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#include <math.h> #include <sunmath.h> main() { union { floatflt; unsignedun; } r; union { doubledbl; unsignedun[2]; } d; /* ÇÜÀºÅÙ */ d.dbl = M_PI; (void) printf("DP Approx pi = %08x %08x = %18.17e \n", d.un[0], d.un[1], d.dbl); /* ñÀºÅÙ */ r.flt = infinityf(); (void) printf("Single Precision %8.7e : %08x \n", r.flt, r.un); exit(0); }

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DP Approx pi = 400921fb 54442d18 = 3.14159265358979312e+00 Single Precision Infinity : 7f800000

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program print_ieee_values c c implicit ʸ¤Ï¡¢f77_floatingpoint µ¿»÷ÁȤ߹þ¤ß´Ø¿ô¤¬¡¢ c Àµ¤·¤¤·¿¤ÇÀë¸À¤µ¤ì¤Æ¤¤¤ë¤«¤ò³Îǧ¤·¤Þ¤¹¡£ c implicit real*16 (q) implicit double precision (d) implicit real (r) real*16 z double precision x real r c z = q_min_normal() write(*,7) z, z 7 format('min normal, quad: ',1pe47.37e4,/,' in hex ',z32.32) c x = d_min_normal() write(*,14) x, x 14 format('min normal, double: ',1pe23.16,' in hex ',z16.16) c r = r_min_normal() write(*,27) r, r 27 format('min normal, single: ',1pe14.7,' in hex ',z8.8) c end

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min normal, quad: 3.3621031431120935062626778173217526026E- 4932 in hex 00010000000000000000000000000000 min normal, double:2.2250738585072014-308 in hex 0010000000000000 min normal, single:1.1754944E-38 in hex 00800000

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#ifdef DP #define GENERIC double precision #else #define GENERIC real #endif #define SIZE 400000 program example c implicit GENERIC (a-h,o-z) GENERIC x(SIZE), y, lb, ub real time(2), u1, u2 c c [-ln2/2,ln2/2] ¤Ë¤ª¤±¤ëÍð¿ô¤Ç¤Î EXP ¤ò·×»» lb = -0.3465735903 ub = 0.3465735903 c c Íð¿ô¤ÎÇÛÎó¤òÀ¸À® #ifdef DP call d_init_addrans() call d_addrans(x,SIZE,lb,ub) #else call r_init_addrans() call r_addrans(x,SIZE,lb,ub) #endif c c ¥¯¥í¥Ã¥¯³«»Ï call dtime(tarray) u1 = tarray(1) c c »Ø¿ô¤ò·×»» do 16 i=1,SIZE y = exp(x(i)) 16 continue c c ·Ð²á»þ´Ö¤Î¼èÆÀ call dtime(time) u2 = tarray(1) print *,'time used by EXP is ',u2-u1 print *,'last values for x and exp(x) are ',x(SIZE),y c call flush(6) end

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/* * sin ¤Ø¤Î SIZE*LOOPS Íð¿ô°ú¿ô¤ò¥Æ¥¹¥È¤·¤Þ¤¹¡£ * ÈÏ°Ï¤Ï [0¡¢¤·¤­¤¤ÃÍ ] ¤È¤·¡¢ * ¤·¤­¤¤ÃÍ = 3E30000000000000 (3.72529029846191406e-09) ¤È¤·¤Þ¤¹¡£ */ #include <math.h> #include <sunmath.h> #define SIZE 10000 #define LOOPS 100 int main() { double x[SIZE], y[SIZE]; int i, j, n; double lb, ub; union { unsignedu[2]; doubled; } upperbound; upperbound.u[0] = 0x3e300000; upperbound.u[1] = 0x00000000; /* Íð¿ô´Ø¿ô¤ò½é´ü²½ */ d_init_addrans_(); /* sin ¤Ë¤Ä¤¤¤Æ (SIZE * LOOPS) ¸Ä¤Î°ú¿ô¤ò¥Æ¥¹¥È */ for (j = 0; j < LOOPS; j++) { /* * Ťµ SIZE ¤ÎÍð¿ô¤Î¥Ù¥¯¥È¥ë x ¤òÀ¸À®¤·¡¢ * »°³Ñ´Ø¿ô¤Î´Ø¿ô¤Ø¤ÎÆþÎϤȤ·¤Æ»ÈÍÑ */ n = SIZE; ub = upperbound.d; lb = 0.0; d_addrans_(x, &n, &lb, &ub); for (i = 0; i < n; i++) y[i] = sin(x[i]); /*sin(x) == x ¡©¡¡ x ¤¬¾®¤µ¤¤¾ì¹ç¤Ë¤Ê¤ë */ for (i = 0; i < n; i++) if (x[i] != y[i]) printf( " OOPS: %d sin(%18.17e)=%18.17e \n", i, x[i], y[i]); } printf(" comparison ended; no differences\n"); ieee_retrospective_(); return(0); }

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c

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cieee_functions(3m)

clibm_double(3f)

clibm_single(3f)

c

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c

cilogb(x): x ¤Î´ð¿ô 2 ¤Ç¤Î¥Ð¥¤¥¢¥¹¤µ¤ì¤Æ¤¤¤Ê¤¤»Ø¿ô¤òÀ°¿ô·Á¼°¤ÇÊÖ¤¹

csignbit(x): 0 ¤Þ¤¿¤Ï 1 ¤ÎÉä¹æ¥Ó¥Ã¥È¤òÊÖ¤¹

ccopysign(x,y): y ¤ÎÉä¹æ¥Ó¥Ã¥È¤òÉÕ¤±¤¿ x ¤òÊÖ¤¹

cnextafter(x,y): x ¤«¤é y Êý¸þ¤Ë¼¡¤Ë½Ð¸½¤¹¤ëɽ¸½²Äǽ¤Ê¿ôÃͤòÊÖ¤¹

cscalbn(x,n): x * 2**n

c

c´Ø¿ôÇÜÀºÅÙñÀºÅÙ

c--------------------------------------------------------

cilogb(x)i = id_ilogb(x)i = ir_ilogb(r)

csignbit(x)i = id_signbit(x)i = ir_signbit(r)

ccopysign(x,y)x = d_copysign(x,y)r = r_copysign(r,s)

cnextafter(x,y)z = d_nextafter(x,y)r = r_nextafter(r,s)

cscalbn(x,n)x = d_scalbn(x,n)r = r_scalbn(r,n)

c

program ieee_functions_demo

implicit double precision (d)

implicit real (r)

double precision x, y, z, direction

real r, s, t, r_direction

integer i, scale

print *

print *, 'DOUBLE PRECISION EXAMPLES:'

print *

x = 32.0d0

i = id_ilogb(x)

write(*,1) x, i

1format(' The base 2 exponent of ', F4.1, ' is ', I2)

x = -5.5d0

y = 12.4d0

z = d_copysign(x,y)

write(*,2) x, y, z

2 format(F5.1, ' was given the sign of ', F4.1,

* ' and is now ', F4.1)

x = -5.5d0

i = id_signbit(x)

print *, 'The sign bit of ', x, ' is ', i

x = d_min_subnormal()

direction = -d_infinity()

y = d_nextafter(x, direction)

write(*,3) x

3format(' Starting from ', 1PE23.16E3,

- ', the next representable number ')

write(*,4) direction, y

4format(' towards ', F4.1, ' is ', 1PE23.16E3)

x = d_min_subnormal()

direction = 1.0d0

y = d_nextafter(x, direction)

write(*,3) x

write(*,4) direction, y

x = 2.0d0

scale = 3

y = d_scalbn(x, scale)

write (*,5) x, scale, y

5format(' Scaling ', F4.1, ' by 2**', I1, ' is ', F4.1)

print *

print *, 'SINGLE PRECISION EXAMPLES:'

print *

r = 32.0

i = ir_ilogb(r)

write (*,1) r, i

r = -5.5

i = ir_signbit(r)

print *, 'The sign bit of ', r, ' is ', i

r = -5.5

s = 12.4

t = r_copysign(r,s)

write (*,2) r, s, t

r = r_min_subnormal()

r_direction = -r_infinity()

s = r_nextafter(r, r_direction)

write(*,3) r

write(*,4) r_direction, s

r = r_min_subnormal()

r_direction = 1.0e0

s = r_nextafter(r, r_direction)

write(*,3) r

write(*,4) r_direction, s

r = 2.0

scale = 3

s = r_scalbn(r, scale)

write (*,5) r, scale, y

print *

end

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DOUBLE PRECISION EXAMPLES: The base 2 exponent of 32.0 is 5 -5.5 was given the sign of 12.4 and is now 5.5 The sign bit of -5.5000000000000 is 1 Starting from 4.9406564584124654-324, the next representable number towards -Inf is 0.0000000000000000E+000 Starting from 4.9406564584124654-324, the next representable number towards 1.0 is 9.8813129168249309-324 Scaling 2.0 by 2**3 is 16.0 SINGLE PRECISION EXAMPLES: The base 2 exponent of 32.0 is 5 The sign bit of -5.50000 is 1 -5.5 was given the sign of 12.4 and is now 5.5 Starting from 1.4012984643248171E-045, the next representable number towards -Inf is 0.0000000000000000E+000 Starting from 1.4012984643248171E-045, the next representable number towards 1.0 is 2.8025969286496341E-045 Scaling 2.0 by 2**3 is 16.0 [ÆüËܸìÌõ] Ãí: °Ê²¼¤Î IEEE ÉâÆ°¾®¿ôÅÀÎã³°¤¬È¯À¸¤·¤Þ¤·¤¿: ÉÔÀµ³Î¡¢¥¢¥ó¥À¡¼¥Õ¥í¡¼ ¾ÜºÙ¤Ï¡¢¡Ø¿ôÃÍ·×»»¥¬¥¤¥É¡Ù¤Î ieee_flags

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#include <math.h> #include <sunmath.h> int main() { double x; float r; x = quiet_nan(0); printf("quiet NaN: %.16e = %08x %08x \n", x, ((int *) &x)[0], ((int *) &x)[1]); x = nextafter(max_subnormal(), 0.0); printf("nextafter(max_subnormal,0) = %.16e\n",x); printf(" = %08x %08x\n", ((int *) &x)[0], ((int *) &x)[1]); r = min_subnormalf(); printf("single precision min subnormal = %.8e = %08x\n", r, ((int *) &r)[0]); return 0; }

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quiet NaN: NaN = 7fffffff ffffffff nextafter(max_subnormal,0) = 2.2250738585072004e-308 = 000fffff fffffffe single precision min subnormal = 1.40129846e-45 = 00000001

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quiet NaN: NaN = ffffffff 7fffffff nextafter(max_subnormal,0) = 2.2250738585072004e-308 = fffffffe 000fffff single precision min subnormal = 1.40129846e-45 = 00000001

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program print_ieee_values c c implicit ʸ¤Ï¡¢f77_floatingpoint ¤Îµ¿»÷ÁȤ߹þ¤ß´Ø¿ô¤¬¡¢ c Àµ¤·¤¤·¿¤ÇÀë¸À¤µ¤ì¤Æ¤¤¤ë¤«¤É¤¦¤«¤ò³Îǧ¤·¤Þ¤¹¡£ c implicit real*16 (q) implicit double precision (d) implicit real (r) real*16 z, zero, one double precision x real r c zero = 0.0 one = 1.0 z = q_nextafter(zero, one) x = d_infinity() r = r_max_normal() c print *, z print *, x print *, r c end

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#include <math.h> #include <sunmath.h> int main() { int i; double x, y; char *out_1, *out_2, *dummy; /* °ìÈÌŪ¤Ê´Ý¤á¤ÎÊý¸þ¤ò¼èÆÀ¤¹¤ë */ i = ieee_flags("get", "direction", "", &out_1); x = sqrt(.5); printf("With rounding direction %s, \n", out_1); printf("sqrt(.5) = 0x%08x 0x%08x = %16.15e\n", ((int *) &x)[0], ((int *) &x)[1], x); /* ´Ý¤áÊý¸þ¤ÎÀßÄê */ if (ieee_flags("set", "direction", "tozero", &dummy) != 0) printf("Not able to change rounding direction!\n"); i = ieee_flags("get", "direction", "", &out_2); x = sqrt(.5); /* * printf ¤â¸½ºß¤Î´Ý¤áÊý¸þ¤Ë±Æ¶Á¤ò¼õ¤±¤ë¤¿¤á¡¢ * printf ¤ÎÁ°¤Ë¸µ¤Î´Ý¤áÊý¸þ¤òÉü¸µ¤¹¤ë */ if (ieee_flags("set", "direction", out_1, &dummy) != 0) printf("Not able to change rounding direction!\n"); printf("\nWith rounding direction %s,\n", out_2); printf("sqrt(.5) = 0x%08x 0x%08x = %16.15e\n", ((int *) &x)[0], ((int *) &x)[1], x); return 0; }

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demo%cc rounding_direction.c -lm demo%a.out With rounding direction nearest, sqrt(.5) = 0x3fe6a09e 0x667f3bcd = 7.071067811865476e-01 With rounding direction tozero, sqrt(.5) = 0x3fe6a09e 0x667f3bcc = 7.071067811865475e-01 demo%

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demo%cc rounding_direction.c -lm demo%a.out With rounding direction nearest, sqrt(.5) = 0x667f3bcd 0x3fe6a09e = 7.071067811865476e-01 With rounding direction tozero, sqrt(.5) = 0x667f3bcc 0x3fe6a09e = 7.071067811865475e-01 demo%

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program ieee_flags_demo character*16 out i = ieee_flags("set", "direction", "tozero", out) if (i.ne.0) print *, 'not able to set rounding direction' i = ieee_flags("get", "direction", "", out) print *, "Rounding direction is: ", out end

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Note: Rounding direction toward zero. See the Numerical Computation Guide, ieee_flags(3M) [ÆüËܸìÌõ] Ãí: ¥¼¥íÊý¸þ¤Ø¤Î´Ý¤á ¾ÜºÙ¤Ï¡¢¡Ø¿ôÃÍ·×»»¥¬¥¤¥É¡Ù¤Îieee_flags

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#include <stdio.h>

#include <math.h>

#include <sunmath.h>

#include <fenv.h>

/*

* Á᤹¤®¤ë¥¢¥ó¥À¡¼¥Õ¥í¡¼¤Þ¤¿¤Ï¥ª¡¼¥Ð¡¼¥Õ¥í¡¼¤òÈò¤±¤ë¥Ù¥¯¥È¥ë x ¤Î

* ¥æ¡¼¥¯¥ê¥Ã¥ÉÀµµ¬·Á¤ò·×»»¤·¤Þ¤¹

*/

double norm(int n, double *x)

{

fenv_t env;

double s, b, d, t;

int i, f;

/* ´Ä¶­¤òÊݸ¤·¤Æ¥Õ¥é¥°¤ò¥¯¥ê¥¢¤·¡¢Ï¢Â³¤·¤¿Îã³°½èÍý¤ò³ÎΩ¤·¤Þ¤¹ */

feholdexcept(&env);

/* ¥É¥Ã¥ÈÀÑ x.x ¤Î·×»»¤ò»î¤ß¤Þ¤¹ */

d = 1.0; /* ·å°ÜÆ°»Ò */

s = 0.0;

for (i = 0; i < n; i++)

s += x[i] * x[i];

/* ¥¢¥ó¥À¡¼¥Õ¥í¡¼¤Þ¤¿¤Ï¥ª¡¼¥Ð¡¼¥Õ¥í¡¼¤òÄ´¤Ù¤Þ¤¹ */

f = fetestexcept(FE_UNDERFLOW | FE_OVERFLOW);

if (f & FE_OVERFLOW) {

/* ºÇ½é¤Î»î¹Ô¤¬¥ª¡¼¥Ð¡¼¥Õ¥í¡¼¤·¤¿¤é¡¢¥¹¥±¡¼¥ê¥ó¥°¤ò ¾®¤µ¤¯¤·¤Æ¤â¤¦°ìÅټ¹Ԥ·¤Æ¤¯¤À¤µ¤¤ */

feclearexcept(FE_OVERFLOW);

b = scalbn(1.0, -640);

d = 1.0 / b;

s = 0.0;

for (i = 0; i < n; i++) {

t = b * x[i];

s += t * t;

}

}

else if (f & FE_UNDERFLOW && s < scalbn(1.0, -970)) {

/* ºÇ½é¤Î»î¹Ô¤¬¥¢¥ó¥À¡¼¥Õ¥í¡¼¤·¤¿¤é¡¢¥¹¥±¡¼¥ê¥ó¥°¤ò Â礭¤¯¤·¤Æ¤â¤¦°ìÅټ¹Ԥ·¤Æ¤¯¤À¤µ¤¤ */

b = scalbn(1.0, 1022);

d = 1.0 / b;

s = 0.0;

for (i = 0; i < n; i++) {

t = b * x[i];

s += t * t;

}

}

/* ¤³¤ì¤Þ¤Ç¤ËȯÀ¸¤·¤¿¥¢¥ó¥À¡¼¥Õ¥í¡¼¤ò¤¹¤Ù¤Æ±£¤·¤Þ¤¹ */

feclearexcept(FE_UNDERFLOW);

/* ´Ä¶­¤òÉü¸µ¤·¡¢¤³¤ì¤Þ¤Ç¤Ëµ¯¤­¤¿¤Û¤«¤ÎÎã³°¤òȯÀ¸¤µ¤»¤Þ¤¹ */

feupdateenv(&env);

/* Ê¿Êýº¬¤ò¼èÆÀ¤·¡¢¥¹¥±¡¼¥ê¥ó¥°¤ò²ò½ü¤·¤Þ¤¹ */

return d * sqrt(s);

}

int main()

{

double x[100], l, u;

int n = 100;

fex_set_log(stdout);

l = 0.0;

u = min_normal();

d_lcrans_(x, &n, &l, &u);

printf("norm: %g\n", norm(n, x));

l = sqrt(max_normal());

u = l * 2.0;

d_lcrans_(x, &n, &l, &u);

printf("norm: %g\n", norm(n, x));

return 0;

}

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demo%cc norm.c -R/opt/SUNWspro/lib -L/opt/SUNWspro/lib -lm9x -lsunmath -lm demo%a.out Floating point underflow at 0x000153a8 __d_lcrans_, nonstop mode 0x000153b4 __d_lcrans_ 0x00011594 main Floating point underflow at 0x00011244 norm, nonstop mode 0x00011248 norm 0x000115b4 main norm: 1.32533e-307 Floating point overflow at 0x00011244 norm, nonstop mode 0x00011248 norm 0x00011660 main norm: 2.02548e+155

¼¡¤ÎÎã¤Ï¡¢x86 ¤Ç¤Îfesetprec ´Ø¿ô¤Î·ë²Ì¤ò¼¨¤·¤Æ¤¤¤Þ¤¹ (¤³¤Î´Ø¿ô¤Ï SPARC ¤Ç¤Ï»ÈÍѤǤ­¤Þ¤»¤ó)¡£while ¥ë¡¼¥×¤Ï¡¢1 ¤Ë²Ã¤¨¤é¤ì¤ë¤È¤­¤Ë´°Á´¤Ë´Ý¤á¤é¤ì¤ë¡¢2 ¤ÎºÇÂç¤ÎÎß¾è¤òõ¤¹¤³¤È¤Ë¤è¤ê¡¢»ÈÍѤǤ­¤ëÀºÅ٤ηèÄê¤ò»î¤ß¤Æ¤¤¤Þ¤¹¡£ºÇ½é¤Î¥ë¡¼¥×¤¬¼¨¤¹¤è¤¦¤Ë¡¢¤¹¤Ù¤Æ¤ÎÃæ´Ö·ë²Ì¤ò³ÈÄ¥ÇÜÀºÅÙ¤Çɾ²Á¤¹¤ë x86 ¤Î¤è¤¦¤Ê¥¢¡¼¥­¥Æ¥¯¥Á¥ã¤Ç¤Ï¡¢¤³¤Î¼êË¡¤¬´üÂԤɤª¤ê¤ËÆ°ºî¤·¤Ê¤¤¾ì¹ç¤¬¤¢¤ê¤Þ¤¹¡£¤½¤Î¤¿¤á¡¢2 ÈÖÌܤΥ롼¥×¤¬¼¨¤¹¤è¤¦¤Ë¡¢fesetprec ´Ø¿ô¤ò»ÈÍѤ·¤Æ¡¢¤¹¤Ù¤Æ¤Î·ë²Ì¤¬´õ˾¤¹¤ëÀºÅ٤˴ݤá¤é¤ì¤ë¤è¤¦¤Ë»ØÄê¤Ç¤­¤Þ¤¹¡£

#include <math.h>

#include <fenv.h>

int main()

{

double x;

x = 1.0;

while (1.0 + x != 1.0)

x *= 0.5;

printf("%d significant bits\n", -ilogb(x));

fesetprec(FE_DBLPREC);

x = 1.0;

while (1.0 + x != 1.0)

x *= 0.5;

printf("%d significant bits\n", -ilogb(x));

return 0;

}

x86 ¥·¥¹¥Æ¥à¤Ç¤Ï¡¢¤³¤Î¥×¥í¥°¥é¥à¤Î½ÐÎϤϼ¡¤Î¤è¤¦¤Ë¤Ê¤ê¤Þ¤¹¡£

64 significant bits 53 significant bits

¼¡¤Î¥³¡¼¥É¥Õ¥é¥°¥á¥ó¥È¤Ï¡¢¥Þ¥ë¥Á¥¹¥ì¥Ã¥ÉÂбþ¤Î¥×¥í¥°¥é¥à¤Ç´Ä¶­´Ø¿ô¤ò»ÈÍѤ·¤Æ¡¢¿Æ¥¹¥ì¥Ã¥É¤«¤é»Ò¥¹¥ì¥Ã¥É¤ËÉâÆ°¾®¿ôÅÀ¥â¡¼¥É¤òÅÁ¤¨¡¢»Ò¥¹¥ì¥Ã¥É¤¬¿Æ¥¹¥ì¥Ã¥É¤Ë·ë¹ç¤µ¤ì¤ë¤È¤­¤Ë»Ò¥¹¥ì¥Ã¥ÉÆâ¤ÇȯÀ¸¤¹¤ëÎã³°¥Õ¥é¥°¤ò²óÉü¤¹¤ëÊýË¡¤ò¼¨¤·¤Æ¤¤¤Þ¤¹ (¥Þ¥ë¥Á¥¹¥ì¥Ã¥ÉÂбþ¤Î¥×¥í¥°¥é¥à¤Îµ­½Ò¤Ë¤Ä¤¤¤Æ¤Ï¡¢¡ØSolaris ¤Ç¤Î¥Þ¥ë¥Á¥¹¥ì¥Ã¥É¤Î¥×¥í¥°¥é¥ß¥ó¥°¡Ù¤ò»²¾È)¡£

#include <thread.h>

#include <fenv.h>

fenv_t env;

void child(void *p)

{

/* Æþ¸ý¤Ç¿Æ¤Î´Ä¶­¤ò·Ñ¾µ¤·¤Þ¤¹ */

fesetenv(&env);

...

/* ½ªÎ»Á°¤Ë»Ò¤Î´Ä¶­¤òÊݸ¤·¤Þ¤¹ */

fegetenv(&env);

}

void parent()

{

thread_t tid;

void *arg;

...

/* »Ò¤òºîÀ®¤¹¤ëÁ°¤Ë¿Æ¤Î´Ä¶­¤òÊݸ¤·¤Þ¤¹ */

fegetenv(&env);

thr_create(NULL, NULL, child, arg, NULL, &tid);

...

/* »Ò¤È·ë¹ç¤·¤Þ¤¹ */

thr_join(tid, NULL, &arg);

/* »Ò¤ÇȯÀ¸¤·¤¿Îã³°¥Õ¥é¥°¤ò¿Æ¤Î´Ä¶­¤Ë¥Þ¡¼¥¸¤·¤Þ¤¹ */

fex_merge_flags(&env);

...

}

°ìÈÌŪ¤Ë¤Ï¡¢Îã³°¥Ó¥Ã¥È¤Î¸¡¾Ú¤ä¥¯¥ê¥¢¤Ï¥æ¡¼¥¶¡¼¥×¥í¥°¥é¥à¤Ç¹Ô¤¤¤Þ¤¹¡£¼«Á³È¯À¸¤·¤¿Îã³°¥Õ¥é¥°¤ò¸¡¾Ú¤¹¤ë C ¥×¥í¥°¥é¥à¤ò¼¨¤·¤Þ¤¹¡£

#include <sunmath.h> #include <sys/ieeefp.h> int main() { int code, inexact, division, underflow, overflow, invalid; double x; char *out; /* ¥¢¥ó¥À¡¼¥Õ¥í¡¼Îã³°¤òȯÀ¸¤µ¤»¤ë */ x = max_subnormal() / 2.0; /* ¤³¤Îʸ¤Ï¡¢Á°¤Îʸ¤¬ºÇŬ²½¤µ¤ì¤Æ¤¤¤Ê¤¤¤³¤È¤ò¼¨¤¹ */ printf("x = %g\n",x); /* ¤É¤ÎÎã³°¤¬È¯À¸¤·¤¿¤«¤òȽÄꤹ¤ë */ code = ieee_flags("get", "exception", "", &out); /* Ìá¤êÃͤò²òÆɤ¹¤ë */ inexact = (code >> fp_inexact) & 0x1; underflow = (code >> fp_underflow) & 0x1; division = (code >> fp_division) & 0x1; overflow = (code >> fp_overflow) & 0x1; invalid = (code >> fp_invalid) & 0x1; /* "out" ¤ÏȯÀ¸¤·¤¿Îã³°¤Î¤¦¤Á¤ÇºÇ¤âÍ¥ÀèÅÙ¤¬¹â¤¤ */ printf(" Highest priority exception is: %s\n", out); /* 1 ¤ÏÎã³°¤¬È¯À¸¤·¤¿¤³¤È¤ò¼¨¤·¡¢ */ /* 0 ¤ÏÎã³°¤¬È¯À¸¤·¤Æ¤¤¤Ê¤¤¤³¤È¤ò¼¨¤¹ */ printf("%d %d %d %d %d\n", invalid, overflow, division, underflow, inexact); ieee_retrospective_(); return(0); }

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demo%a.out x = 1.11254e-308 Highest priority exception is: underflow 0 0 0 1 1 Inexact; Underflow; See the Numerical Computation Guide, ieee_flags(3M) [ÆüËܸìÌõ] Ãí: °Ê²¼¤Î IEEE ÉâÆ°¾®¿ôÅÀÎã³°¤¬È¯À¸¤·¤Þ¤·¤¿: ÉÔÀµ³Î¡¢¥¢¥ó¥À¡¼¥Õ¥í¡¼ ¾ÜºÙ¤Ï¡¢¡Ø¿ôÃÍ·×»»¥¬¥¤¥É¡Ù¤Îieee_flags

FORTRAN ¤Ç¤âƱ¤¸¤³¤È¤¬¹Ô¤¨¤Þ¤¹¡£

/* ¤³¤ì¤Ï¡¢¼¡¤Î¤è¤¦¤ÊFORTRAN ¥×¥í¥°¥é¥àÎ㠤Ǥ¹¡§ * ¥¢¥ó¥À¡¼¥Õ¥í¡¼Îã³°¤òȯÀ¸¤µ¤»¤ë *ieee_flags ¤ò»ÈÍѤ·¤Æ¡¢¤É¤ÎÎã³°¤¬È¯À¸¤·¤¿¤«¤òȽÄꤹ¤ë *ieee_flags ¤¬ÊÖ¤¹À°¿ôÃͤò²òÆɤ¹¤ë * ̤½èÍý¤ÎÎã³°¤ò¤¹¤Ù¤Æ¥¯¥ê¥¢¤Ë¤¹¤ë C ¤Î¥×¥ê¥×¥í¥»¥Ã¥µ¤¬µ¯Æ°¤µ¤ì¡¢¥Ø¥Ã¥À¡¼¥Õ¥¡¥¤¥ëf77_floatingpoint.h ¤¬¼è¤ê¹þ¤Þ¤ì¤ë¤è¤¦¤Ë¤¹¤ë¤¿¤á¤Ë¡¢¤³¤Î¥×¥í¥°¥é¥à¤ÏÀÜÈø¼­.F ¤Î¥Õ¥¡¥¤¥ë¤ËÊݸ¤·¤Æ¤¯¤À¤µ¤¤¡£ */ #include <f77_floatingpoint.h> program decode_accrued_exceptions double precision x integer accrued, inx, div, under, over, inv character*16 out double precision d_max_subnormal c ¥¢¥ó¥À¡¼¥Õ¥í¡¼Îã³°¤òȯÀ¸¤µ¤»¤ë x = d_max_subnormal() / 2.0 c ¤É¤ÎÎã³°¤¬È¯À¸¤·¤¿¤«¤òȽÄꤹ¤ë accrued = ieee_flags('get', 'exception', '', out) c ¥Ó¥Ã¥È¥·¥Õ¥È¤ÎÁȤ߹þ¤ß¤ò»ÈÍѤ·¤Æ¡¢ieee_flags ¤¬ÊÖ¤·¤¿Ãͤò²òÆɤ¹¤ë inx = and(rshift(accrued, fp_inexact) , 1) under = and(rshift(accrued, fp_underflow), 1) div = and(rshift(accrued, fp_division) , 1) over = and(rshift(accrued, fp_overflow) , 1) inv = and(rshift(accrued, fp_invalid) , 1) c ºÇ¹â¤ÎÍ¥ÀèÅÙ¤ò¤â¤ÄÎã³°¤¬ "out" ¤ËÊÖ¤µ¤ì¤ë print *, "Highest priority exception is ", out c1 ¤ÏÎã³°¤¬È¯À¸¤·¤¿¤³¤È¤ò¼¨¤·¡¢0 ¤ÏÎã³°¤¬È¯À¸¤·¤Æ¤¤¤Ê¤¤¤³¤È¤ò¼¨¤¹ print *, inv, over, div, under, inx c ̤½èÍý¤ÎÎã³°¤ò¤¹¤Ù¤Æ¥¯¥ê¥¢¤Ë¤¹¤ë i = ieee_flags('clear', 'exception', 'all', out) end

¤³¤Î FORTRAN ¥×¥í¥°¥é¥à¤Î½ÐÎϤǤ¹¡£

Highest priority exception is underflow 0 0 0 1 1

Ä̾¥æ¡¼¥¶¡¼¥×¥í¥°¥é¥à¤ÇÎã³°¥Õ¥é¥°¤òÀßÄꤹ¤ë¤³¤È¤Ï¤¢¤ê¤Þ¤»¤ó¤¬¡¢¤Ç¤­¤Ê¤¤¤³¤È¤Ç¤Ï¤¢¤ê¤Þ¤»¤ó¡£°Ê²¼¤Î C ¤ÎÎã¤Ç¤½¤ì¤ò¼¨¤·¤Þ¤¹¡£

#include <sunmath.h> int main() { int code; char *out; if (ieee_flags("clear", "exception", "all", &out) != 0) printf("could not clear exceptions\n"); if (ieee_flags("set", "exception", "division", &out) != 0) printf("could not get exception\n"); code = ieee_flags("get", "exception", "", &out); printf("out is: %s , fp exception code is: %X \n", out, code); return(0); }

SPARC ¤Ç¤Ï¡¢Á°µ­¤Î¥×¥í¥°¥é¥à¤Î½ÐÎϤϼ¡¤Î¤è¤¦¤Ë¤Ê¤ê¤Þ¤¹¡£

out is: division , fp exception code is: 2

x86 ¤Ç¤Ï¡¢½ÐÎϤϼ¡¤Î¤è¤¦¤Ë¤Ê¤ê¤Þ¤¹¡£

out is: division , fp exception code is: 4

---

Ãí -°Ê²¼¤ÎÎã¤Ï¡¢Solaris ¥ª¥Ú¥ì¡¼¥Æ¥£¥ó¥°´Ä¶­¤À¤±¤ËŬÍѤǤ­¤Þ¤¹¡£

---

SPARC ¤Ç¤Ï¡¢Îã³°¤òÆÃÄꤹ¤ë¤¿¤á¤Î¥·¥°¥Ê¥ë¥Ï¥ó¥É¥é¤òÀßÄꤹ¤ë FORTRAN ¤ª¤è¤Ó C ¥×¥í¥°¥é¥à¤ÎÎã¤ò¼¨¤·¤Þ¤¹¡£

FORTRAN ¥×¥í¥°¥é¥àÎã

program demo

c ¥·¥°¥Ê¥ë¥Ï¥ó¥É¥é´Ø¿ô¤ÎÀë¸À

external fp_exc_hdl

double precision d_min_normal

double precision x

c ¥·¥°¥Ê¥ë¥Ï¥ó¥É¥é¤ÎÀßÄê

i = ieee_handler(`set', `common', fp_exc_hdl)

if (i.ne.0) print *, `ieee trapping not supported here'

c ¥¢¥ó¥À¡¼¥Õ¥í¡¼Îã³°¤òȯÀ¸¤µ¤»¤ë (¥È¥é¥Ã¥×¤Ï¤µ¤ì¤Ê¤¤)

x = d_min_normal() / 13.0

print *, `d_min_normal() / 13.0 = `, x

c ¥ª¡¼¥Ð¡¼¥Õ¥í¡¼Îã³°¤òȯÀ¸¤µ¤»¤ë

c ½ÐÎϤµ¤ì¤¿ÃͤϷë²Ì¤È¤Ï´Ø·¸¤Ê¤¤

x = 1.0d300*1.0d300

print *, `1.0d300*1.0d300 = `, x

end

c

c ÉâÆ°¾®¿ôÅÀÎã³°½èÍý´Ø¿ô

c

integer function fp_exc_hdl(sig, sip, uap)

integer sig, code, addr

character label*16

c

c ¹½Â¤ÂÎ /siginfo/ ¤Ï <sys/siginfo.h> ¤Ë¤è¤ë siginfo_t ¤Î²ò¼á

c

structure /fault/

integer address

end structure

structure /siginfo/

integer si_signo

integer si_code

integer si_errno

record /fault/ fault

end structure

record /siginfo/ sip

c FPE ¥³¡¼¥É¤Î°ìÍ÷¤Ï <sys/machsig.h> ¤ò»²¾È

c SIGFPE ̾¤ò¸¡½Ð

code = sip.si_code

if (code.eq.3) label = 'division'

if (code.eq.4) label = 'overflow'

if (code.eq.5) label = 'underflow'

if (code.eq.6) label = 'inexact'

if (code.eq.7) label = 'invalid'

addr = sip.fault.address

c ȯÀ¸¤·¤¿¥·¥°¥Ê¥ë¤Ë´Ø¤¹¤ë¾ðÊó¤ò½ÐÎÏ

write (*,77) code, label, addr

77 format ('floating-point exception code ', i2, ',',

* a17, ',', ' at address ', z8 )

end

½ÐÎϤϼ¡¤Î¤È¤ª¤ê¤Ç¤¹¡£

d_min_normal() / 13.0 = 1.7115952757748-309 floating-point exception code 4, overflow , at address 1131C 1.0d300*1.0d300 = 1.0000000000000+300 Inexact; Underflow; IEEE floating-point exception traps enabled: overflow; division by zero; invalid operation; [ÆüËܸìÌõ] Ãí: °Ê²¼¤Î IEEE ÉâÆ°¾®¿ôÅÀÎã³°¤¬È¯À¸¤·¤Þ¤·¤¿: ÉÔÀµ³Î¡¢¥¢¥ó¥À¡¼¥Õ¥í¡¼ °Ê²¼¤Î IEEE ÉâÆ°¾®¿ôÅÀÎã³°¤Î¥È¥é¥Ã¥×¤¬Í­¸ú¤Ç¤¹: ¥ª¡¼¥Ð¡¼¥Õ¥í¡¼¡¢¥¼¥í¤Ë¤è¤ë½ü»»¡¢Ìµ¸ú¤Ê±é»» ¾ÜºÙ¤Ï¡¢¡Ø¿ôÃÍ·×»»¥¬¥¤¥É¡Ù¤Îieee_flagsieee_handler

C ¥×¥í¥°¥é¥àÎã

/*

* 5 ¤Ä¤Î IEEE Îã³° (̵¸ú¤Ê±é»»¡¢¥¼¥í½ü»»¡¢¥ª¡¼¥Ð¡¼¥Õ¥í¡¼¡¢

* ¥¢¥ó¥À¡¼¥Õ¥í¡¼¡¢ÉÔÀµ³Î·ë²Ì) ¤òȯÀ¸¤µ¤»¤Þ¤¹¡£

* ¤¹¤Ù¤Æ¤ÎÉâÆ°¾®¿ôÅÀÎã³°¤ò¥È¥é¥Ã¥×¤·¡¢ * ¥á¥Ã¥»¡¼¥¸¤ò½ÐÎϤ·¤Æ¤«¤é½èÍý¤ò³¤±¤Þ¤¹¡£ * i = ieee("get","exception","",&out); * ¾åµ­¤Î¼°¤Ë¤è¤Ã¤ÆȯÀ¸¤·¤¿Îã³°¤Ë¤Ä¤¤¤ÆÌ䤤¹ç¤ï¤»¤ò¤¹¤ë¤³¤È¤â²Äǽ¤Ç¤¹¡£ * ȯÀ¸¤·¤¿ºÇ¹â¤ÎÎ㳰̾¤¬´Þ¤Þ¤ì¡¢i ¤Ï¤¹¤Ù¤Æ¤ÎÎã³°¤ò¸¡½Ð¤¹¤ë¤è¤¦¤Ë * ²ò¼á¤µ¤ì¤Þ¤¹¡£

*/

#include <sunmath.h>

#include <signal.h>

#include <siginfo.h>

#include <ucontext.h>

extern void trap_all_fp_exc(int sig, siginfo_t *sip,

ucontext_t *uap);

int main()

{

doublex, y, z;

char*out;

/*

* Use ieee_handler to establish "trap_all_fp_exc"

* as the signal handler to use whenever any floating

* point exception occurs.

*/

if (ieee_handler("set", "all", trap_all_fp_exc) != 0)

printf(" IEEE trapping not supported here.\n");

/* disable trapping (uninteresting) inexact exceptions */

if (ieee_handler("set", "inexact", SIGFPE_IGNORE) != 0)

printf("Trap handler for inexact not cleared.\n");

/* raise invalid */

if (ieee_flags("clear", "exception", "all", &out) != 0)

printf(" could not clear exceptions\n");

printf("1. Invalid: signaling_nan(0) * 2.5\n");

x = signaling_nan(0);

y = 2.5;

z = x * y;

/* raise division */

if (ieee_flags("clear", "exception", "all", &out) != 0)

printf(" could not clear exceptions\n");

printf("2. Div0: 1.0 / 0.0\n");

x = 1.0;

y = 0.0;

z = x / y;

/* raise overflow */

if (ieee_flags("clear", "exception", "all", &out) != 0)

printf(" could not clear exceptions\n");

printf("3. Overflow: -max_normal() - 1.0e294\n");

x = -max_normal();

y = -1.0e294;

z = x + y;

/* raise underflow */

if (ieee_flags("clear", "exception", "all", &out) != 0)

printf(" could not clear exceptions\n");

printf("4. Underflow: min_normal() * min_normal()\n");

x = min_normal();

y = x;

z = x * y;

/* enable trapping on inexact exception */

if (ieee_handler("set", "inexact", trap_all_fp_exc) != 0)

printf("Could not set trap handler for inexact.\n");

/* raise inexact */

if (ieee_flags("clear", "exception", "all", &out) != 0)

printf(" could not clear exceptions\n");

printf("5. Inexact: 2.0 / 3.0\n");

x = 2.0;

y = 3.0;

z = x / y;

/* don't trap on inexact */

if (ieee_handler("set", "inexact", SIGFPE_IGNORE) != 0)

printf(" could not reset inexact trap\n");

/* check that we're not trapping on inexact anymore */

if (ieee_flags("clear", "exception", "all", &out) != 0)

printf(" could not clear exceptions\n");

printf("6. Inexact trapping disabled; 2.0 / 3.0\n");

x = 2.0;

y = 3.0;

z = x / y;

/* find out if there are any outstanding exceptions */

ieee_retrospective_();

/* exit gracefully */

return 0;

}

void trap_all_fp_exc(int sig, siginfo_t *sip, ucontext_t *uap) {

char*label = "undefined";

/* see /usr/include/sys/machsig.h for SIGFPE codes */

switch (sip->si_code) {

case FPE_FLTRES:

label = "inexact";

break;

case FPE_FLTDIV:

label = "division";

break;

case FPE_FLTUND:

label = "underflow";

break;

case FPE_FLTINV:

label = "invalid";

break;

case FPE_FLTOVF:

label = "overflow";

break;

}

printf(

" signal %d, sigfpe code %d: %s exception at address %x\n",

sig, sip->si_code, label, sip->_data._fault._addr);

}

¤³¤Î C ¥×¥í¥°¥é¥à¤Î·ë²Ì¤Ï¼¡¤Î¤è¤¦¤Ë¤Ê¤ê¤Þ¤¹¡£

1. Invalid: signaling_nan(0) * 2.5 signal 8, sigfpe code 7: invalid exception at address 10da8 2. Div0: 1.0 / 0.0 signal 8, sigfpe code 3: division exception at address 10e44 3. Overflow: -max_normal() - 1.0e294 signal 8, sigfpe code 4: overflow exception at address 10ee8 4. Underflow: min_normal() * min_normal() signal 8, sigfpe code 5: underflow exception at address 10f80 5. Inexact: 2.0 / 3.0 signal 8, sigfpe code 6: inexact exception at address 1106c 6. Inexact trapping disabled; 2.0 / 3.0 Ãí: °Ê²¼¤Î IEEE ÉâÆ°¾®¿ôÅÀÎã³°¤¬È¯À¸¤·¤Þ¤·¤¿: ÉÔÀµ³Î¡¢¥¢¥ó¥À¡¼¥Õ¥í¡¼ °Ê²¼¤Î IEEE ÉâÆ°¾®¿ôÅÀÎã³°¤Î¥È¥é¥Ã¥×¤¬Í­¸ú¤Ç¤¹: ¥ª¡¼¥Ð¡¼¥Õ¥í¡¼¡¢¥¼¥í¤Ë¤è¤ë½ü»»¡¢Ìµ¸ú¤Ê±é»» ¾ÜºÙ¤Ï¡¢¡Ø¿ôÃÍ·×»»¥¬¥¤¥É¡Ù¤Îieee_flagsieee_handler

SPARC ¤Ç¤Ï¡¢¼¡¤Î¥×¥í¥°¥é¥à¤Ï¡¢¤¢¤ëÎã³°¤Î¾õ¶·¤«¤éµ¯¤­¤¿¥Ç¥Õ¥©¥ë¥È·ë²Ì¤ò½¤Àµ¤¹¤ë¤¿¤á¤Îieee_handler ¤È¥¤¥ó¥¯¥ë¡¼¥É¥Õ¥¡¥¤¥ë¤Î»ÈÍÑÊýË¡¤ò¼¨¤·¤Æ¤¤¤Þ¤¹¡£

/*

* ¥¼¥í½ü»»¤ÎÎã³°¤òȯÀ¸¤µ¤»¡¢¥·¥°¥Ê¥ë¥Ï¥ó¥É¥é¤ò»ÈÍѤ·¤Æ·ë²Ì¤ò MAXDOUBLE

* (¤Þ¤¿¤Ï MAXFLOAT) ¤ÇÃÖ¤­´¹¤¨¤Þ¤¹¡£

*

* ¥Õ¥é¥° -Xa ¤Ç¥³¥ó¥Ñ¥¤¥ë¤·¤Þ¤¹¡£

*/

#include <values.h>

#include <siginfo.h>

#include <ucontext.h>

void division_handler(int sig, siginfo_t *sip, ucontext_t *uap);

int main() {

doublex, y, z;

floatr, s, t;

char*out;

/*

* ieee_handler ¤ò»ÈÍѤ·¡¢division_handler ¤ò IEEE Îã³°

* "division" ¤¬È¯À¸¤·¤¿¤È¤­¤Ë»ÈÍѤ¹¤ë¥·¥°¥Ê¥ë¥Ï¥ó¥É¥é¤ËÀßÄꤹ¤ë

*/

if (ieee_handler("set","division",division_handler)!=0) {

printf(" IEEE trapping not supported here.\n");

}

/* ¥¼¥í½ü»»¤ÎÎã³°¤òȯÀ¸¤µ¤»¤ë */

x = 1.0;

y = 0.0;

z = x / y;

/*

* ¥æ¡¼¥¶¡¼¤¬Ä󶡤·¤¿ÃÍ MAXDOUBLE ¤¬ IEEE ¤Î¥Ç¥Õ¥©¥ë¥ÈÃͤǤ¢¤ë

* ̵¸ÂÂç¤ÎÂå¤ï¤ê¤ËÃÖ´¹¤µ¤ì¤Æ¤¤¤ë¤«¤É¤¦¤«¤ò³Îǧ¤¹¤ë

*/

printf("double precision division: %g/%g = %g \n",x,y,z);

/* ¥¼¥í½ü»»¤ÎÎã³°¤òȯÀ¸¤µ¤»¤ë */

r = 1.0;

s = 0.0;

t = r / s;

/*

* ¥æ¡¼¥¶¡¼¤¬Ä󶡤·¤¿ÃÍ MAXFLOAT ¤¬ IEEE ¤Î¥Ç¥Õ¥©¥ë¥ÈÃͤǤ¢¤ë

* ̵¸ÂÂç¤ÎÂå¤ï¤ê¤ËÃÖ´¹¤µ¤ì¤Æ¤¤¤ë¤«¤É¤¦¤«¤ò³Îǧ¤¹¤ë

*/

printf("single precision division: %g/%g = %g \n",r,s,t);

ieee_retrospective_();

return 0;

}

void division_handler(int sig, siginfo_t *sip, ucontext_t *uap) {

intinst;

unsignedrd, mask, single_prec=0;

floatf_val = MAXFLOAT;

doubled_val = MAXDOUBLE;

long*f_val_p = (long *) &f_val;

/* Îã³°¤òȯÀ¸¤µ¤»¤ëÌ¿Îá */

inst = uap->uc_mcontext.fpregs.fpu_q->FQu.fpq.fpq_instr;

/*

* °ÜÆ°Àè¤Î¥ì¥¸¥¹¥¿¤òÉü¹æ²½¤¹¤ë¡£

* ¥Ó¥Ã¥È 29:25 ¤Ï¡¢SPARC ÉâÆ°¾®¿ôÅÀÌ¿Îá¤ËÂФ·¤Æ

* °ÜÆ°Àè¥ì¥¸¥¹¥¿¤òÉä¹æ²½¤¹¤ë

*/

mask = 0x1f;

rd = (mask & (inst >> 25));

/*

* ¤³¤ì¤ÏñÀºÅÙÌ¿ÎᤫÇÜÀºÅÙÌ¿Îᤫ?

* ¥Ó¥Ã¥È 5:6 ¤Ï opcode ¤ÎÀºÅÙ¤òÉä¹æ²½¤¹¤ë¡£

* ¥Ó¥Ã¥È 5 ¤¬ 1 ¤Ç¤¢¤ì¤Ð sp ¤È¤Ê¤ê¡¢¤½¤ì°Ê³°¤Ï dp ¤È¤Ê¤ë

*/

mask = 0x1;

single_prec = (mask & (inst >> 5));

/* ¥æ¡¼¥¶¡¼¤¬ÄêµÁ¤·¤¿Ãͤò°ÜÆ°Àè¥ì¥¸¥¹¥¿¤ËÆþ¤ì¤ë */

if (single_prec) {

uap->uc_mcontext.fpregs.fpu_fr.fpu_regs[rd] =

f_val_p[0];

} else {

uap->uc_mcontext.fpregs.fpu_fr.fpu_dregs[rd/2] = d_val;

}

}

´üÂÔ¤µ¤ì¤ë½ÐÎϤϼ¡¤Î¤È¤ª¤ê¤Ç¤¹¡£

double precision division: 1/0 = 1.79769e+308 single precision division: 1/0 = 3.40282e+38 division by zero; See the Numerical Computation Guide, ieee_handler(3M) [ÆüËܸìÌõ] Ãí: °Ê²¼¤Î IEEE ÉâÆ°¾®¿ôÅÀÎã³°¤Î¥È¥é¥Ã¥×¤¬Í­¸ú¤Ç¤¹: ¥¼¥í¤Ë¤è¤ë½ü»» ¾ÜºÙ¤Ï¡¢¡Ø¿ôÃÍ·×»»¥¬¥¤¥É¡Ù¤Îieee_handler

ÆÃÄê¤ÎÉâÆ°¾®¿ôÅÀÎã³°¤Î¾ì¹ç¡¢ieee_handler ¤ò»ÈÍѤ·¤Æ¶¯À©Åª¤Ë¥×¥í¥°¥é¥à¤ò°Û¾ï½ªÎ»¤µ¤»¤ë¤³¤È¤¬¤Ç¤­¤Þ¤¹¡£

#include <f77_floatingpoint.h> program abort c ieeer = ieee_handler('set', 'division', SIGFPE_ABORT) if (ieeer .ne. 0) print *, ' ieee trapping not supported' r = 14.2 s = 0.0 r = r/s c print *, 'you should not see this; system should abort' c end

¤³¤ÎÀá¤Ç¤Ï¡¢libm9x.so ¤¬Ä󶡤¹¤ëÎã³°½èÍýµ¡Ç½¤Î°ìÉô¤Î»ÈÍÑÊýË¡¤ò¼¨¤·¤¿Îã¤ò¼è¤ê¤¢¤²¤Þ¤¹¡£ºÇ½é¤ÎÎã¤Ï¡¢¿ôÃÍx ¤È·¸¿ôa₀¡¢a₁¡¢...a_N¡¢¤ª¤è¤Ób₀¡¢b₁¡¢....b_N-1 ¤Î¾ì¹ç¤Ë¡¢´Ø¿ôf(x) ¤È¤½¤ÎºÇ½é¤ÎƳ´Ø¿ôf'(x) ¤òɾ²Á¤¹¤ë¥¿¥¹¥¯¤Ë´ð¤Å¤¤¤Æ¤¤¤Þ¤¹¡£¤³¤Î¾ì¹çf ¤ÏϢʬ¿ôf(x) =a₀ +b₀/(x +a₁ +b₁/(x + ... /(x +a_N-1 +b_N-1/(x +a_N))...))¤Ç¤¹¡£IEEE ±é»»¤Ç¤Ï¡¢f ¤Î·×»»¤Ï´Êñ¤Ç¤¹¡£Ãæ´Ö½ü»»¤Î 1 ¤Ä¤¬¥ª¡¼¥Ð¡¼¥Õ¥í¡¼¤·¤¿¤ê¥¼¥í½ü»»¤ò¹Ô¤¦¾ì¹ç¤Ç¤â¡¢É¸½à¤Ë¤è¤Ã¤Æ»ØÄꤵ¤ì¤¿¥Ç¥Õ¥©¥ë¥È¤ÎÃÍ (Àµ¤·¤¯Éä¹æ¤¬ÉÕ¤¤¤¿Ìµ¸ÂÂç) ¤¬Àµ¤·¤¤·ë²Ì¤ò½Ð¤·¤Þ¤¹¡£°ìÊý¡¢f' ¤Î·×»»¤Ï¡¢¤³¤ì¤òɾ²Á¤¹¤ë¤â¤Ã¤È¤â´Ê·é¤Ê¥Õ¥©¡¼¥à¤¬¡¢ºï½ü²Äǽ¤ÊÆðÛÅÀ¤ò»ý¤Æ¤ë¤¿¤á¡¢f ¤Î·×»»¤è¤ê¤âº¤Æñ¤Ç¤¹¡£·×»»Ãæ¤Ë¤³¤ì¤é¤ÎÆðÛÅÀ¤Î 1 ¤Ä¤¬½Ð¸½¤¹¤ë¤È¡¢ÉÔÄê·Á 0/0¡¢0*̵¸ÂÂ硢̵¸ÂÂç/̵¸ÂÂç¤Î 1 ¤Ä¤Îɾ²Á¤¬»î¤ß¤é¤ì¤Þ¤¹ (¤³¤ì¤é¤ÎÉÔÄê·Á¤Ï¤¹¤Ù¤Æ̵¸ú¤Ê±é»»Îã³°¤òȯÀ¸¤·¤Þ¤¹)¡£W. Kahan ¤Ï¡¢¡ÖÁ°ÃÖ´¹¡×¤È¤¤¤¦µ¡Ç½¤Ë¤è¤ê¤³¤ì¤é¤ÎÎã³°¤ò½èÍý¤¹¤ëÊýË¡¤òÄ󾧤·¤Æ¤¤¤Þ¤¹¡£

Á°ÃÖ´¹¤Ï¡¢Îã³°¤ËÂФ¹¤ë IEEE ¤Î¥Ç¥Õ¥©¥ë¥È¤Î±þÅú (Îã³°±é»»¤Î·ë²Ì¤òÃÖ´¹¤¹¤ëÃͤò¥æ¡¼¥¶¡¼¤¬Á°¤â¤Ã¤Æ»ØÄꤹ¤ë¤³¤È¤òµö²Ä¤¹¤ë) ¤ò³ÈÄ¥¤·¤¿¤â¤Î¤Ç¤¹¡£libm9x.so ¤ò»ÈÍѤ¹¤ë¤È¡¢FEX_CUSTOM Îã³°½èÍý¥â¡¼¥É¤Ç¥Ï¥ó¥É¥é¤ò¥¤¥ó¥¹¥È¡¼¥ë¤·¡¢¥×¥í¥°¥é¥à¤Ç´Êñ¤ËÁ°ÃÖ´¹¤ò¼ÂÁõ¤Ç¤­¤Þ¤¹¡£¤³¤Î¥â¡¼¥É¤Ç¤Ï¡¢¥Ï¥ó¥É¥é¤ËÅϤµ¤ì¤¿info ¥Ñ¥é¥á¡¼¥¿¤¬¼¨¤¹¥Ç¡¼¥¿¹½Â¤¤ËÃͤò³ÊǼ¤¹¤ë¤À¤±¤Ç¡¢¥Ï¥ó¥É¥é¤ÏÎã³°±é»»¤Î·ë²Ì¤ËÂФ·¤ÆǤ°Õ¤ÎÃͤò¶¡µë¤Ç¤­¤Þ¤¹¡£¼¡¤Ë¡¢FEX_CUSTOM ¥Ï¥ó¥É¥é¤Ë¤è¤Ã¤Æ¼ÂÁõ¤·¤¿Á°ÃÖ´¹¤ò»ÈÍѤ·¤ÆϢʬ¿ô¤È¤½¤ÎƳ´Ø¿ô¤ò·×»»¤¹¤ë¥×¥í¥°¥é¥àÎã¤ò¼¨¤·¤Þ¤¹¡£

#include <stdio.h>

#include <sunmath.h>

#include <fenv.h>

volatile double p;

void handler(int ex, fex_info_t *info)

{

info->res.type = fex_double;

if (ex == FEX_INV_ZMI)

info->res.val.d = p;

else

info->res.val.d = infinity();

}

/*

* x ¤Î°ÌÃ֤Ƿ¸¿ô a[j] ¤È b[j] ¤Ë¤è¤Ã¤Æ»ØÄꤵ¤ì¤ëϢʬ¿ô¤ò

* ɾ²Á¤·¡¢´Ø¿ôÃͤò *pf¡¢Æ³´Ø¿ô¤ÎÃͤò *pf1 ¤ÇÊÖ¤·¤Þ¤¹

*/

void continued_fraction(int N, double *a, double *b,

double x, double *pf, double *pf1)

{

fex_handler_t oldhdl; /* ¥Ï¥ó¥É¥é¤ÎÊݸ¤Þ¤¿¤ÏÉü¸µ¤Î¤¿¤á */

volatile double t;

double f, f1, d, d1, q;

int j;

fex_getexcepthandler(&oldhdl, FEX_DIVBYZERO | FEX_INVALID);

fex_set_handling(FEX_DIVBYZERO, FEX_NONSTOP, NULL);

fex_set_handling(FEX_INV_ZDZ | FEX_INV_IDI | FEX_INV_ZMI,

FEX_CUSTOM, handler);

f1 = 0.0;

f = a[N];

for (j = N - 1; j >= 0; j--) {

d = x + f;

d1 = 1.0 + f1;

q = b[j] / d;

/* ÊÑ¿ô p ¤ËÂФ¹¤ëÂåÆþ¤È f1 ¤Îɾ²Á¤Î´Ö¤ÇÀµ¤·¤¤½ç½ø¤Å¤±¤ò °Ý»ý¤¹¤ë¤¿¤á¡¢volatile ÊÑ¿ô t ¤ËÂФ¹¤ë¼¡¤ÎÂåÆþ¤¬É¬ÍפǤ¹ */

t = f1 = (-d1 / d) * q;

p = b[j-1] * d1 / b[j];

f = a[j] + q;

}

fex_setexcepthandler(&oldhdl, FEX_DIVBYZERO | FEX_INVALID);

*pf = f;

*pf1 = f1;

}

/* ¼¡¤Î·¸¿ô¡¢x = -3¡¢1¡¢4¡¢¤ª¤è¤Ó 5 ¤Ï¤¹¤Ù¤ÆÃæ´ÖÎã³°¤Ë½Ð²ñ¤¤¤Þ¤¹ */

double a[] = { -1.0, 2.0, -3.0, 4.0, -5.0 };

double b[] = { 2.0, 4.0, 6.0, 8.0 };

int main()

{

double x, f, f1;

int i;

feraiseexcept(FE_INEXACT); /* ÉÔÀµ³Î¤Ê±é»»¤Î¥í¥°µ­Ï¿¤òËɻߤ·¤Þ¤¹ */

fex_set_log(stdout);

fex_set_handling(FEX_COMMON, FEX_ABORT, NULL);

for (i = -5; i <= 5; i++) {

x = i;

continued_fraction(4, a, b, x, &f, &f1);

printf("f(% g) = %12g, f'(% g) = %12g\n", x, f, x, f1);

}

return 0;

}

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¥á¥¤¥ó¥×¥í¥°¥é¥à¤Ï¡¢fex_set_log ´Ø¿ô¤ò¸Æ¤Ó½Ð¤·¤Æ¡¢Á̵ڿÇÃÇ¤Î¥í¥°µ­Ï¿¤òÍ­¸ú¤Ë¤·¤Þ¤¹¡£¤³¤Î¸Æ¤Ó½Ð¤·¤ò¹Ô¤¦Á°¤Ë¡¢¥á¥¤¥ó¥×¥í¥°¥é¥à¤ÏÉÔÀµ³Î¥Õ¥é¥°¤òȯÀ¸¤µ¤»¤Þ¤¹¡£¤³¤ì¤Ë¤è¤ê¡¢ÉÔÀµ³Î±é»»¤Îµ­Ï¿¤òËɤ®¤Þ¤¹ (¡ÖÁ̵ڿÇÃǡפÇÀâÌÀ¤·¤Æ¤¤¤ë¤è¤¦¤Ë¡¢FEX_NONSTOP ¥â¡¼¥É¤Ç¤ÏÎã³°¤Î¥Õ¥é¥°¤¬È¯À¸¤¹¤ë¤ÈÎã³°¤¬¥í¥°¤Ëµ­Ï¿¤µ¤ì¤Þ¤»¤ó)¡£¥á¥¤¥ó¥×¥í¥°¥é¥à¤Ï¤Þ¤¿¡¢¶¦ÄÌÎã³°¤ËÂФ·¤Æ FEX_ABORT ¥â¡¼¥É¤ò³ÎΩ¤·¡¢continued_fraction ¤Ë¤è¤Ã¤ÆÌÀ¼¨Åª¤Ë½èÍý¤µ¤ì¤Ê¤¤°Û¾ï¤ÊÎã³°¤¬¥×¥í¥°¥é¥à¤òÄä»ß¤¹¤ë¤³¤È¤òËɤ®¤Þ¤¹¡£¥×¥í¥°¥é¥à¤Ï¡¢ºÇ¸å¤Ë¿ô¤«½ê¤ÇÆÃÄê¤ÎϢʬ¿ô¤òɾ²Á¤·¤Þ¤¹¡£¼¡¤Î½ÐÎÏÎ㤬¼¨¤¹¤è¤¦¤Ë¡¢±é»»¤ÏÃæ´ÖÎã³°¤Ë½Ð²ñ¤¤¤Þ¤¹¡£

f(-5) = -1.59649, f'(-5) = -0.1818 f(-4) = -1.87302, f'(-4) = -0.428193 Floating point division by zero at 0x08048dbe continued_fraction, nonstop mode 0x08048dc1 continued_fraction 0x08048eda main Floating point invalid operation (inf/inf) at 0x08048dcf continued_fraction, handler: handler 0x08048dd2 continued_fraction 0x08048eda main Floating point invalid operation (0*inf) at 0x08048dd2 continued_fraction, handler: handler 0x08048dd8 continued_fraction 0x08048eda main f(-3) = -3, f'(-3) = -3.16667 f(-2) = -4.44089e-16, f'(-2) = -3.41667 f(-1) = -1.22222, f'(-1) = -0.444444 f( 0) = -1.33333, f'( 0) = 0.203704 f( 1) = -1, f'( 1) = 0.333333 f( 2) = -0.777778, f'( 2) = 0.12037 f( 3) = -0.714286, f'( 3) = 0.0272109 f( 4) = -0.666667, f'( 4) = 0.203704 f( 5) = -0.777778, f'( 5) = 0.0185185

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#include <stdio.h>

#include <math.h>

#include <fenv.h>

/*

* x ¤Î°ÌÃ֤Ƿ¸¿ô a[j] ¤È b[j] ¤Ë¤è¤Ã¤Æ»ØÄꤵ¤ì¤ëϢʬ¿ô¤ò * ɾ²Á¤·¡¢´Ø¿ôÃͤò *pf¡¢Æ³´Ø¿ô¤ÎÃͤò *pf1 ¤ÇÊÖ¤·¤Þ¤¹

*/

void continued_fraction(int N, double *a, double *b,

double x, double *pf, double *pf1)

{

fex_handler_t oldhdl;

fexcept_t oldinvflag;

double f, f1, d, d1, pd1, q;

int j;

fex_getexcepthandler(&oldhdl, FEX_DIVBYZERO | FEX_INVALID);

fegetexceptflag(&oldinvflag, FE_INVALID);

fex_set_handling(FEX_DIVBYZERO | FEX_INV_ZDZ | FEX_INV_IDI |

FEX_INV_ZMI, FEX_NONSTOP, NULL);

feclearexcept(FE_INVALID);

f1 = 0.0;

f = a[N];

for (j = N - 1; j >= 0; j--) {

d = x + f;

d1 = 1.0 + f1;

q = b[j] / d;

f1 = (-d1 / d) * q;

f = a[j] + q;

}

if (fetestexcept(FE_INVALID)) {

/* recompute and test for NaN */

f1 = pd1 = 0.0;

f = a[N];

for (j = N - 1; j >= 0; j--) {

d = x + f;

d1 = 1.0 + f1;

q = b[j] / d;

f1 = (-d1 / d) * q;

if (isnan(f1))

f1 = b[j] * pd1 / b[j+1];

pd1 = d1;

f = a[j] + q;

}

}

fesetexceptflag(&oldinvflag, FE_INVALID);

fex_setexcepthandler(&oldhdl, FEX_DIVBYZERO | FEX_INVALID);

*pf = f;

*pf1 = f1;

}

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c

c Á°ÃÖ´¹¥Ï¥ó¥É¥é

c

subroutine handler(ex, info)

structure /fex_numeric_t/

integer type

union

map

integer i

end map

map

integer*8 l

end map

map

real f

end map

map

real*8 d

end map

map

real*16 q

end map

end union

end structure

structure /fex_info_t/

integer op, flags

record /fex_numeric_t/ op1, op2, res

end structure

integer ex

record /fex_info_t/ info

common /presub/ p

double precision p, d_infinity

volatile p

c 4 = fex_double; Äê¿ô¤Ë¤Ä¤¤¤Æ¤Ï <fenv.h> ¤ò»²¾È¤·¤Æ¤¯¤À¤µ¤¤¡£

info.res.type = 4

c x'80' = FEX_INV_ZMI

if (loc(ex) .eq. x'80') then

info.res.d = p

else

info.res.d = d_infinity()

endif

return

end

c

c x ¤Î°ÌÃ֤Ƿ¸¿ô a[j] ¤È b[j] ¤Ë¤è¤Ã¤Æ»ØÄꤵ¤ì¤ëϢʬ¿ô¤ò c ɾ²Á¤·¡¢´Ø¿ôÃͤò f¡¢Æ³´Ø¿ô¤ÎÃͤò f1 ¤ÇÊÖ¤·¤Þ¤¹

c

subroutine continued_fraction(n, a, b, x, f, f1)

integer n

double precision a(*), b(*), x, f, f1

common /presub/ p

integer j, oldhdl

dimension oldhdl(24)

double precision d, d1, q, p, t

volatile p, t

external fex_getexcepthandler, fex_setexcepthandler

external fex_set_handling, handler

c$pragma c(fex_getexcepthandler, fex_setexcepthandler)

c$pragma c(fex_set_handling)

c x'ff2' = FEX_DIVBYZERO | FEX_INVALID

call fex_getexcepthandler(oldhdl, %val(x'ff2'))

c x'2' = FEX_DIVBYZERO, 0 = FEX_NONSTOP

call fex_set_handling(%val(x'2'), %val(0), %val(0))

c x'b0' = FEX_INV_ZDZ | FEX_INV_IDI | FEX_INV_ZMI, 3 = FEX_CUSTOM

call fex_set_handling(%val(x'b0'), %val(3), handler)

f1 = 0.0d0

f = a(n+1)

do j = n, 1, -1

d = x + f

d1 = 1.0d0 + f1

q = b(j) / d

f1 = (-d1 / d) * q

c

c ÊÑ¿ô p ¤ËÂФ¹¤ëÂåÆþ¤È f1 ¤Îɾ²Á¤Î´Ö¤ÇÀµ¤·¤¤½ç½ø¤Å¤± c ¤ò°Ý»ý¤¹¤ë¤¿¤á¡¢volatile ÊÑ¿ô t ¤ËÂФ¹¤ë¼¡¤ÎÂåÆþ¤¬ c ɬÍפǤ¹

t = f1

p = b(j-1) * d1 / b(j)

f = a(j) + q

end do

call fex_setexcepthandler(oldhdl, %val(x'ff2'))

return

end

c

c ¥á¥¤¥ó¥×¥í¥°¥é¥à

c

program cf

integer i

double precision a, b, x, f, f1

dimension a(5), b(4)

data a /-1.0d0, 2.0d0, -3.0d0, 4.0d0, -5.0d0/

data b /2.0d0, 4.0d0, 6.0d0, 8.0d0/

external fex_set_handling

c$pragma c(fex_set_handling)

c x'ffa' = FEX_COMMON, 1 = FEX_ABORT

call fex_set_handling(%val(x'ffa'), %val(1), %val(0))

do i = -5, 5

x = dble(i)

call continued_fraction(4, a, b, x, f, f1)

write (*, 1) i, f, i, f1

end do

1 format('f(', I2, ') = ', G12.6, ', f''(', I2, ') = ', G12.6)

end

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f(-5) = -1.59649 , f'(-5) = -.181800 f(-4) = -1.87302 , f'(-4) = -.428193 f(-3) = -3.00000 , f'(-3) = -3.16667 f(-2) = -.444089E-15, f'(-2) = -3.41667 f(-1) = -1.22222 , f'(-1) = -.444444 f( 0) = -1.33333 , f'( 0) = 0.203704 f( 1) = -1.00000 , f'( 1) = 0.333333 f( 2) = -.777778 , f'( 2) = 0.120370 f( 3) = -.714286 , f'( 3) = 0.272109E-01 f( 4) = -.666667 , f'( 4) = 0.203704 f( 5) = -.777778 , f'( 5) = 0.185185E-01 Note: IEEE floating-point exception flags raised: Inexact; Division by Zero; Invalid Operation; IEEE floating-point exception traps enabled: overflow; division by zero; invalid operation; See the Numerical Computation Guide, ieee_flags(3M), ieee_handler(3M) [ÆüËܸìÌõ] Ãí: °Ê²¼¤Î IEEE ÉâÆ°¾®¿ôÅÀÎã³°¤¬È¯À¸¤·¤Þ¤·¤¿: ÉÔÀµ³Î¡¢¥¼¥í¤Ë¤è¤ë½ü»»¡¢Ìµ¸ú¤Ê±é»» °Ê²¼¤Î IEEE ÉâÆ°¾®¿ôÅÀÎã³°¤Î¥È¥é¥Ã¥×¤¬Í­¸ú¤Ç¤¹: ¥ª¡¼¥Ð¡¼¥Õ¥í¡¼¡¢¥¼¥í¤Ë¤è¤ë½ü»»¡¢Ìµ¸ú¤Ê±é»» ¾ÜºÙ¤Ï¡¢¡Ø¿ôÃÍ·×»»¥¬¥¤¥É¡Ù¤Îieee_flagsieee_handler

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/* À°¿ô¤Î¥¼¥í½ü»»Îã³°¤òÀ¸À®¤¹¤ë */ #include <siginfo.h> #include <ucontext.h> #include <signal.h> void int_handler(int sig, siginfo_t *sip, ucontext_t *uap); int main() { int a, b, c; /* * sigfpe(3) ¤ò»ÈÍѤ·¤Æ¡¢À°¿ô¥¼¥í½ü»»¤Ç»ÈÍѤ¹¤ë¥·¥°¥Ê¥ë¥Ï¥ó¥É¥é¤È¤·¤Æ * int_handler ¤òÀßÄꤹ¤ë */ /* * À°¿ô¤Î¥¼¥í½ü»»¤ËÂФ¹¤ë¥·¥°¥Ê¥ë¥Ï¥ó¥É¥é¤¬ÀßÄꤵ¤ì¤Æ¤¤¤Ê¤¤¤È¡¢ * À°¿ô¤Î¥¼¥í½ü»»¤Ï°Û¾ï½ªÎ»¤¹¤ë */ sigfpe(FPE_INTDIV, int_handler); a = 4; b = 0; c = a / b; printf("%d / %d = %d\n\n", a, b, c); return(0); } void int_handler(int sig, siginfo_t *sip, ucontext_t *uap) { printf("Signal %d, code %d, at addr %x\n", sig, sip->si_code, sip->_data._fault._addr); /* * ¥×¥í¥°¥é¥à¥«¥¦¥ó¥¿¤òÁý¤ä¤¹¡£¥ª¥Ú¥ì¡¼¥Æ¥£¥ó¥°¥·¥¹¥Æ¥à¤Ï¡¢ * ¤³¤ì¤òÉâÆ°¾®¿ôÅÀÎã³°¤ËÂФ·¤Æ¼«Æ°Åª¤Ë¹Ô¤¦¡£ * À°¿ô¤Î¥¼¥í½ü»»¤ËÂФ·¤Æ¤Ï¹Ô¤ï¤Ê¤¤¡£ */ uap->uc_mcontext.gregs[REG_PC] = uap->uc_mcontext.gregs[REG_nPC]; }

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/* * °Ê²¼¤ÎÊýË¡¤ò¼¨¤¹¥Ç¥â¥×¥í¥°¥é¥à¤Ç¤¹¡£ * * 1. ÇÛÎó°ú¿ô¤òÅϤ·¤Æ¡¢f77 ¥µ¥Ö¥ë¡¼¥Á¥ó¤ò C ¤«¤é¸Æ¤Ó½Ð¤¹ÊýË¡ * 2. ñÀºÅÙ f77 ´Ø¿ô¤ò C ¤«¤é¸Æ¤Ó½Ð¤¹ÊýË¡ * 3. ÇÜÀºÅÙ f77 ´Ø¿ô¤ò C ¤«¤é¸Æ¤Ó½Ð¤¹ÊýË¡ */ extern int demo_one_(double *); extern float demo_two_(float *); extern double demo_three_(double *); int main() { double array[3][4]; float f, g; double x, y; int i, j; for (i = 0; i < 3; i++) for (j = 0; j < 4; j++) array[i][j] = i + 2*j; g = 1.5; y = g; /* ÇÛÎó¤ò fortran ´Ø¿ô¤ËÅϤ¹ (ÇÛÎó¤Î½ÐÎÏ) */ demo_one_(&array[0][0]); printf(" from the driver\n"); for (i = 0; i < 3; i++) { for (j = 0; j < 4; j++) printf(" array[%d][%d] = %e\n", i, j, array[i][j]); printf("\n"); } /* ñÀºÅÙ fortran ´Ø¿ô¤ò¸Æ¤Ó½Ð¤¹ */ f = demo_two_(&g); printf( " f = sin(g) from a single precision fortran function\n"); printf(" f, g: %8.7e, %8.7e\n", f, g); printf("\n"); /* ÇÜÀºÅÙ fortran ´Ø¿ô¤ò¸Æ¤Ó½Ð¤¹ */ x = demo_three_(&y); printf( " x = sin(y) from a double precision fortran function\n"); printf(" x, y: %18.17e, %18.17e\n", x, y); ieee_retrospective_(); return(0); }

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subroutine demo_one(array) double precision array(4,3) print *, 'from the fortran routine:' do 10 i =1,4 do 20 j = 1,3 print *, ' array[', i, '][', j, '] = ', array(i,j) 20 continue print * 10 continue return end real function demo_two(number) real number demo_two = sin(number) return end double precision function demo_three(number) double precision number demo_three = sin(number) return end

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cc -c driver.c f77 -c drivee.f demo_one: demo_two: demo_three: f77 -o driver driver.o drivee.o

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from the fortran routine: array[ 1][ 1] = 0. array[ 1][ 2] = 1.0000000000000 array[ 1][ 3] = 2.0000000000000 array[ 2][ 1] = 2.0000000000000 array[ 2][ 2] = 3.0000000000000 array[ 2][ 3] = 4.0000000000000 array[ 3][ 1] = 4.0000000000000 array[ 3][ 2] = 5.0000000000000 array[ 3][ 3] = 6.0000000000000 array[ 4][ 1] = 6.0000000000000 array[ 4][ 2] = 7.0000000000000 array[ 4][ 3] = 8.0000000000000 from the driver array[0][0] = 0.000000e+00 array[0][1] = 2.000000e+00 array[0][2] = 4.000000e+00 array[0][3] = 6.000000e+00 array[1][0] = 1.000000e+00 array[1][1] = 3.000000e+00 array[1][2] = 5.000000e+00 array[1][3] = 7.000000e+00 array[2][0] = 2.000000e+00 array[2][1] = 4.000000e+00 array[2][2] = 6.000000e+00 array[2][3] = 8.000000e+00 f = sin(g) from a single precision fortran function f, g: 9.9749500e-01, 1.5000000e+00 x = sin(y) from a double precision fortran function x, y: 9.97494986604054446e-01, 1.50000000000000000e+00

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