The pow(x) method appears to not be particularly accurate on HD4600 running Mesa 10.2.7.
What I am seeing is the following:
What I should be seeing is:
The customer is unable to upgrade so a software pow implementation was required.
The software version is based on code by Sun Microsystems, as such the copyright is reproduced here for the code.
A copysignf function is required to be implemented:
The scalebnf function is required as well:
Finally the powf function:
This will potentially kill performance so this should only be used for the offending driver/GPU.
The problem with Mesa has been resolved by around Mesa version 12.1.
Notes: The GIS data is from Natural Earth.
What I am seeing is the following:
What I should be seeing is:
The customer is unable to upgrade so a software pow implementation was required.
The software version is based on code by Sun Microsystems, as such the copyright is reproduced here for the code.
/* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */
A copysignf function is required to be implemented:
float copysignf(in float x, in float y)
{
uint ix = floatBitsToUint(x);
uint iy = floatBitsToUint(y);
return uintBitsToFloat((ix&uint(0x7fffffff))|(iy&uint(0x80000000)));
}
The scalebnf function is required as well:
float scalbnf (in float x, in int n)
{
const float
two25 = 3.355443200e+07, /* 0x4c000000 */
twom25 = 2.9802322388e-08, /* 0x33000000 */
huge = 1.0e+30,
tiny = 1.0e-30;
int ix = floatBitsToInt(x);
int k = (ix&0x7f800000)>>23; /* extract exponent */
if (k==0) { /* 0 or subnormal x */
if ((ix&0x7fffffff)==0) return x; /* +-0 */
x *= two25;
ix = floatBitsToInt(x);
k = ((ix&0x7f800000)>>23) - 25;
if (n< -50000) return tiny*x; /*underflow*/
}
if (k==0xff) return x+x; /* NaN or Inf */
k = k+n;
if (k > 0xfe) return huge*copysignf(huge,x); /* overflow */
if (k > 0) /* normal result */
{
x = intBitsToFloat((ix&0x807fffff)|(k<<23));
return x;
}
if (k <= -25) {
if (n > 50000) /* in case integer overflow in n+k */
return huge*copysignf(huge,x); /*overflow*/
else return tiny*copysignf(tiny,x); /*underflow*/
}
k += 25; /* subnormal result */
x = intBitsToFloat((ix&0x807fffff)|(k<<23));
return x*twom25;
}
Finally the powf function:
float powf(in float x, in float y)
{
const float huge = 1.0e+30, tiny = 1.0e-30;
const float
zero = 0.0,
one = 1.0,
two = 2.0,
two24 = 16777216.0, //0x1.0p+24, /* 16777216.0, 0x4b800000 */
/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
L1 = 6.0000002384e-01, /* 0x3f19999a */
L2 = 4.2857143283e-01, /* 0x3edb6db7 */
L3 = 3.3333334327e-01, /* 0x3eaaaaab */
L4 = 2.7272811532e-01, /* 0x3e8ba305 */
L5 = 2.3066075146e-01, /* 0x3e6c3255 */
L6 = 2.0697501302e-01, /* 0x3e53f142 */
P1 = 1.6666667163e-01, /* 0x3e2aaaab */
P2 = -2.7777778450e-03, /* 0xbb360b61 */
P3 = 6.6137559770e-05, /* 0x388ab355 */
P4 = -1.6533901999e-06, /* 0xb5ddea0e */
P5 = 4.1381369442e-08, /* 0x3331bb4c */
lg2 = 6.9314718246e-01, /* 0x3f317218 */
lg2_h = 6.93145752e-01, /* 0x3f317200 */
lg2_l = 1.42860654e-06, /* 0x35bfbe8c */
ovt = 4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
cp = 9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
cp_h = 9.6179199219e-01, /* 0x3f763800 =head of cp */
cp_l = 4.7017383622e-06, /* 0x369dc3a0 =tail of cp_h */
ivln2 = 1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
ivln2_h = 1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
ivln2_l = 7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/
float bp[2];
bp[0] = 1.0;
bp[1] = 1.5;
float dp_h[2], dp_l[2];
dp_h[0] = 0.0;
dp_h[1] = 5.84960938e-01; /* 0x3f15c000 */
dp_l[0] = 0.0;
dp_l[1] = 1.56322085e-06; /* 0x35d1cfdc */
float z,ax,z_h,z_l,p_h,p_l;
float y1,t1,t2,r,s,t,u,v,w;
int i,j,k,yisint,n;
int hx,hy,ix,iy,is;
hx = floatBitsToInt(x);
hy = floatBitsToInt(y);
ix = hx&0x7fffffff;
iy = hy&0x7fffffff;
/* y==zero: x**0 = 1 */
if(iy==0)
return one;
/* +-NaN return x+y */
if(ix > int(0x7f800000) ||
iy > int(0x7f800000))
return x+y;
/* determine if y is an odd int when x < 0
* yisint = 0 ... y is not an integer
* yisint = 1 ... y is an odd int
* yisint = 2 ... y is an even int
*/
yisint = 0;
if(hx<0) {
if(iy>=0x4b800000)
yisint = 2; /* even integer y */
else if(iy>=0x3f800000) {
k = (iy>>23)-0x7f; /* exponent */
j = iy>>(23-k);
if((j<<(23-k))==iy) yisint = 2-(j&1);
}
}
/* special value of y */
if (iy==0x7f800000) { /* y is +-inf */
if (ix==0x3f800000)
return y - y; /* inf**+-1 is NaN */
else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */
return (hy>=0)? y: zero;
else /* (|x|<1)**-,+inf = inf,0 */
return (hy<0)?-y: zero;
}
if(iy==0x3f800000) { /* y is +-1 */
if(hy<0)
return one/x; else return x;
}
if(hy==0x40000000) return x*x; /* y is 2 */
if(hy==0x3f000000) { /* y is 0.5 */
if(hx>=0) /* x >= +0 */
return sqrt(x);
}
ax = abs(x);
/* special value of x */
if(ix==0x7f800000||ix==0||ix==0x3f800000){
z = ax; /*x is +-0,+-inf,+-1*/
if(hy<0) z = one/z; /* z = (1/|x|) */
if(hx<0) {
if(((ix-0x3f800000)|yisint)==0) {
z = (z-z)/(z-z); /* (-1)**non-int is NaN */
} else if(yisint==1)
z = -z; /* (x<0)**odd = -(|x|**odd) */
}
return z;
}
uint hX = uint(hx);
hX = hX >> 31;
int nN = int(hX) -1;
/* (x<0)**(non-int) is NaN */
if((nN|yisint)==0) return (x-x)/(x-x);
/* |y| is huge */
if(iy>0x4d000000) { /* if |y| > 2**27 */
/* over/underflow if x is not close to one */
if(ix<0x3f7ffff8) return (hy<0)? huge*huge:tiny*tiny;
if(ix>0x3f800007) return (hy>0)? huge*huge:tiny*tiny;
/* now |1-x| is tiny <= 2**-20, suffice to compute
log(x) by x-x^2/2+x^3/3-x^4/4 */
t = x-1; /* t has 20 trailing zeros */
w = (t*t)*(0.5-t*(0.333333333333-t*0.25));
u = ivln2_h*t; /* ivln2_h has 16 sig. bits */
v = t*ivln2_l-w*ivln2;
t1 = u+v;
is = floatBitsToInt(t1);
t1 = intBitsToFloat(is&0xfffff000);
t2 = v-(t1-u);
} else {
float s2,s_h,s_l,t_h,t_l;
n = 0;
/* take care subnormal number */
if(ix<0x00800000)
{
ax *= two24; n -= 24; ix = floatBitsToInt(ax);
}
n += ((ix)>>23)-0x7f;
j = ix&0x007fffff;
/* determine interval */
ix = j|0x3f800000; /* normalize ix */
if(j<=0x1cc471) k=0; /* |x|<sqrt(3/2) */
else if(j<0x5db3d7) k=1; /* |x|<sqrt(3) */
else {k=0;n+=1;ix -= 0x00800000;}
ax = intBitsToFloat(ix);
/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
v = one/(ax+bp[k]);
s = u*v;
s_h = s;
is = floatBitsToInt(s_h);
s_h = intBitsToFloat(is&0xfffff000);
/* t_h=ax+bp[k] High */
t_h = intBitsToFloat(((ix>>1)|0x20000000)+0x0040000+(k<<21));
t_l = ax - (t_h-bp[k]);
s_l = v*((u-s_h*t_h)-s_h*t_l);
/* compute log(ax) */
s2 = s*s;
r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
r += s_l*(s_h+s);
s2 = s_h*s_h;
t_h = 3.0+s2+r;
is = floatBitsToInt(t_h);
t_h = intBitsToFloat(is&0xfffff000);
t_l = r-((t_h-3.0)-s2);
/* u+v = s*(1+...) */
u = s_h*t_h;
v = s_l*t_h+t_l*s;
/* 2/(3log2)*(s+...) */
p_h = u+v;
is = floatBitsToInt(p_h);
p_h = intBitsToFloat(is&0xfffff000);
p_l = v-(p_h-u);
z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
z_l = cp_l*p_h+p_l*cp+dp_l[k];
/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
t = n;
t1 = (((z_h+z_l)+dp_h[k])+t);
is = floatBitsToInt(t1);
t1 = intBitsToFloat(is&0xfffff000);
t2 = z_l-(((t1-t)-dp_h[k])-z_h);
}
s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
hX = uint(hx);
hX = hX >> 31;
nN = int(hX) -1;
if((nN|(yisint-1))==0)
s = -one; /* (-ve)**(odd int) */
/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
is = floatBitsToInt(y);
y1 = intBitsToFloat(is&0xfffff000);
p_l = (y-y1)*t1+y*t2;
p_h = y1*t1;
z = p_l+p_h;
j = floatBitsToInt(z);
if (j>0x43000000) /* if z > 128 */
return s*huge*huge; /* overflow */
else if (j==0x43000000) { /* if z == 128 */
if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
}
else if ((j&0x7fffffff)>0x43160000) /* z <= -150 */
return s*tiny*tiny; /* underflow */
else if (uint(j)==0xc3160000u){ /* z == -150 */
if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
}
/*
* compute 2**(p_h+p_l)
*/
i = j&0x7fffffff;
k = (i>>23)-0x7f;
n = 0;
if(i>0x3f000000) { /* if |z| > 0.5, set n = [z+0.5] */
n = j+(0x00800000>>(k+1));
k = ((n&0x7fffffff)>>23)-0x7f; /* new k for n */
t = intBitsToFloat(n&~(0x007fffff>>k));
n = ((n&0x007fffff)|0x00800000)>>(23-k);
if(j<0) n = -n;
p_h -= t;
}
t = p_l+p_h;
is = floatBitsToInt(t);
t = intBitsToFloat(is&0xfffff000);
u = t*lg2_h;
v = (p_l-(t-p_h))*lg2+t*lg2_l;
z = u+v;
w = v-(z-u);
t = z*z;
t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
r = (z*t1)/(t1-two)-(w+z*w);
z = one-(r-z);
j = floatBitsToInt(z);
j += (n<<23);
if((j>>23)<=0) z = scalbnf(z,n); /* subnormal output */
else z = intBitsToFloat(j);
return s*z;
}
This will potentially kill performance so this should only be used for the offending driver/GPU.
The problem with Mesa has been resolved by around Mesa version 12.1.
Notes: The GIS data is from Natural Earth.
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