[337] | 1 | /* origin: FreeBSD /usr/src/lib/msun/src/s_expm1.c */
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| 2 | /*
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| 3 | * ====================================================
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| 4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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| 5 | *
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| 6 | * Developed at SunPro, a Sun Microsystems, Inc. business.
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| 7 | * Permission to use, copy, modify, and distribute this
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| 8 | * software is freely granted, provided that this notice
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| 9 | * is preserved.
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| 10 | * ====================================================
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| 11 | */
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| 12 | /* expm1(x)
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| 13 | * Returns exp(x)-1, the exponential of x minus 1.
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| 14 | *
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| 15 | * Method
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| 16 | * 1. Argument reduction:
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| 17 | * Given x, find r and integer k such that
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| 18 | *
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| 19 | * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
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| 20 | *
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| 21 | * Here a correction term c will be computed to compensate
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| 22 | * the error in r when rounded to a floating-point number.
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| 23 | *
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| 24 | * 2. Approximating expm1(r) by a special rational function on
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| 25 | * the interval [0,0.34658]:
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| 26 | * Since
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| 27 | * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
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| 28 | * we define R1(r*r) by
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| 29 | * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
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| 30 | * That is,
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| 31 | * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
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| 32 | * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
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| 33 | * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
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| 34 | * We use a special Remez algorithm on [0,0.347] to generate
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| 35 | * a polynomial of degree 5 in r*r to approximate R1. The
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| 36 | * maximum error of this polynomial approximation is bounded
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| 37 | * by 2**-61. In other words,
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| 38 | * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
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| 39 | * where Q1 = -1.6666666666666567384E-2,
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| 40 | * Q2 = 3.9682539681370365873E-4,
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| 41 | * Q3 = -9.9206344733435987357E-6,
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| 42 | * Q4 = 2.5051361420808517002E-7,
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| 43 | * Q5 = -6.2843505682382617102E-9;
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| 44 | * z = r*r,
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| 45 | * with error bounded by
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| 46 | * | 5 | -61
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| 47 | * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
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| 48 | * | |
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| 49 | *
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| 50 | * expm1(r) = exp(r)-1 is then computed by the following
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| 51 | * specific way which minimize the accumulation rounding error:
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| 52 | * 2 3
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| 53 | * r r [ 3 - (R1 + R1*r/2) ]
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| 54 | * expm1(r) = r + --- + --- * [--------------------]
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| 55 | * 2 2 [ 6 - r*(3 - R1*r/2) ]
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| 56 | *
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| 57 | * To compensate the error in the argument reduction, we use
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| 58 | * expm1(r+c) = expm1(r) + c + expm1(r)*c
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| 59 | * ~ expm1(r) + c + r*c
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| 60 | * Thus c+r*c will be added in as the correction terms for
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| 61 | * expm1(r+c). Now rearrange the term to avoid optimization
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| 62 | * screw up:
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| 63 | * ( 2 2 )
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| 64 | * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
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| 65 | * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
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| 66 | * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
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| 67 | * ( )
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| 68 | *
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| 69 | * = r - E
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| 70 | * 3. Scale back to obtain expm1(x):
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| 71 | * From step 1, we have
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| 72 | * expm1(x) = either 2^k*[expm1(r)+1] - 1
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| 73 | * = or 2^k*[expm1(r) + (1-2^-k)]
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| 74 | * 4. Implementation notes:
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| 75 | * (A). To save one multiplication, we scale the coefficient Qi
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| 76 | * to Qi*2^i, and replace z by (x^2)/2.
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| 77 | * (B). To achieve maximum accuracy, we compute expm1(x) by
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| 78 | * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
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| 79 | * (ii) if k=0, return r-E
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| 80 | * (iii) if k=-1, return 0.5*(r-E)-0.5
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| 81 | * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
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| 82 | * else return 1.0+2.0*(r-E);
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| 83 | * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
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| 84 | * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
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| 85 | * (vii) return 2^k(1-((E+2^-k)-r))
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| 86 | *
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| 87 | * Special cases:
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| 88 | * expm1(INF) is INF, expm1(NaN) is NaN;
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| 89 | * expm1(-INF) is -1, and
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| 90 | * for finite argument, only expm1(0)=0 is exact.
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| 91 | *
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| 92 | * Accuracy:
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| 93 | * according to an error analysis, the error is always less than
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| 94 | * 1 ulp (unit in the last place).
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| 95 | *
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| 96 | * Misc. info.
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| 97 | * For IEEE double
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| 98 | * if x > 7.09782712893383973096e+02 then expm1(x) overflow
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| 99 | *
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| 100 | * Constants:
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| 101 | * The hexadecimal values are the intended ones for the following
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| 102 | * constants. The decimal values may be used, provided that the
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| 103 | * compiler will convert from decimal to binary accurately enough
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| 104 | * to produce the hexadecimal values shown.
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| 105 | */
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| 106 |
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| 107 | #include "libm.h"
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| 108 |
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| 109 | static const double
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| 110 | o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
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| 111 | ln2_hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
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| 112 | ln2_lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
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| 113 | invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
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| 114 | /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */
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| 115 | Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
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| 116 | Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
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| 117 | Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
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| 118 | Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
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| 119 | Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
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| 120 |
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| 121 | double expm1(double x)
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| 122 | {
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| 123 | double_t y,hi,lo,c,t,e,hxs,hfx,r1,twopk;
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| 124 | union {double f; uint64_t i;} u = {x};
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| 125 | uint32_t hx = u.i>>32 & 0x7fffffff;
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| 126 | int k, sign = u.i>>63;
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| 127 |
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| 128 | /* filter out huge and non-finite argument */
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| 129 | if (hx >= 0x4043687A) { /* if |x|>=56*ln2 */
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| 130 | if (isnan(x))
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| 131 | return x;
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| 132 | if (sign)
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| 133 | return -1;
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| 134 | if (x > o_threshold) {
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| 135 | x *= 0x1p1023;
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| 136 | return x;
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| 137 | }
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| 138 | }
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| 139 |
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| 140 | /* argument reduction */
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| 141 | if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
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| 142 | if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
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| 143 | if (!sign) {
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| 144 | hi = x - ln2_hi;
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| 145 | lo = ln2_lo;
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| 146 | k = 1;
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| 147 | } else {
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| 148 | hi = x + ln2_hi;
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| 149 | lo = -ln2_lo;
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| 150 | k = -1;
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| 151 | }
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| 152 | } else {
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| 153 | k = invln2*x + (sign ? -0.5 : 0.5);
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| 154 | t = k;
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| 155 | hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
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| 156 | lo = t*ln2_lo;
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| 157 | }
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| 158 | x = hi-lo;
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| 159 | c = (hi-x)-lo;
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| 160 | } else if (hx < 0x3c900000) { /* |x| < 2**-54, return x */
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| 161 | if (hx < 0x00100000)
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| 162 | FORCE_EVAL((float)x);
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| 163 | return x;
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| 164 | } else
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| 165 | k = 0;
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| 166 |
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| 167 | /* x is now in primary range */
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| 168 | hfx = 0.5*x;
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| 169 | hxs = x*hfx;
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| 170 | r1 = 1.0+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
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| 171 | t = 3.0-r1*hfx;
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| 172 | e = hxs*((r1-t)/(6.0 - x*t));
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| 173 | if (k == 0) /* c is 0 */
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| 174 | return x - (x*e-hxs);
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| 175 | e = x*(e-c) - c;
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| 176 | e -= hxs;
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| 177 | /* exp(x) ~ 2^k (x_reduced - e + 1) */
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| 178 | if (k == -1)
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| 179 | return 0.5*(x-e) - 0.5;
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| 180 | if (k == 1) {
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| 181 | if (x < -0.25)
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| 182 | return -2.0*(e-(x+0.5));
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| 183 | return 1.0+2.0*(x-e);
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| 184 | }
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| 185 | u.i = (uint64_t)(0x3ff + k)<<52; /* 2^k */
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| 186 | twopk = u.f;
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| 187 | if (k < 0 || k > 56) { /* suffice to return exp(x)-1 */
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| 188 | y = x - e + 1.0;
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| 189 | if (k == 1024)
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| 190 | y = y*2.0*0x1p1023;
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| 191 | else
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| 192 | y = y*twopk;
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| 193 | return y - 1.0;
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| 194 | }
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| 195 | u.i = (uint64_t)(0x3ff - k)<<52; /* 2^-k */
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| 196 | if (k < 20)
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| 197 | y = (x-e+(1-u.f))*twopk;
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| 198 | else
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| 199 | y = (x-(e+u.f)+1)*twopk;
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| 200 | return y;
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| 201 | }
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