log2_inline (ix);
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double_t ylogx = y * logx; /* Note: cannot overflow, y is single prec. */
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if (__builtin_expect ((asuint64 (ylogx) >> 47 & 0xffff)
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>= asuint64 (126.0 * POWF_SCALE) >> 47,
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0))
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{
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/* |y*log(x)| >= 126. */
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if (ylogx > 0x1.fffffffd1d571p+6 * POWF_SCALE)
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return __math_oflowf (sign_bias);
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if (ylogx <= -150.0 * POWF_SCALE)
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return __math_uflowf (sign_bias);
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#if WANT_ERRNO_UFLOW
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if (ylogx < -149.0 * POWF_SCALE)
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return __math_may_uflowf (sign_bias);
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#endif
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}
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return (float) exp2_inline (ylogx, sign_bias);
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}
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#endif /* !__OBSOLETE_MATH */
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