105a0b428SJohn Marino /* e_j1f.c -- float version of e_j1.c.
205a0b428SJohn Marino * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
305a0b428SJohn Marino */
405a0b428SJohn Marino
505a0b428SJohn Marino /*
605a0b428SJohn Marino * ====================================================
705a0b428SJohn Marino * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
805a0b428SJohn Marino *
905a0b428SJohn Marino * Developed at SunPro, a Sun Microsystems, Inc. business.
1005a0b428SJohn Marino * Permission to use, copy, modify, and distribute this
1105a0b428SJohn Marino * software is freely granted, provided that this notice
1205a0b428SJohn Marino * is preserved.
1305a0b428SJohn Marino * ====================================================
1405a0b428SJohn Marino */
1505a0b428SJohn Marino
1605a0b428SJohn Marino #include "math.h"
1705a0b428SJohn Marino #include "math_private.h"
1805a0b428SJohn Marino
1905a0b428SJohn Marino static float ponef(float), qonef(float);
2005a0b428SJohn Marino
2105a0b428SJohn Marino static const float
2205a0b428SJohn Marino huge = 1e30,
2305a0b428SJohn Marino one = 1.0,
2405a0b428SJohn Marino invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
2505a0b428SJohn Marino tpi = 6.3661974669e-01, /* 0x3f22f983 */
2605a0b428SJohn Marino /* R0/S0 on [0,2] */
2705a0b428SJohn Marino r00 = -6.2500000000e-02, /* 0xbd800000 */
2805a0b428SJohn Marino r01 = 1.4070566976e-03, /* 0x3ab86cfd */
2905a0b428SJohn Marino r02 = -1.5995563444e-05, /* 0xb7862e36 */
3005a0b428SJohn Marino r03 = 4.9672799207e-08, /* 0x335557d2 */
3105a0b428SJohn Marino s01 = 1.9153760746e-02, /* 0x3c9ce859 */
3205a0b428SJohn Marino s02 = 1.8594678841e-04, /* 0x3942fab6 */
3305a0b428SJohn Marino s03 = 1.1771846857e-06, /* 0x359dffc2 */
3405a0b428SJohn Marino s04 = 5.0463624390e-09, /* 0x31ad6446 */
3505a0b428SJohn Marino s05 = 1.2354227016e-11; /* 0x2d59567e */
3605a0b428SJohn Marino
3705a0b428SJohn Marino static const float zero = 0.0;
3805a0b428SJohn Marino
3905a0b428SJohn Marino float
j1f(float x)4005a0b428SJohn Marino j1f(float x)
4105a0b428SJohn Marino {
4205a0b428SJohn Marino float z, s,c,ss,cc,r,u,v,y;
4305a0b428SJohn Marino int32_t hx,ix;
4405a0b428SJohn Marino
4505a0b428SJohn Marino GET_FLOAT_WORD(hx,x);
4605a0b428SJohn Marino ix = hx&0x7fffffff;
4705a0b428SJohn Marino if(ix>=0x7f800000) return one/x;
4805a0b428SJohn Marino y = fabsf(x);
4905a0b428SJohn Marino if(ix >= 0x40000000) { /* |x| >= 2.0 */
5005a0b428SJohn Marino s = sinf(y);
5105a0b428SJohn Marino c = cosf(y);
5205a0b428SJohn Marino ss = -s-c;
5305a0b428SJohn Marino cc = s-c;
5405a0b428SJohn Marino if(ix<0x7f000000) { /* make sure y+y not overflow */
5505a0b428SJohn Marino z = cosf(y+y);
5605a0b428SJohn Marino if ((s*c)>zero) cc = z/ss;
5705a0b428SJohn Marino else ss = z/cc;
5805a0b428SJohn Marino }
5905a0b428SJohn Marino /*
6005a0b428SJohn Marino * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
6105a0b428SJohn Marino * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
6205a0b428SJohn Marino */
63*74b7c7a8SJohn Marino if((u_int64_t)ix>0x80000000U) z = (invsqrtpi*cc)/sqrtf(y);
6405a0b428SJohn Marino else {
6505a0b428SJohn Marino u = ponef(y); v = qonef(y);
6605a0b428SJohn Marino z = invsqrtpi*(u*cc-v*ss)/sqrtf(y);
6705a0b428SJohn Marino }
6805a0b428SJohn Marino if(hx<0) return -z;
6905a0b428SJohn Marino else return z;
7005a0b428SJohn Marino }
7105a0b428SJohn Marino if(ix<0x32000000) { /* |x|<2**-27 */
7205a0b428SJohn Marino if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */
7305a0b428SJohn Marino }
7405a0b428SJohn Marino z = x*x;
7505a0b428SJohn Marino r = z*(r00+z*(r01+z*(r02+z*r03)));
7605a0b428SJohn Marino s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
7705a0b428SJohn Marino r *= x;
7805a0b428SJohn Marino return(x*(float)0.5+r/s);
7905a0b428SJohn Marino }
8005a0b428SJohn Marino
8105a0b428SJohn Marino static const float U0[5] = {
8205a0b428SJohn Marino -1.9605709612e-01, /* 0xbe48c331 */
8305a0b428SJohn Marino 5.0443872809e-02, /* 0x3d4e9e3c */
8405a0b428SJohn Marino -1.9125689287e-03, /* 0xbafaaf2a */
8505a0b428SJohn Marino 2.3525259166e-05, /* 0x37c5581c */
8605a0b428SJohn Marino -9.1909917899e-08, /* 0xb3c56003 */
8705a0b428SJohn Marino };
8805a0b428SJohn Marino static const float V0[5] = {
8905a0b428SJohn Marino 1.9916731864e-02, /* 0x3ca3286a */
9005a0b428SJohn Marino 2.0255257550e-04, /* 0x3954644b */
9105a0b428SJohn Marino 1.3560879779e-06, /* 0x35b602d4 */
9205a0b428SJohn Marino 6.2274145840e-09, /* 0x31d5f8eb */
9305a0b428SJohn Marino 1.6655924903e-11, /* 0x2d9281cf */
9405a0b428SJohn Marino };
9505a0b428SJohn Marino
9605a0b428SJohn Marino float
y1f(float x)9705a0b428SJohn Marino y1f(float x)
9805a0b428SJohn Marino {
9905a0b428SJohn Marino float z, s,c,ss,cc,u,v;
10005a0b428SJohn Marino int32_t hx,ix;
10105a0b428SJohn Marino
10205a0b428SJohn Marino GET_FLOAT_WORD(hx,x);
10305a0b428SJohn Marino ix = 0x7fffffff&hx;
10405a0b428SJohn Marino /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
10505a0b428SJohn Marino if(ix>=0x7f800000) return one/(x+x*x);
10605a0b428SJohn Marino if(ix==0) return -one/zero;
10705a0b428SJohn Marino if(hx<0) return zero/zero;
10805a0b428SJohn Marino if(ix >= 0x40000000) { /* |x| >= 2.0 */
10905a0b428SJohn Marino s = sinf(x);
11005a0b428SJohn Marino c = cosf(x);
11105a0b428SJohn Marino ss = -s-c;
11205a0b428SJohn Marino cc = s-c;
11305a0b428SJohn Marino if(ix<0x7f000000) { /* make sure x+x not overflow */
11405a0b428SJohn Marino z = cosf(x+x);
11505a0b428SJohn Marino if ((s*c)>zero) cc = z/ss;
11605a0b428SJohn Marino else ss = z/cc;
11705a0b428SJohn Marino }
11805a0b428SJohn Marino /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
11905a0b428SJohn Marino * where x0 = x-3pi/4
12005a0b428SJohn Marino * Better formula:
12105a0b428SJohn Marino * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
12205a0b428SJohn Marino * = 1/sqrt(2) * (sin(x) - cos(x))
12305a0b428SJohn Marino * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
12405a0b428SJohn Marino * = -1/sqrt(2) * (cos(x) + sin(x))
12505a0b428SJohn Marino * To avoid cancellation, use
12605a0b428SJohn Marino * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
12705a0b428SJohn Marino * to compute the worse one.
12805a0b428SJohn Marino */
12905a0b428SJohn Marino if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x);
13005a0b428SJohn Marino else {
13105a0b428SJohn Marino u = ponef(x); v = qonef(x);
13205a0b428SJohn Marino z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
13305a0b428SJohn Marino }
13405a0b428SJohn Marino return z;
13505a0b428SJohn Marino }
13605a0b428SJohn Marino if(ix<=0x24800000) { /* x < 2**-54 */
13705a0b428SJohn Marino return(-tpi/x);
13805a0b428SJohn Marino }
13905a0b428SJohn Marino z = x*x;
14005a0b428SJohn Marino u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
14105a0b428SJohn Marino v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
14205a0b428SJohn Marino return(x*(u/v) + tpi*(j1f(x)*logf(x)-one/x));
14305a0b428SJohn Marino }
14405a0b428SJohn Marino
14505a0b428SJohn Marino /* For x >= 8, the asymptotic expansions of pone is
14605a0b428SJohn Marino * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
14705a0b428SJohn Marino * We approximate pone by
14805a0b428SJohn Marino * pone(x) = 1 + (R/S)
14905a0b428SJohn Marino * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
15005a0b428SJohn Marino * S = 1 + ps0*s^2 + ... + ps4*s^10
15105a0b428SJohn Marino * and
15205a0b428SJohn Marino * | pone(x)-1-R/S | <= 2 ** ( -60.06)
15305a0b428SJohn Marino */
15405a0b428SJohn Marino
15505a0b428SJohn Marino static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
15605a0b428SJohn Marino 0.0000000000e+00, /* 0x00000000 */
15705a0b428SJohn Marino 1.1718750000e-01, /* 0x3df00000 */
15805a0b428SJohn Marino 1.3239480972e+01, /* 0x4153d4ea */
15905a0b428SJohn Marino 4.1205184937e+02, /* 0x43ce06a3 */
16005a0b428SJohn Marino 3.8747453613e+03, /* 0x45722bed */
16105a0b428SJohn Marino 7.9144794922e+03, /* 0x45f753d6 */
16205a0b428SJohn Marino };
16305a0b428SJohn Marino static const float ps8[5] = {
16405a0b428SJohn Marino 1.1420736694e+02, /* 0x42e46a2c */
16505a0b428SJohn Marino 3.6509309082e+03, /* 0x45642ee5 */
16605a0b428SJohn Marino 3.6956207031e+04, /* 0x47105c35 */
16705a0b428SJohn Marino 9.7602796875e+04, /* 0x47bea166 */
16805a0b428SJohn Marino 3.0804271484e+04, /* 0x46f0a88b */
16905a0b428SJohn Marino };
17005a0b428SJohn Marino
17105a0b428SJohn Marino static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
17205a0b428SJohn Marino 1.3199052094e-11, /* 0x2d68333f */
17305a0b428SJohn Marino 1.1718749255e-01, /* 0x3defffff */
17405a0b428SJohn Marino 6.8027510643e+00, /* 0x40d9b023 */
17505a0b428SJohn Marino 1.0830818176e+02, /* 0x42d89dca */
17605a0b428SJohn Marino 5.1763616943e+02, /* 0x440168b7 */
17705a0b428SJohn Marino 5.2871520996e+02, /* 0x44042dc6 */
17805a0b428SJohn Marino };
17905a0b428SJohn Marino static const float ps5[5] = {
18005a0b428SJohn Marino 5.9280597687e+01, /* 0x426d1f55 */
18105a0b428SJohn Marino 9.9140142822e+02, /* 0x4477d9b1 */
18205a0b428SJohn Marino 5.3532670898e+03, /* 0x45a74a23 */
18305a0b428SJohn Marino 7.8446904297e+03, /* 0x45f52586 */
18405a0b428SJohn Marino 1.5040468750e+03, /* 0x44bc0180 */
18505a0b428SJohn Marino };
18605a0b428SJohn Marino
18705a0b428SJohn Marino static const float pr3[6] = {
18805a0b428SJohn Marino 3.0250391081e-09, /* 0x314fe10d */
18905a0b428SJohn Marino 1.1718686670e-01, /* 0x3defffab */
19005a0b428SJohn Marino 3.9329774380e+00, /* 0x407bb5e7 */
19105a0b428SJohn Marino 3.5119403839e+01, /* 0x420c7a45 */
19205a0b428SJohn Marino 9.1055007935e+01, /* 0x42b61c2a */
19305a0b428SJohn Marino 4.8559066772e+01, /* 0x42423c7c */
19405a0b428SJohn Marino };
19505a0b428SJohn Marino static const float ps3[5] = {
19605a0b428SJohn Marino 3.4791309357e+01, /* 0x420b2a4d */
19705a0b428SJohn Marino 3.3676245117e+02, /* 0x43a86198 */
19805a0b428SJohn Marino 1.0468714600e+03, /* 0x4482dbe3 */
19905a0b428SJohn Marino 8.9081134033e+02, /* 0x445eb3ed */
20005a0b428SJohn Marino 1.0378793335e+02, /* 0x42cf936c */
20105a0b428SJohn Marino };
20205a0b428SJohn Marino
20305a0b428SJohn Marino static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
20405a0b428SJohn Marino 1.0771083225e-07, /* 0x33e74ea8 */
20505a0b428SJohn Marino 1.1717621982e-01, /* 0x3deffa16 */
20605a0b428SJohn Marino 2.3685150146e+00, /* 0x401795c0 */
20705a0b428SJohn Marino 1.2242610931e+01, /* 0x4143e1bc */
20805a0b428SJohn Marino 1.7693971634e+01, /* 0x418d8d41 */
20905a0b428SJohn Marino 5.0735230446e+00, /* 0x40a25a4d */
21005a0b428SJohn Marino };
21105a0b428SJohn Marino static const float ps2[5] = {
21205a0b428SJohn Marino 2.1436485291e+01, /* 0x41ab7dec */
21305a0b428SJohn Marino 1.2529022980e+02, /* 0x42fa9499 */
21405a0b428SJohn Marino 2.3227647400e+02, /* 0x436846c7 */
21505a0b428SJohn Marino 1.1767937469e+02, /* 0x42eb5bd7 */
21605a0b428SJohn Marino 8.3646392822e+00, /* 0x4105d590 */
21705a0b428SJohn Marino };
21805a0b428SJohn Marino
21905a0b428SJohn Marino static float
ponef(float x)22005a0b428SJohn Marino ponef(float x)
22105a0b428SJohn Marino {
22205a0b428SJohn Marino const float *p,*q;
22305a0b428SJohn Marino float z,r,s;
22405a0b428SJohn Marino int32_t ix;
22505a0b428SJohn Marino GET_FLOAT_WORD(ix,x);
22605a0b428SJohn Marino ix &= 0x7fffffff;
22705a0b428SJohn Marino if(ix>=0x41000000) {p = pr8; q= ps8;}
22805a0b428SJohn Marino else if(ix>=0x40f71c58){p = pr5; q= ps5;}
22905a0b428SJohn Marino else if(ix>=0x4036db68){p = pr3; q= ps3;}
230*74b7c7a8SJohn Marino else /*if(ix>=0x40000000)*/ {p = pr2; q= ps2;}
23105a0b428SJohn Marino z = one/(x*x);
23205a0b428SJohn Marino r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
23305a0b428SJohn Marino s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
23405a0b428SJohn Marino return one+ r/s;
23505a0b428SJohn Marino }
23605a0b428SJohn Marino
23705a0b428SJohn Marino
23805a0b428SJohn Marino /* For x >= 8, the asymptotic expansions of qone is
23905a0b428SJohn Marino * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
24005a0b428SJohn Marino * We approximate pone by
24105a0b428SJohn Marino * qone(x) = s*(0.375 + (R/S))
24205a0b428SJohn Marino * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
24305a0b428SJohn Marino * S = 1 + qs1*s^2 + ... + qs6*s^12
24405a0b428SJohn Marino * and
24505a0b428SJohn Marino * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
24605a0b428SJohn Marino */
24705a0b428SJohn Marino
24805a0b428SJohn Marino static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
24905a0b428SJohn Marino 0.0000000000e+00, /* 0x00000000 */
25005a0b428SJohn Marino -1.0253906250e-01, /* 0xbdd20000 */
25105a0b428SJohn Marino -1.6271753311e+01, /* 0xc1822c8d */
25205a0b428SJohn Marino -7.5960174561e+02, /* 0xc43de683 */
25305a0b428SJohn Marino -1.1849806641e+04, /* 0xc639273a */
25405a0b428SJohn Marino -4.8438511719e+04, /* 0xc73d3683 */
25505a0b428SJohn Marino };
25605a0b428SJohn Marino static const float qs8[6] = {
25705a0b428SJohn Marino 1.6139537048e+02, /* 0x43216537 */
25805a0b428SJohn Marino 7.8253862305e+03, /* 0x45f48b17 */
25905a0b428SJohn Marino 1.3387534375e+05, /* 0x4802bcd6 */
26005a0b428SJohn Marino 7.1965775000e+05, /* 0x492fb29c */
26105a0b428SJohn Marino 6.6660125000e+05, /* 0x4922be94 */
26205a0b428SJohn Marino -2.9449025000e+05, /* 0xc88fcb48 */
26305a0b428SJohn Marino };
26405a0b428SJohn Marino
26505a0b428SJohn Marino static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
26605a0b428SJohn Marino -2.0897993405e-11, /* 0xadb7d219 */
26705a0b428SJohn Marino -1.0253904760e-01, /* 0xbdd1fffe */
26805a0b428SJohn Marino -8.0564479828e+00, /* 0xc100e736 */
26905a0b428SJohn Marino -1.8366960144e+02, /* 0xc337ab6b */
27005a0b428SJohn Marino -1.3731937256e+03, /* 0xc4aba633 */
27105a0b428SJohn Marino -2.6124443359e+03, /* 0xc523471c */
27205a0b428SJohn Marino };
27305a0b428SJohn Marino static const float qs5[6] = {
27405a0b428SJohn Marino 8.1276550293e+01, /* 0x42a28d98 */
27505a0b428SJohn Marino 1.9917987061e+03, /* 0x44f8f98f */
27605a0b428SJohn Marino 1.7468484375e+04, /* 0x468878f8 */
27705a0b428SJohn Marino 4.9851425781e+04, /* 0x4742bb6d */
27805a0b428SJohn Marino 2.7948074219e+04, /* 0x46da5826 */
27905a0b428SJohn Marino -4.7191835938e+03, /* 0xc5937978 */
28005a0b428SJohn Marino };
28105a0b428SJohn Marino
28205a0b428SJohn Marino static const float qr3[6] = {
28305a0b428SJohn Marino -5.0783124372e-09, /* 0xb1ae7d4f */
28405a0b428SJohn Marino -1.0253783315e-01, /* 0xbdd1ff5b */
28505a0b428SJohn Marino -4.6101160049e+00, /* 0xc0938612 */
28605a0b428SJohn Marino -5.7847221375e+01, /* 0xc267638e */
28705a0b428SJohn Marino -2.2824453735e+02, /* 0xc3643e9a */
28805a0b428SJohn Marino -2.1921012878e+02, /* 0xc35b35cb */
28905a0b428SJohn Marino };
29005a0b428SJohn Marino static const float qs3[6] = {
29105a0b428SJohn Marino 4.7665153503e+01, /* 0x423ea91e */
29205a0b428SJohn Marino 6.7386511230e+02, /* 0x4428775e */
29305a0b428SJohn Marino 3.3801528320e+03, /* 0x45534272 */
29405a0b428SJohn Marino 5.5477290039e+03, /* 0x45ad5dd5 */
29505a0b428SJohn Marino 1.9031191406e+03, /* 0x44ede3d0 */
29605a0b428SJohn Marino -1.3520118713e+02, /* 0xc3073381 */
29705a0b428SJohn Marino };
29805a0b428SJohn Marino
29905a0b428SJohn Marino static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
30005a0b428SJohn Marino -1.7838172539e-07, /* 0xb43f8932 */
30105a0b428SJohn Marino -1.0251704603e-01, /* 0xbdd1f475 */
30205a0b428SJohn Marino -2.7522056103e+00, /* 0xc0302423 */
30305a0b428SJohn Marino -1.9663616180e+01, /* 0xc19d4f16 */
30405a0b428SJohn Marino -4.2325313568e+01, /* 0xc2294d1f */
30505a0b428SJohn Marino -2.1371921539e+01, /* 0xc1aaf9b2 */
30605a0b428SJohn Marino };
30705a0b428SJohn Marino static const float qs2[6] = {
30805a0b428SJohn Marino 2.9533363342e+01, /* 0x41ec4454 */
30905a0b428SJohn Marino 2.5298155212e+02, /* 0x437cfb47 */
31005a0b428SJohn Marino 7.5750280762e+02, /* 0x443d602e */
31105a0b428SJohn Marino 7.3939318848e+02, /* 0x4438d92a */
31205a0b428SJohn Marino 1.5594900513e+02, /* 0x431bf2f2 */
31305a0b428SJohn Marino -4.9594988823e+00, /* 0xc09eb437 */
31405a0b428SJohn Marino };
31505a0b428SJohn Marino
31605a0b428SJohn Marino static float
qonef(float x)31705a0b428SJohn Marino qonef(float x)
31805a0b428SJohn Marino {
31905a0b428SJohn Marino const float *p,*q;
32005a0b428SJohn Marino float s,r,z;
32105a0b428SJohn Marino int32_t ix;
32205a0b428SJohn Marino GET_FLOAT_WORD(ix,x);
32305a0b428SJohn Marino ix &= 0x7fffffff;
32405a0b428SJohn Marino if(ix>=0x40200000) {p = qr8; q= qs8;}
32505a0b428SJohn Marino else if(ix>=0x40f71c58){p = qr5; q= qs5;}
32605a0b428SJohn Marino else if(ix>=0x4036db68){p = qr3; q= qs3;}
327*74b7c7a8SJohn Marino else /*if(ix>=0x40000000)*/ {p = qr2; q= qs2;}
32805a0b428SJohn Marino z = one/(x*x);
32905a0b428SJohn Marino r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
33005a0b428SJohn Marino s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
33105a0b428SJohn Marino return ((float).375 + r/s)/x;
33205a0b428SJohn Marino }
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