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/* @(#)s_expm1.c 5.1 93/09/24 */ |
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/* |
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* ==================================================== |
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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* |
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* Developed at SunPro, a Sun Microsystems, Inc. business. |
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* Permission to use, copy, modify, and distribute this |
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* software is freely granted, provided that this notice |
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* is preserved. |
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* ==================================================== |
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*/ |
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/* expm1(x) |
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* Returns exp(x)-1, the exponential of x minus 1. |
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* |
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* Method |
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* 1. Argument reduction: |
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* Given x, find r and integer k such that |
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* |
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* x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 |
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* |
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* Here a correction term c will be computed to compensate |
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* the error in r when rounded to a floating-point number. |
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* |
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* 2. Approximating expm1(r) by a special rational function on |
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* the interval [0,0.34658]: |
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* Since |
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* r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... |
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* we define R1(r*r) by |
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* r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) |
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* That is, |
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* R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) |
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* = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) |
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* = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... |
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* We use a special Remes algorithm on [0,0.347] to generate |
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* a polynomial of degree 5 in r*r to approximate R1. The |
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* maximum error of this polynomial approximation is bounded |
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* by 2**-61. In other words, |
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* R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 |
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* where Q1 = -1.6666666666666567384E-2, |
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* Q2 = 3.9682539681370365873E-4, |
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* Q3 = -9.9206344733435987357E-6, |
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* Q4 = 2.5051361420808517002E-7, |
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* Q5 = -6.2843505682382617102E-9; |
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* (where z=r*r, and the values of Q1 to Q5 are listed below) |
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* with error bounded by |
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* | 5 | -61 |
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* | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 |
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* | | |
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* |
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* expm1(r) = exp(r)-1 is then computed by the following |
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* specific way which minimize the accumulation rounding error: |
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* 2 3 |
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* r r [ 3 - (R1 + R1*r/2) ] |
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* expm1(r) = r + --- + --- * [--------------------] |
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* 2 2 [ 6 - r*(3 - R1*r/2) ] |
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* |
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* To compensate the error in the argument reduction, we use |
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* expm1(r+c) = expm1(r) + c + expm1(r)*c |
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* ~ expm1(r) + c + r*c |
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* Thus c+r*c will be added in as the correction terms for |
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* expm1(r+c). Now rearrange the term to avoid optimization |
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* screw up: |
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* ( 2 2 ) |
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* ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) |
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* expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) |
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* ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) |
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* ( ) |
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* |
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* = r - E |
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* 3. Scale back to obtain expm1(x): |
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* From step 1, we have |
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* expm1(x) = either 2^k*[expm1(r)+1] - 1 |
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* = or 2^k*[expm1(r) + (1-2^-k)] |
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* 4. Implementation notes: |
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* (A). To save one multiplication, we scale the coefficient Qi |
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* to Qi*2^i, and replace z by (x^2)/2. |
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* (B). To achieve maximum accuracy, we compute expm1(x) by |
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* (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) |
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* (ii) if k=0, return r-E |
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* (iii) if k=-1, return 0.5*(r-E)-0.5 |
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* (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) |
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* else return 1.0+2.0*(r-E); |
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* (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) |
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* (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else |
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* (vii) return 2^k(1-((E+2^-k)-r)) |
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* |
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* Special cases: |
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* expm1(INF) is INF, expm1(NaN) is NaN; |
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* expm1(-INF) is -1, and |
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* for finite argument, only expm1(0)=0 is exact. |
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* |
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* Accuracy: |
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* according to an error analysis, the error is always less than |
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* 1 ulp (unit in the last place). |
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* |
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* Misc. info. |
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* For IEEE double |
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* if x > 7.09782712893383973096e+02 then expm1(x) overflow |
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* |
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* Constants: |
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* The hexadecimal values are the intended ones for the following |
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* constants. The decimal values may be used, provided that the |
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* compiler will convert from decimal to binary accurately enough |
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* to produce the hexadecimal values shown. |
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*/ |
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#include <float.h> |
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#include <math.h> |
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#include "math_private.h" |
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static const double |
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one = 1.0, |
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huge = 1.0e+300, |
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tiny = 1.0e-300, |
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o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ |
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ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ |
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ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ |
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invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ |
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/* scaled coefficients related to expm1 */ |
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Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ |
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Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ |
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Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ |
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Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ |
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Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ |
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double |
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expm1(double x) |
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{ |
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double y,hi,lo,c,t,e,hxs,hfx,r1; |
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int32_t k,xsb; |
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u_int32_t hx; |
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GET_HIGH_WORD(hx,x); |
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xsb = hx&0x80000000; /* sign bit of x */ |
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✓✓ |
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if(xsb==0) y=x; else y= -x; /* y = |x| */ |
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hx &= 0x7fffffff; /* high word of |x| */ |
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/* filter out huge and non-finite argument */ |
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✗✓ |
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if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ |
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if(hx >= 0x40862E42) { /* if |x|>=709.78... */ |
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if(hx>=0x7ff00000) { |
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u_int32_t low; |
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GET_LOW_WORD(low,x); |
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if(((hx&0xfffff)|low)!=0) |
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return x+x; /* NaN */ |
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else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ |
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} |
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if(x > o_threshold) return huge*huge; /* overflow */ |
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} |
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if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ |
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if(x+tiny<0.0) /* raise inexact */ |
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return tiny-one; /* return -1 */ |
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} |
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} |
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/* argument reduction */ |
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✓✓ |
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if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ |
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✓✓ |
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if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ |
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✓✓ |
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if(xsb==0) |
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{hi = x - ln2_hi; lo = ln2_lo; k = 1;} |
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else |
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{hi = x + ln2_hi; lo = -ln2_lo; k = -1;} |
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} else { |
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k = invln2*x+((xsb==0)?0.5:-0.5); |
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t = k; |
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hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ |
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lo = t*ln2_lo; |
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} |
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x = hi - lo; |
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c = (hi-x)-lo; |
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} |
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✓✓ |
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else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ |
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t = huge+x; /* return x with inexact flags when x!=0 */ |
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return x - (t-(huge+x)); |
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} |
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else k = 0; |
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/* x is now in primary range */ |
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hfx = 0.5*x; |
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hxs = x*hfx; |
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r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); |
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t = 3.0-r1*hfx; |
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e = hxs*((r1-t)/(6.0 - x*t)); |
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✓✓ |
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if(k==0) return x - (x*e-hxs); /* c is 0 */ |
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else { |
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e = (x*(e-c)-c); |
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e -= hxs; |
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✓✓ |
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if(k== -1) return 0.5*(x-e)-0.5; |
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✓✓ |
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if(k==1) { |
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✓✓ |
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if(x < -0.25) return -2.0*(e-(x+0.5)); |
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else return one+2.0*(x-e); |
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} |
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✓✓ |
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if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ |
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u_int32_t high; |
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y = one-(e-x); |
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GET_HIGH_WORD(high,y); |
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SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ |
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return y-one; |
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} |
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t = one; |
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✓✓ |
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if(k<20) { |
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u_int32_t high; |
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SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */ |
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y = t-(e-x); |
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GET_HIGH_WORD(high,y); |
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SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ |
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} else { |
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u_int32_t high; |
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SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */ |
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y = x-(e+t); |
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y += one; |
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GET_HIGH_WORD(high,y); |
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SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ |
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} |
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} |
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return y; |
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} |
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DEF_STD(expm1); |
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LDBL_MAYBE_CLONE(expm1); |