| File: | src/lib/libm/src/s_expm1.c |
| Warning: | line 137, column 13 Value stored to 'y' is never read |
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| 1 | /* @(#)s_expm1.c 5.1 93/09/24 */ |
| 2 | /* |
| 3 | * ==================================================== |
| 4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
| 5 | * |
| 6 | * Developed at SunPro, a Sun Microsystems, Inc. business. |
| 7 | * Permission to use, copy, modify, and distribute this |
| 8 | * software is freely granted, provided that this notice |
| 9 | * is preserved. |
| 10 | * ==================================================== |
| 11 | */ |
| 12 | |
| 13 | /* expm1(x) |
| 14 | * Returns exp(x)-1, the exponential of x minus 1. |
| 15 | * |
| 16 | * Method |
| 17 | * 1. Argument reduction: |
| 18 | * Given x, find r and integer k such that |
| 19 | * |
| 20 | * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 |
| 21 | * |
| 22 | * Here a correction term c will be computed to compensate |
| 23 | * the error in r when rounded to a floating-point number. |
| 24 | * |
| 25 | * 2. Approximating expm1(r) by a special rational function on |
| 26 | * the interval [0,0.34658]: |
| 27 | * Since |
| 28 | * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... |
| 29 | * we define R1(r*r) by |
| 30 | * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) |
| 31 | * That is, |
| 32 | * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) |
| 33 | * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) |
| 34 | * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... |
| 35 | * We use a special Remes algorithm on [0,0.347] to generate |
| 36 | * a polynomial of degree 5 in r*r to approximate R1. The |
| 37 | * maximum error of this polynomial approximation is bounded |
| 38 | * by 2**-61. In other words, |
| 39 | * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 |
| 40 | * where Q1 = -1.6666666666666567384E-2, |
| 41 | * Q2 = 3.9682539681370365873E-4, |
| 42 | * Q3 = -9.9206344733435987357E-6, |
| 43 | * Q4 = 2.5051361420808517002E-7, |
| 44 | * Q5 = -6.2843505682382617102E-9; |
| 45 | * (where z=r*r, and the values of Q1 to Q5 are listed below) |
| 46 | * with error bounded by |
| 47 | * | 5 | -61 |
| 48 | * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 |
| 49 | * | | |
| 50 | * |
| 51 | * expm1(r) = exp(r)-1 is then computed by the following |
| 52 | * specific way which minimize the accumulation rounding error: |
| 53 | * 2 3 |
| 54 | * r r [ 3 - (R1 + R1*r/2) ] |
| 55 | * expm1(r) = r + --- + --- * [--------------------] |
| 56 | * 2 2 [ 6 - r*(3 - R1*r/2) ] |
| 57 | * |
| 58 | * To compensate the error in the argument reduction, we use |
| 59 | * expm1(r+c) = expm1(r) + c + expm1(r)*c |
| 60 | * ~ expm1(r) + c + r*c |
| 61 | * Thus c+r*c will be added in as the correction terms for |
| 62 | * expm1(r+c). Now rearrange the term to avoid optimization |
| 63 | * screw up: |
| 64 | * ( 2 2 ) |
| 65 | * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) |
| 66 | * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) |
| 67 | * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) |
| 68 | * ( ) |
| 69 | * |
| 70 | * = r - E |
| 71 | * 3. Scale back to obtain expm1(x): |
| 72 | * From step 1, we have |
| 73 | * expm1(x) = either 2^k*[expm1(r)+1] - 1 |
| 74 | * = or 2^k*[expm1(r) + (1-2^-k)] |
| 75 | * 4. Implementation notes: |
| 76 | * (A). To save one multiplication, we scale the coefficient Qi |
| 77 | * to Qi*2^i, and replace z by (x^2)/2. |
| 78 | * (B). To achieve maximum accuracy, we compute expm1(x) by |
| 79 | * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) |
| 80 | * (ii) if k=0, return r-E |
| 81 | * (iii) if k=-1, return 0.5*(r-E)-0.5 |
| 82 | * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) |
| 83 | * else return 1.0+2.0*(r-E); |
| 84 | * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) |
| 85 | * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else |
| 86 | * (vii) return 2^k(1-((E+2^-k)-r)) |
| 87 | * |
| 88 | * Special cases: |
| 89 | * expm1(INF) is INF, expm1(NaN) is NaN; |
| 90 | * expm1(-INF) is -1, and |
| 91 | * for finite argument, only expm1(0)=0 is exact. |
| 92 | * |
| 93 | * Accuracy: |
| 94 | * according to an error analysis, the error is always less than |
| 95 | * 1 ulp (unit in the last place). |
| 96 | * |
| 97 | * Misc. info. |
| 98 | * For IEEE double |
| 99 | * if x > 7.09782712893383973096e+02 then expm1(x) overflow |
| 100 | * |
| 101 | * Constants: |
| 102 | * The hexadecimal values are the intended ones for the following |
| 103 | * constants. The decimal values may be used, provided that the |
| 104 | * compiler will convert from decimal to binary accurately enough |
| 105 | * to produce the hexadecimal values shown. |
| 106 | */ |
| 107 | |
| 108 | #include <float.h> |
| 109 | #include <math.h> |
| 110 | |
| 111 | #include "math_private.h" |
| 112 | |
| 113 | static const double |
| 114 | one = 1.0, |
| 115 | huge = 1.0e+300, |
| 116 | tiny = 1.0e-300, |
| 117 | o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ |
| 118 | ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ |
| 119 | ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ |
| 120 | invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ |
| 121 | /* scaled coefficients related to expm1 */ |
| 122 | Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ |
| 123 | Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ |
| 124 | Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ |
| 125 | Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ |
| 126 | Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ |
| 127 | |
| 128 | double |
| 129 | expm1(double x) |
| 130 | { |
| 131 | double y,hi,lo,c,t,e,hxs,hfx,r1; |
| 132 | int32_t k,xsb; |
| 133 | u_int32_t hx; |
| 134 | |
| 135 | GET_HIGH_WORD(hx,x)do { ieee_double_shape_type gh_u; gh_u.value = (x); (hx) = gh_u .parts.msw; } while (0); |
| 136 | xsb = hx&0x80000000; /* sign bit of x */ |
| 137 | if(xsb==0) y=x; else y= -x; /* y = |x| */ |
Value stored to 'y' is never read | |
| 138 | hx &= 0x7fffffff; /* high word of |x| */ |
| 139 | |
| 140 | /* filter out huge and non-finite argument */ |
| 141 | if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ |
| 142 | if(hx >= 0x40862E42) { /* if |x|>=709.78... */ |
| 143 | if(hx>=0x7ff00000) { |
| 144 | u_int32_t low; |
| 145 | GET_LOW_WORD(low,x)do { ieee_double_shape_type gl_u; gl_u.value = (x); (low) = gl_u .parts.lsw; } while (0); |
| 146 | if(((hx&0xfffff)|low)!=0) |
| 147 | return x+x; /* NaN */ |
| 148 | else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ |
| 149 | } |
| 150 | if(x > o_threshold) return huge*huge; /* overflow */ |
| 151 | } |
| 152 | if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ |
| 153 | if(x+tiny<0.0) /* raise inexact */ |
| 154 | return tiny-one; /* return -1 */ |
| 155 | } |
| 156 | } |
| 157 | |
| 158 | /* argument reduction */ |
| 159 | if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ |
| 160 | if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ |
| 161 | if(xsb==0) |
| 162 | {hi = x - ln2_hi; lo = ln2_lo; k = 1;} |
| 163 | else |
| 164 | {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} |
| 165 | } else { |
| 166 | k = invln2*x+((xsb==0)?0.5:-0.5); |
| 167 | t = k; |
| 168 | hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ |
| 169 | lo = t*ln2_lo; |
| 170 | } |
| 171 | x = hi - lo; |
| 172 | c = (hi-x)-lo; |
| 173 | } |
| 174 | else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ |
| 175 | t = huge+x; /* return x with inexact flags when x!=0 */ |
| 176 | return x - (t-(huge+x)); |
| 177 | } |
| 178 | else k = 0; |
| 179 | |
| 180 | /* x is now in primary range */ |
| 181 | hfx = 0.5*x; |
| 182 | hxs = x*hfx; |
| 183 | r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); |
| 184 | t = 3.0-r1*hfx; |
| 185 | e = hxs*((r1-t)/(6.0 - x*t)); |
| 186 | if(k==0) return x - (x*e-hxs); /* c is 0 */ |
| 187 | else { |
| 188 | e = (x*(e-c)-c); |
| 189 | e -= hxs; |
| 190 | if(k== -1) return 0.5*(x-e)-0.5; |
| 191 | if(k==1) { |
| 192 | if(x < -0.25) return -2.0*(e-(x+0.5)); |
| 193 | else return one+2.0*(x-e); |
| 194 | } |
| 195 | if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ |
| 196 | u_int32_t high; |
| 197 | y = one-(e-x); |
| 198 | GET_HIGH_WORD(high,y)do { ieee_double_shape_type gh_u; gh_u.value = (y); (high) = gh_u .parts.msw; } while (0); |
| 199 | SET_HIGH_WORD(y,high+(k<<20))do { ieee_double_shape_type sh_u; sh_u.value = (y); sh_u.parts .msw = (high+(k<<20)); (y) = sh_u.value; } while (0); /* add k to y's exponent */ |
| 200 | return y-one; |
| 201 | } |
| 202 | t = one; |
| 203 | if(k<20) { |
| 204 | u_int32_t high; |
| 205 | SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k))do { ieee_double_shape_type sh_u; sh_u.value = (t); sh_u.parts .msw = (0x3ff00000 - (0x200000>>k)); (t) = sh_u.value; } while (0); /* t=1-2^-k */ |
| 206 | y = t-(e-x); |
| 207 | GET_HIGH_WORD(high,y)do { ieee_double_shape_type gh_u; gh_u.value = (y); (high) = gh_u .parts.msw; } while (0); |
| 208 | SET_HIGH_WORD(y,high+(k<<20))do { ieee_double_shape_type sh_u; sh_u.value = (y); sh_u.parts .msw = (high+(k<<20)); (y) = sh_u.value; } while (0); /* add k to y's exponent */ |
| 209 | } else { |
| 210 | u_int32_t high; |
| 211 | SET_HIGH_WORD(t,((0x3ff-k)<<20))do { ieee_double_shape_type sh_u; sh_u.value = (t); sh_u.parts .msw = (((0x3ff-k)<<20)); (t) = sh_u.value; } while (0); /* 2^-k */ |
| 212 | y = x-(e+t); |
| 213 | y += one; |
| 214 | GET_HIGH_WORD(high,y)do { ieee_double_shape_type gh_u; gh_u.value = (y); (high) = gh_u .parts.msw; } while (0); |
| 215 | SET_HIGH_WORD(y,high+(k<<20))do { ieee_double_shape_type sh_u; sh_u.value = (y); sh_u.parts .msw = (high+(k<<20)); (y) = sh_u.value; } while (0); /* add k to y's exponent */ |
| 216 | } |
| 217 | } |
| 218 | return y; |
| 219 | } |
| 220 | DEF_STD(expm1)__asm__(".global " "expm1" " ; " "expm1" " = " "_libm_expm1"); |
| 221 | LDBL_MAYBE_CLONE(expm1)__asm(""); |