1 | /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_expm1l.c */
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2 | /*
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3 | * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
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4 | *
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5 | * Permission to use, copy, modify, and distribute this software for any
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6 | * purpose with or without fee is hereby granted, provided that the above
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7 | * copyright notice and this permission notice appear in all copies.
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8 | *
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9 | * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
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10 | * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
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11 | * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
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12 | * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
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13 | * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
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14 | * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
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15 | * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
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16 | */
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17 | /*
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18 | * Exponential function, minus 1
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19 | * Long double precision
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20 | *
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21 | *
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22 | * SYNOPSIS:
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23 | *
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24 | * long double x, y, expm1l();
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25 | *
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26 | * y = expm1l( x );
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27 | *
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28 | *
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29 | * DESCRIPTION:
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30 | *
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31 | * Returns e (2.71828...) raised to the x power, minus 1.
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32 | *
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33 | * Range reduction is accomplished by separating the argument
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34 | * into an integer k and fraction f such that
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35 | *
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36 | * x k f
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37 | * e = 2 e.
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38 | *
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39 | * An expansion x + .5 x^2 + x^3 R(x) approximates exp(f) - 1
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40 | * in the basic range [-0.5 ln 2, 0.5 ln 2].
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41 | *
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42 | *
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43 | * ACCURACY:
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44 | *
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45 | * Relative error:
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46 | * arithmetic domain # trials peak rms
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47 | * IEEE -45,+maxarg 200,000 1.2e-19 2.5e-20
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48 | */
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49 |
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50 | #include "libm.h"
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51 |
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52 | #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
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53 | long double expm1l(long double x)
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54 | {
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55 | return expm1(x);
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56 | }
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57 | #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
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58 |
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59 | /* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x)
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60 | -.5 ln 2 < x < .5 ln 2
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61 | Theoretical peak relative error = 3.4e-22 */
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62 | static const long double
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63 | P0 = -1.586135578666346600772998894928250240826E4L,
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64 | P1 = 2.642771505685952966904660652518429479531E3L,
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65 | P2 = -3.423199068835684263987132888286791620673E2L,
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66 | P3 = 1.800826371455042224581246202420972737840E1L,
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67 | P4 = -5.238523121205561042771939008061958820811E-1L,
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68 | Q0 = -9.516813471998079611319047060563358064497E4L,
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69 | Q1 = 3.964866271411091674556850458227710004570E4L,
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70 | Q2 = -7.207678383830091850230366618190187434796E3L,
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71 | Q3 = 7.206038318724600171970199625081491823079E2L,
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72 | Q4 = -4.002027679107076077238836622982900945173E1L,
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73 | /* Q5 = 1.000000000000000000000000000000000000000E0 */
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74 | /* C1 + C2 = ln 2 */
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75 | C1 = 6.93145751953125E-1L,
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76 | C2 = 1.428606820309417232121458176568075500134E-6L,
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77 | /* ln 2^-65 */
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78 | minarg = -4.5054566736396445112120088E1L,
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79 | /* ln 2^16384 */
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80 | maxarg = 1.1356523406294143949492E4L;
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81 |
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82 | long double expm1l(long double x)
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83 | {
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84 | long double px, qx, xx;
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85 | int k;
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86 |
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87 | if (isnan(x))
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88 | return x;
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89 | if (x > maxarg)
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90 | return x*0x1p16383L; /* overflow, unless x==inf */
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91 | if (x == 0.0)
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92 | return x;
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93 | if (x < minarg)
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94 | return -1.0;
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95 |
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96 | xx = C1 + C2;
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97 | /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
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98 | px = floorl(0.5 + x / xx);
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99 | k = px;
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100 | /* remainder times ln 2 */
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101 | x -= px * C1;
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102 | x -= px * C2;
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103 |
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104 | /* Approximate exp(remainder ln 2).*/
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105 | px = (((( P4 * x + P3) * x + P2) * x + P1) * x + P0) * x;
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106 | qx = (((( x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0;
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107 | xx = x * x;
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108 | qx = x + (0.5 * xx + xx * px / qx);
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109 |
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110 | /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
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111 | We have qx = exp(remainder ln 2) - 1, so
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112 | exp(x) - 1 = 2^k (qx + 1) - 1 = 2^k qx + 2^k - 1. */
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113 | px = scalbnl(1.0, k);
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114 | x = px * qx + (px - 1.0);
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115 | return x;
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116 | }
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117 | #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
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118 | // TODO: broken implementation to make things compile
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119 | long double expm1l(long double x)
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120 | {
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121 | return expm1(x);
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122 | }
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123 | #endif
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