### Bug Summary

File: | real/expm1.c
Location: | line 167, column 10
Description: | Value stored to 'y' is never read

### Annotated Source Code

1 | /* @(#)s_expm1.c 1.5 04/04/22 */
2 |
3 | /*
4 | * Copyright (c) 2015-2017 Carsten Sonne Larsen <cs@innolan.dk>
5 | * All rights reserved.
6 | *
7 | * Redistribution and use in source and binary forms, with or without
8 | * modification, are permitted provided that the following conditions
9 | * are met:
10 | * 1. Redistributions of source code must retain the above copyright
11 | * notice, this list of conditions and the following disclaimer.
12 | * 2. Redistributions in binary form must reproduce the above copyright
13 | * notice, this list of conditions and the following disclaimer in the
14 | * documentation and/or other materials provided with the distribution.
15 | *
16 | * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
17 | * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
18 | * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
19 | * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
20 | * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
21 | * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
22 | * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
23 | * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
24 | * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
25 | * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
26 | *
27 | * The origin source code can be obtained from:
28 | * http://www.netlib.org/fdlibm/s_expm1.c
29 | *
30 | */
31 |
32 | /*
33 | * ====================================================
34 | * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
35 | *
36 | * Permission to use, copy, modify, and distribute this
37 | * software is freely granted, provided that this notice
38 | * is preserved.
39 | * ====================================================
40 | *
41 | */
42 |
43 | #include "prim.h"
44 | #include "math.h"
45 |
46 | /* expm1(x)
47 | * Returns exp(x)-1, the exponential of x minus 1.
48 | *
49 | * Method
50 | * 1. Argument reduction:
51 | * Given x, find r and integer k such that
52 | *
53 | * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
54 | *
55 | * Here a correction term c will be computed to compensate
56 | * the error in r when rounded to a floating-point number.
57 | *
58 | * 2. Approximating expm1(r) by a special rational function on
59 | * the interval [0,0.34658]:
60 | * Since
61 | * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
62 | * we define R1(r*r) by
63 | * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
64 | * That is,
65 | * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
66 | * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
67 | * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
68 | * We use a special Remes algorithm on [0,0.347] to generate
69 | * a polynomial of degree 5 in r*r to approximate R1. The
70 | * maximum error of this polynomial approximation is bounded
71 | * by 2**-61. In other words,
72 | * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
73 | * where Q1 = -1.6666666666666567384E-2,
74 | * Q2 = 3.9682539681370365873E-4,
75 | * Q3 = -9.9206344733435987357E-6,
76 | * Q4 = 2.5051361420808517002E-7,
77 | * Q5 = -6.2843505682382617102E-9;
78 | * (where z=r*r, and the values of Q1 to Q5 are listed below)
79 | * with error bounded by
80 | * | 5 | -61
81 | * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
82 | * | |
83 | *
84 | * expm1(r) = exp(r)-1 is then computed by the following
85 | * specific way which minimize the accumulation rounding error:
86 | * 2 3
87 | * r r [ 3 - (R1 + R1*r/2) ]
88 | * expm1(r) = r + --- + --- * [--------------------]
89 | * 2 2 [ 6 - r*(3 - R1*r/2) ]
90 | *
91 | * To compensate the error in the argument reduction, we use
92 | * expm1(r+c) = expm1(r) + c + expm1(r)*c
93 | * ~ expm1(r) + c + r*c
94 | * Thus c+r*c will be added in as the correction terms for
95 | * expm1(r+c). Now rearrange the term to avoid optimization
96 | * screw up:
97 | * ( 2 2 )
98 | * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
99 | * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
100 | * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
101 | * ( )
102 | *
103 | * = r - E
104 | * 3. Scale back to obtain expm1(x):
105 | * From step 1, we have
106 | * expm1(x) = either 2^k*[expm1(r)+1] - 1
107 | * = or 2^k*[expm1(r) + (1-2^-k)]
108 | * 4. Implementation notes:
109 | * (A). To save one multiplication, we scale the coefficient Qi
110 | * to Qi*2^i, and replace z by (x^2)/2.
111 | * (B). To achieve maximum accuracy, we compute expm1(x) by
112 | * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
113 | * (ii) if k=0, return r-E
114 | * (iii) if k=-1, return 0.5*(r-E)-0.5
115 | * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
116 | * else return 1.0+2.0*(r-E);
117 | * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
118 | * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
119 | * (vii) return 2^k(1-((E+2^-k)-r))
120 | *
121 | * Special cases:
122 | * expm1(INF) is INF, expm1(NaN) is NaN;
123 | * expm1(-INF) is -1, and
124 | * for finite argument, only expm1(0)=0 is exact.
125 | *
126 | * Accuracy:
127 | * according to an error analysis, the error is always less than
128 | * 1 ulp (unit in the last place).
129 | *
130 | * Misc. info.
131 | * For IEEE double
132 | * if x > 7.09782712893383973096e+02 then expm1(x) overflow
133 | *
134 | * Constants:
135 | * The hexadecimal values are the intended ones for the following
136 | * constants. The decimal values may be used, provided that the
137 | * compiler will convert from decimal to binary accurately enough
138 | * to produce the hexadecimal values shown.
139 | */
140 |
141 | static const double
142 | one = 1.0,
143 | huge = 1.0e+300,
144 | tiny = 1.0e-300,
145 | o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
146 | ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
147 | ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
148 | invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
149 | /* scaled coefficients related to expm1 */
150 | Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
151 | Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
152 | Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
153 | Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
154 | Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
155 |
156 | double expm1(double x)
157 | {
158 | double y,hi,lo,c,t,e,hxs,hfx,r1;
159 | sword k,xsb;
160 | uword hx;
161 |
162 | c = 0.0;
163 |
164 | GET_HIGH_WORD(hx,x)do { ieee_double_shape_type gh_u; gh_u.value = (x); (hx) = gh_u .parts.msw; } while (0); /* high word of x */
165 | xsb = hx&0x80000000; /* sign bit of x */
166 | if(xsb==0) y=x;
167 | else y= -x; /* y = |x| */
| Value stored to 'y' is never read
168 | hx &= 0x7fffffff; /* high word of |x| */
169 |
170 | /* filter out huge and non-finite argument */
171 | if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */
172 | if(hx >= 0x40862E42) { /* if |x|>=709.78... */
173 | if(hx>=0x7ff00000) {
174 | uword low;
175 | GET_LOW_WORD(low,x)do { ieee_double_shape_type gl_u; gl_u.value = (x); (low) = gl_u .parts.lsw; } while (0);
176 | if(((hx&0xfffff)|low)!=0)
177 | return x+x; /* NaN */
178 | else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
179 | }
180 | if(x > o_threshold) return huge*huge; /* overflow */
181 | }
182 | if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
183 | if(x+tiny<0.0) /* raise inexact */
184 | return tiny-one; /* return -1 */
185 | }
186 | }
187 |
188 | /* argument reduction */
189 | if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
190 | if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
191 | if(xsb==0)
192 | {
193 | hi = x - ln2_hi;
194 | lo = ln2_lo;
195 | k = 1;
196 | }
197 | else
198 | {
199 | hi = x + ln2_hi;
200 | lo = -ln2_lo;
201 | k = -1;
202 | }
203 | } else {
204 | k = (sword)(invln2*x+((xsb==0)?0.5:-0.5));
205 | t = k;
206 | hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
207 | lo = t*ln2_lo;
208 | }
209 | x = hi - lo;
210 | c = (hi-x)-lo;
211 | }
212 | else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
213 | t = huge+x; /* return x with inexact flags when x!=0 */
214 | return x - (t-(huge+x));
215 | }
216 | else k = 0;
217 |
218 | /* x is now in primary range */
219 | hfx = 0.5*x;
220 | hxs = x*hfx;
221 | r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
222 | t = 3.0-r1*hfx;
223 | e = hxs*((r1-t)/(6.0 - x*t));
224 | if(k==0) return x - (x*e-hxs); /* c is 0 */
225 | else {
226 | e = (x*(e-c)-c);
227 | e -= hxs;
228 | if(k== -1) return 0.5*(x-e)-0.5;
229 | if(k==1) {
230 | if(x < -0.25) return -2.0*(e-(x+0.5));
231 | else return one+2.0*(x-e);
232 | }
233 | if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
234 | uword hy;
235 |
236 | y = one-(e-x);
237 | GET_HIGH_WORD(hy,y)do { ieee_double_shape_type gh_u; gh_u.value = (y); (hy) = gh_u .parts.msw; } while (0);
238 | SET_HIGH_WORD(y, hy + (k<<20))do { ieee_double_shape_type sh_u; sh_u.value = (y); sh_u.parts .msw = (hy + (k<<20)); (y) = sh_u.value; } while (0); /* add k to y's exponent */
239 | return y-one;
240 | }
241 | t = one;
242 | if(k<20) {
243 | uword hy;
244 |
245 | 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 */
246 | y = t-(e-x);
247 | GET_HIGH_WORD(hy, y)do { ieee_double_shape_type gh_u; gh_u.value = (y); (hy) = gh_u .parts.msw; } while (0);
248 | SET_HIGH_WORD(y, hy + (k<<20))do { ieee_double_shape_type sh_u; sh_u.value = (y); sh_u.parts .msw = (hy + (k<<20)); (y) = sh_u.value; } while (0); /* add k to y's exponent */
249 | } else {
250 | uword hy;
251 |
252 | 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 */
253 | y = x-(e+t);
254 | y += one;
255 | GET_HIGH_WORD(hy, y)do { ieee_double_shape_type gh_u; gh_u.value = (y); (hy) = gh_u .parts.msw; } while (0);
256 | SET_HIGH_WORD(y, hy + (k<<20))do { ieee_double_shape_type sh_u; sh_u.value = (y); sh_u.parts .msw = (hy + (k<<20)); (y) = sh_u.value; } while (0); /* add k to y's exponent */
257 | }
258 | }
259 | return y;
260 | }