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       1                 :             : /*-
       2                 :             :  * Copyright (c) 2011 David Schultz <das@FreeBSD.ORG>
       3                 :             :  * All rights reserved.
       4                 :             :  *
       5                 :             :  * Redistribution and use in source and binary forms, with or without
       6                 :             :  * modification, are permitted provided that the following conditions
       7                 :             :  * are met:
       8                 :             :  * 1. Redistributions of source code must retain the above copyright
       9                 :             :  *    notice, this list of conditions and the following disclaimer.
      10                 :             :  * 2. Redistributions in binary form must reproduce the above copyright
      11                 :             :  *    notice, this list of conditions and the following disclaimer in the
      12                 :             :  *    documentation and/or other materials provided with the distribution.
      13                 :             :  *
      14                 :             :  * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
      15                 :             :  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
      16                 :             :  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
      17                 :             :  * ARE DISCLAIMED.  IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
      18                 :             :  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
      19                 :             :  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
      20                 :             :  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
      21                 :             :  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
      22                 :             :  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
      23                 :             :  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
      24                 :             :  * SUCH DAMAGE.
      25                 :             :  */
      26                 :             : 
      27                 :             : #include "cdefs-compat.h"
      28                 :             : //__FBSDID("$FreeBSD: src/lib/msun/src/k_exp.c,v 1.1 2011/10/21 06:27:56 das Exp $");
      29                 :             : 
      30                 :             : #include <openlibm_complex.h>
      31                 :             : #include <openlibm_math.h>
      32                 :             : 
      33                 :             : #include "math_private.h"
      34                 :             : 
      35                 :             : static const u_int32_t k = 1799;                /* constant for reduction */
      36                 :             : static const double kln2 =  1246.97177782734161156;     /* k * ln2 */
      37                 :             : 
      38                 :             : /*
      39                 :             :  * Compute exp(x), scaled to avoid spurious overflow.  An exponent is
      40                 :             :  * returned separately in 'expt'.
      41                 :             :  *
      42                 :             :  * Input:  ln(DBL_MAX) <= x < ln(2 * DBL_MAX / DBL_MIN_DENORM) ~= 1454.91
      43                 :             :  * Output: 2**1023 <= y < 2**1024
      44                 :             :  */
      45                 :             : static double
      46                 :           0 : __frexp_exp(double x, int *expt)
      47                 :             : {
      48                 :             :         double exp_x;
      49                 :             :         u_int32_t hx;
      50                 :             : 
      51                 :             :         /*
      52                 :             :          * We use exp(x) = exp(x - kln2) * 2**k, carefully chosen to
      53                 :             :          * minimize |exp(kln2) - 2**k|.  We also scale the exponent of
      54                 :             :          * exp_x to MAX_EXP so that the result can be multiplied by
      55                 :             :          * a tiny number without losing accuracy due to denormalization.
      56                 :             :          */
      57                 :           0 :         exp_x = exp(x - kln2);
      58                 :           0 :         GET_HIGH_WORD(hx, exp_x);
      59                 :           0 :         *expt = (hx >> 20) - (0x3ff + 1023) + k;
      60                 :           0 :         SET_HIGH_WORD(exp_x, (hx & 0xfffff) | ((0x3ff + 1023) << 20));
      61                 :           0 :         return (exp_x);
      62                 :             : }
      63                 :             : 
      64                 :             : /*
      65                 :             :  * __ldexp_exp(x, expt) and __ldexp_cexp(x, expt) compute exp(x) * 2**expt.
      66                 :             :  * They are intended for large arguments (real part >= ln(DBL_MAX))
      67                 :             :  * where care is needed to avoid overflow.
      68                 :             :  *
      69                 :             :  * The present implementation is narrowly tailored for our hyperbolic and
      70                 :             :  * exponential functions.  We assume expt is small (0 or -1), and the caller
      71                 :             :  * has filtered out very large x, for which overflow would be inevitable.
      72                 :             :  */
      73                 :             : 
      74                 :             : OLM_DLLEXPORT double
      75                 :           0 : __ldexp_exp(double x, int expt)
      76                 :             : {
      77                 :             :         double exp_x, scale;
      78                 :             :         int ex_expt;
      79                 :             : 
      80                 :           0 :         exp_x = __frexp_exp(x, &ex_expt);
      81                 :           0 :         expt += ex_expt;
      82                 :           0 :         INSERT_WORDS(scale, (0x3ff + expt) << 20, 0);
      83                 :           0 :         return (exp_x * scale);
      84                 :             : }
      85                 :             : 
      86                 :             : OLM_DLLEXPORT double complex
      87                 :           0 : __ldexp_cexp(double complex z, int expt)
      88                 :             : {
      89                 :             :         double x, y, exp_x, scale1, scale2;
      90                 :             :         int ex_expt, half_expt;
      91                 :             : 
      92                 :           0 :         x = creal(z);
      93                 :           0 :         y = cimag(z);
      94                 :           0 :         exp_x = __frexp_exp(x, &ex_expt);
      95                 :           0 :         expt += ex_expt;
      96                 :             : 
      97                 :             :         /*
      98                 :             :          * Arrange so that scale1 * scale2 == 2**expt.  We use this to
      99                 :             :          * compensate for scalbn being horrendously slow.
     100                 :             :          */
     101                 :           0 :         half_expt = expt / 2;
     102                 :           0 :         INSERT_WORDS(scale1, (0x3ff + half_expt) << 20, 0);
     103                 :           0 :         half_expt = expt - half_expt;
     104                 :           0 :         INSERT_WORDS(scale2, (0x3ff + half_expt) << 20, 0);
     105                 :             : 
     106                 :           0 :         return (CMPLX(cos(y) * exp_x * scale1 * scale2,
     107                 :             :             sin(y) * exp_x * scale1 * scale2));
     108                 :             : }