[352] | 1 | /* origin: FreeBSD /usr/src/lib/msun/src/e_rem_pio2.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 SunSoft, 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 | * Optimized by Bruce D. Evans.
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| 13 | */
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| 14 | /* __rem_pio2(x,y)
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| 15 | *
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| 16 | * return the remainder of x rem pi/2 in y[0]+y[1]
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| 17 | * use __rem_pio2_large() for large x
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| 18 | */
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| 19 |
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| 20 | #include "libm.h"
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| 21 |
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| 22 | #if FLT_EVAL_METHOD==0 || FLT_EVAL_METHOD==1
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| 23 | #define EPS DBL_EPSILON
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| 24 | #elif FLT_EVAL_METHOD==2
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| 25 | #define EPS LDBL_EPSILON
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| 26 | #endif
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| 27 |
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| 28 | /*
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| 29 | * invpio2: 53 bits of 2/pi
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| 30 | * pio2_1: first 33 bit of pi/2
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| 31 | * pio2_1t: pi/2 - pio2_1
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| 32 | * pio2_2: second 33 bit of pi/2
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| 33 | * pio2_2t: pi/2 - (pio2_1+pio2_2)
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| 34 | * pio2_3: third 33 bit of pi/2
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| 35 | * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
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| 36 | */
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| 37 | static const double
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| 38 | toint = 1.5/EPS,
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| 39 | invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
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| 40 | pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
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| 41 | pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
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| 42 | pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
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| 43 | pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
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| 44 | pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
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| 45 | pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
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| 46 |
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| 47 | /* caller must handle the case when reduction is not needed: |x| ~<= pi/4 */
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| 48 | int __rem_pio2(double x, double *y)
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| 49 | {
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| 50 | union {double f; uint64_t i;} u = {x};
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| 51 | double_t z,w,t,r,fn;
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| 52 | double tx[3],ty[2];
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| 53 | uint32_t ix;
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| 54 | int sign, n, ex, ey, i;
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| 55 |
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| 56 | sign = u.i>>63;
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| 57 | ix = u.i>>32 & 0x7fffffff;
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| 58 | if (ix <= 0x400f6a7a) { /* |x| ~<= 5pi/4 */
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| 59 | if ((ix & 0xfffff) == 0x921fb) /* |x| ~= pi/2 or 2pi/2 */
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| 60 | goto medium; /* cancellation -- use medium case */
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| 61 | if (ix <= 0x4002d97c) { /* |x| ~<= 3pi/4 */
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| 62 | if (!sign) {
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| 63 | z = x - pio2_1; /* one round good to 85 bits */
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| 64 | y[0] = z - pio2_1t;
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| 65 | y[1] = (z-y[0]) - pio2_1t;
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| 66 | return 1;
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| 67 | } else {
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| 68 | z = x + pio2_1;
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| 69 | y[0] = z + pio2_1t;
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| 70 | y[1] = (z-y[0]) + pio2_1t;
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| 71 | return -1;
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| 72 | }
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| 73 | } else {
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| 74 | if (!sign) {
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| 75 | z = x - 2*pio2_1;
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| 76 | y[0] = z - 2*pio2_1t;
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| 77 | y[1] = (z-y[0]) - 2*pio2_1t;
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| 78 | return 2;
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| 79 | } else {
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| 80 | z = x + 2*pio2_1;
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| 81 | y[0] = z + 2*pio2_1t;
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| 82 | y[1] = (z-y[0]) + 2*pio2_1t;
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| 83 | return -2;
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| 84 | }
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| 85 | }
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| 86 | }
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| 87 | if (ix <= 0x401c463b) { /* |x| ~<= 9pi/4 */
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| 88 | if (ix <= 0x4015fdbc) { /* |x| ~<= 7pi/4 */
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| 89 | if (ix == 0x4012d97c) /* |x| ~= 3pi/2 */
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| 90 | goto medium;
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| 91 | if (!sign) {
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| 92 | z = x - 3*pio2_1;
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| 93 | y[0] = z - 3*pio2_1t;
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| 94 | y[1] = (z-y[0]) - 3*pio2_1t;
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| 95 | return 3;
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| 96 | } else {
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| 97 | z = x + 3*pio2_1;
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| 98 | y[0] = z + 3*pio2_1t;
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| 99 | y[1] = (z-y[0]) + 3*pio2_1t;
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| 100 | return -3;
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| 101 | }
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| 102 | } else {
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| 103 | if (ix == 0x401921fb) /* |x| ~= 4pi/2 */
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| 104 | goto medium;
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| 105 | if (!sign) {
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| 106 | z = x - 4*pio2_1;
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| 107 | y[0] = z - 4*pio2_1t;
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| 108 | y[1] = (z-y[0]) - 4*pio2_1t;
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| 109 | return 4;
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| 110 | } else {
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| 111 | z = x + 4*pio2_1;
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| 112 | y[0] = z + 4*pio2_1t;
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| 113 | y[1] = (z-y[0]) + 4*pio2_1t;
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| 114 | return -4;
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| 115 | }
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| 116 | }
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| 117 | }
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| 118 | if (ix < 0x413921fb) { /* |x| ~< 2^20*(pi/2), medium size */
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| 119 | medium:
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| 120 | /* rint(x/(pi/2)), Assume round-to-nearest. */
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| 121 | fn = (double_t)x*invpio2 + toint - toint;
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| 122 | n = (int32_t)fn;
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| 123 | r = x - fn*pio2_1;
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| 124 | w = fn*pio2_1t; /* 1st round, good to 85 bits */
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| 125 | y[0] = r - w;
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| 126 | u.f = y[0];
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| 127 | ey = u.i>>52 & 0x7ff;
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| 128 | ex = ix>>20;
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| 129 | if (ex - ey > 16) { /* 2nd round, good to 118 bits */
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| 130 | t = r;
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| 131 | w = fn*pio2_2;
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| 132 | r = t - w;
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| 133 | w = fn*pio2_2t - ((t-r)-w);
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| 134 | y[0] = r - w;
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| 135 | u.f = y[0];
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| 136 | ey = u.i>>52 & 0x7ff;
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| 137 | if (ex - ey > 49) { /* 3rd round, good to 151 bits, covers all cases */
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| 138 | t = r;
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| 139 | w = fn*pio2_3;
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| 140 | r = t - w;
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| 141 | w = fn*pio2_3t - ((t-r)-w);
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| 142 | y[0] = r - w;
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| 143 | }
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| 144 | }
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| 145 | y[1] = (r - y[0]) - w;
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| 146 | return n;
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| 147 | }
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| 148 | /*
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| 149 | * all other (large) arguments
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| 150 | */
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| 151 | if (ix >= 0x7ff00000) { /* x is inf or NaN */
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| 152 | y[0] = y[1] = x - x;
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| 153 | return 0;
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| 154 | }
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| 155 | /* set z = scalbn(|x|,-ilogb(x)+23) */
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| 156 | u.f = x;
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| 157 | u.i &= (uint64_t)-1>>12;
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| 158 | u.i |= (uint64_t)(0x3ff + 23)<<52;
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| 159 | z = u.f;
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| 160 | for (i=0; i < 2; i++) {
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| 161 | tx[i] = (double)(int32_t)z;
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| 162 | z = (z-tx[i])*0x1p24;
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| 163 | }
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| 164 | tx[i] = z;
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| 165 | /* skip zero terms, first term is non-zero */
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| 166 | while (tx[i] == 0.0)
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| 167 | i--;
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| 168 | n = __rem_pio2_large(tx,ty,(int)(ix>>20)-(0x3ff+23),i+1,1);
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| 169 | if (sign) {
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| 170 | y[0] = -ty[0];
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| 171 | y[1] = -ty[1];
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| 172 | return -n;
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| 173 | }
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| 174 | y[0] = ty[0];
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| 175 | y[1] = ty[1];
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| 176 | return n;
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| 177 | }
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