1 /* @(#)e_hypotl.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 /* __ieee754_hypotl(x,y)
14 *
15 * Method :
16 * If (assume round-to-nearest) z=x*x+y*y
17 * has error less than sqrtl(2)/2 ulp, than
18 * sqrtl(z) has error less than 1 ulp (exercise).
19 *
20 * So, compute sqrtl(x*x+y*y) with some care as
21 * follows to get the error below 1 ulp:
22 *
23 * Assume x>y>0;
24 * (if possible, set rounding to round-to-nearest)
25 * 1. if x > 2y use
26 * x1*x1+(y*y+(x2*(x+x1))) for x*x+y*y
27 * where x1 = x with lower 53 bits cleared, x2 = x-x1; else
28 * 2. if x <= 2y use
29 * t1*y1+((x-y)*(x-y)+(t1*y2+t2*y))
30 * where t1 = 2x with lower 53 bits cleared, t2 = 2x-t1,
31 * y1= y with lower 53 bits chopped, y2 = y-y1.
32 *
33 * NOTE: scaling may be necessary if some argument is too
34 * large or too tiny
35 *
36 * Special cases:
37 * hypotl(x,y) is INF if x or y is +INF or -INF; else
38 * hypotl(x,y) is NAN if x or y is NAN.
39 *
40 * Accuracy:
41 * hypotl(x,y) returns sqrtl(x^2+y^2) with error less
42 * than 1 ulps (units in the last place)
43 */
44
45 #include <math.h>
46 #include <math_private.h>
47 #include <math-underflow.h>
48 #include <libm-alias-finite.h>
49
50 long double
51 __ieee754_hypotl(long double x, long double y)
52 {
53 long double a,b,a1,a2,b1,b2,w,kld;
54 int64_t j,k,ha,hb;
55 double xhi, yhi, hi, lo;
56
57 xhi = ldbl_high (x);
58 EXTRACT_WORDS64 (ha, xhi);
59 yhi = ldbl_high (y);
60 EXTRACT_WORDS64 (hb, yhi);
61 ha &= 0x7fffffffffffffffLL;
62 hb &= 0x7fffffffffffffffLL;
63 if(hb > ha) {a=y;b=x;j=ha; ha=hb;hb=j;} else {a=x;b=y;}
64 a = fabsl(a); /* a <- |a| */
65 b = fabsl(b); /* b <- |b| */
66 if((ha-hb)>0x0780000000000000LL) {return a+b;} /* x/y > 2**120 */
67 k=0;
68 kld = 1.0L;
69 if(ha > 0x5f30000000000000LL) { /* a>2**500 */
70 if(ha >= 0x7ff0000000000000LL) { /* Inf or NaN */
71 w = a+b; /* for sNaN */
72 if (issignaling (a) || issignaling (b))
73 return w;
74 if(ha == 0x7ff0000000000000LL)
75 w = a;
76 if(hb == 0x7ff0000000000000LL)
77 w = b;
78 return w;
79 }
80 /* scale a and b by 2**-600 */
81 a *= 0x1p-600L;
82 b *= 0x1p-600L;
83 k = 600;
84 kld = 0x1p+600L;
85 }
86 else if(hb < 0x23d0000000000000LL) { /* b < 2**-450 */
87 if(hb <= 0x000fffffffffffffLL) { /* subnormal b or 0 */
88 if(hb==0) return a;
89 a *= 0x1p+1022L;
90 b *= 0x1p+1022L;
91 k = -1022;
92 kld = 0x1p-1022L;
93 } else { /* scale a and b by 2^600 */
94 a *= 0x1p+600L;
95 b *= 0x1p+600L;
96 k = -600;
97 kld = 0x1p-600L;
98 }
99 }
100 /* medium size a and b */
101 w = a-b;
102 if (w>b) {
103 ldbl_unpack (a, &hi, &lo);
104 a1 = hi;
105 a2 = lo;
106 /* a*a + b*b
107 = (a1+a2)*a + b*b
108 = a1*a + a2*a + b*b
109 = a1*(a1+a2) + a2*a + b*b
110 = a1*a1 + a1*a2 + a2*a + b*b
111 = a1*a1 + a2*(a+a1) + b*b */
112 w = sqrtl(a1*a1-(b*(-b)-a2*(a+a1)));
113 } else {
114 a = a+a;
115 ldbl_unpack (b, &hi, &lo);
116 b1 = hi;
117 b2 = lo;
118 ldbl_unpack (a, &hi, &lo);
119 a1 = hi;
120 a2 = lo;
121 /* a*a + b*b
122 = a*a + (a-b)*(a-b) - (a-b)*(a-b) + b*b
123 = a*a + w*w - (a*a - 2*a*b + b*b) + b*b
124 = w*w + 2*a*b
125 = w*w + (a1+a2)*b
126 = w*w + a1*b + a2*b
127 = w*w + a1*(b1+b2) + a2*b
128 = w*w + a1*b1 + a1*b2 + a2*b */
129 w = sqrtl(a1*b1-(w*(-w)-(a1*b2+a2*b)));
130 }
131 if(k!=0)
132 {
133 w *= kld;
134 math_check_force_underflow_nonneg (w);
135 return w;
136 }
137 else
138 return w;
139 }
140 libm_alias_finite (__ieee754_hypotl, __hypotl)