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/*--------------------------------------------------------------------
 * TITLE: Plasma Floating Point Library
 * AUTHOR: Steve Rhoads (rhoadss@yahoo.com)
 * DATE CREATED: 3/2/06
 * FILENAME: math.c
 * PROJECT: Plasma CPU core
 * COPYRIGHT: Software placed into the public domain by the author.
 *    Software 'as is' without warranty.  Author liable for nothing.
 * DESCRIPTION:
 *    Plasma Floating Point Library
 *--------------------------------------------------------------------
 * IEEE_fp = sign(1) | exponent(8) | fraction(23)
 * cos(x)=1-x^2/2!+x^4/4!-x^6/6!+...
 * exp(x)=1+x+x^2/2!+x^3/3!+...
 * ln(1+x)=x-x^2/2+x^3/3-x^4/4+...
 * atan(x)=x-x^3/3+x^5/5-x^7/7+...
 * pow(x,y)=exp(y*ln(x))
 * x=tan(a+b)=(tan(a)+tan(b))/(1-tan(a)*tan(b))
 * atan(x)=b+atan((x-atan(b))/(1+x*atan(b)))
 * ln(a*x)=ln(a)+ln(x); ln(x^n)=n*ln(x)
 *--------------------------------------------------------------------*/
#include "rtos.h"
 
//#define USE_SW_MULT
#if !defined(WIN32) && !defined(USE_SW_MULT)
#define USE_MULT64
#endif
 
#define PI ((float)3.1415926)
#define PI_2 ((float)(PI/2.0))
#define PI2 ((float)(PI*2.0))
 
#define FtoL(X) (*(unsigned long*)&(X))
#define LtoF(X) (*(float*)&(X))
 
 
float FP_Neg(float a_fp)
{
   unsigned long a;
   a = FtoL(a_fp);
   a ^= 0x80000000;
   return LtoF(a);
}
 
 
float FP_Add(float a_fp, float b_fp)
{
   unsigned long a, b, c;
   unsigned long as, bs, cs;     //sign
   long ae, af, be, bf, ce, cf;  //exponent and fraction
   a = FtoL(a_fp);
   b = FtoL(b_fp);
   as = a >> 31;                        //sign
   ae = (a >> 23) & 0xff;               //exponent
   af = 0x00800000 | (a & 0x007fffff);  //fraction
   bs = b >> 31;
   be = (b >> 23) & 0xff;
   bf = 0x00800000 | (b & 0x007fffff);
   if(ae > be)
   {
      if(ae - be < 30)
         bf >>= ae - be;
      else
         bf = 0;
      ce = ae;
   }
   else
   {
      if(be - ae < 30)
         af >>= be - ae;
      else
         af = 0;
      ce = be;
   }
   cf = (as ? -af : af) + (bs ? -bf : bf);
   cs = cf < 0;
   cf = cf>=0 ? cf : -cf;
   if(cf == 0)
      return LtoF(cf);
   while(cf & 0xff000000)
   {
      ++ce;
      cf >>= 1;
   }
   while((cf & 0xff800000) == 0)
   {
      --ce;
      cf <<= 1;
   }
   c = (cs << 31) | (ce << 23) | (cf & 0x007fffff);
   if(ce < 1)
      c = 0;
   return LtoF(c);
}
 
 
float FP_Sub(float a_fp, float b_fp)
{
   return FP_Add(a_fp, FP_Neg(b_fp));
}
 
 
float FP_Mult(float a_fp, float b_fp)
{
   unsigned long a, b, c;
   unsigned long as, af, bs, bf, cs, cf;
   long ae, be, ce;
#ifndef USE_MULT64
   unsigned long a2, a1, b2, b1, med1, med2;
#endif
   unsigned long hi, lo;
   a = FtoL(a_fp);
   b = FtoL(b_fp);
   as = a >> 31;
   ae = (a >> 23) & 0xff;
   af = 0x00800000 | (a & 0x007fffff);
   bs = b >> 31;
   be = (b >> 23) & 0xff;
   bf = 0x00800000 | (b & 0x007fffff);
   cs = as ^ bs;
#ifndef USE_MULT64
   a1 = af & 0xffff;
   a2 = af >> 16;
   b1 = bf & 0xffff;
   b2 = bf >> 16;
   lo = a1 * b1;
   med1 = a2 * b1 + (lo >> 16);
   med2 = a1 * b2;
   hi = a2 * b2 + (med1 >> 16) + (med2 >> 16);
   med1 = (med1 & 0xffff) + (med2 & 0xffff);
   hi += (med1 >> 16);
   lo = (med1 << 16) | (lo & 0xffff);
#else
   lo = OS_AsmMult(af, bf, &hi);
#endif
   cf = (hi << 9) | (lo >> 23);
   ce = ae + be - 0x80 + 1;
   if(cf == 0)
      return LtoF(cf);
   while(cf & 0xff000000)
   {
      ++ce;
      cf >>= 1;
   }
   c = (cs << 31) | (ce << 23) | (cf & 0x007fffff);
   if(ce < 1)
      c = 0;
   return LtoF(c);
}
 
 
float FP_Div(float a_fp, float b_fp)
{
   unsigned long a, b, c;
   unsigned long as, af, bs, bf, cs, cf;
   unsigned long a1, b1;
#ifndef USE_MULT64
   unsigned long a2, b2, med1, med2;
#endif
   unsigned long hi, lo;
   long ae, be, ce, d;
   a = FtoL(a_fp);
   b = FtoL(b_fp);
   as = a >> 31;
   ae = (a >> 23) & 0xff;
   af = 0x00800000 | (a & 0x007fffff);
   bs = b >> 31;
   be = (b >> 23) & 0xff;
   bf = 0x00800000 | (b & 0x007fffff);
   cs = as ^ bs;
   ce = ae - (be - 0x80) + 6 - 8;
   a1 = af << 4; //8
   b1 = bf >> 8;
   cf = a1 / b1;
   cf <<= 12; //8
#if 1                  /*non-quick*/
#ifndef USE_MULT64
   a1 = cf & 0xffff;
   a2 = cf >> 16;
   b1 = bf & 0xffff;
   b2 = bf >> 16;
   lo = a1 * b1;
   med1 =a2 * b1 + (lo >> 16);
   med2 = a1 * b2;
   hi = a2 * b2 + (med1 >> 16) + (med2 >> 16);
   med1 = (med1 & 0xffff) + (med2 & 0xffff);
   hi += (med1 >> 16);
   lo = (med1 << 16) | (lo & 0xffff);
#else
   lo = OS_AsmMult(cf, bf, &hi);
#endif
   lo = (hi << 8) | (lo >> 24);
   d = af - lo;    //remainder
   assert(-0xffff < d && d < 0xffff);
   d <<= 16;
   b1 = bf >> 8;
   d = d / (long)b1;
   cf += d;
#endif
   if(cf == 0)
      return LtoF(cf);
   while(cf & 0xff000000)
   {
      ++ce;
      cf >>= 1;
   }
   if(ce < 0)
      ce = 0;
   c = (cs << 31) | (ce << 23) | (cf & 0x007fffff);
   if(ce < 1)
      c = 0;
   return LtoF(c);
}
 
 
long FP_ToLong(float a_fp)
{
   unsigned long a;
   unsigned long as;
   long ae;
   long af, shift;
   a = FtoL(a_fp);
   as = a >> 31;
   ae = (a >> 23) & 0xff;
   af = 0x00800000 | (a & 0x007fffff);
   af <<= 7;
   shift = -(ae - 0x80 - 29);
   if(shift > 0)
   {
      if(shift < 31)
         af >>= shift;
      else
         af = 0;
   }
   af = as ? -af: af;
   return af;
}
 
 
float FP_ToFloat(long af)
{
   unsigned long a;
   unsigned long as, ae;
   as = af>=0 ? 0: 1;
   af = af>=0 ? af: -af;
   ae = 0x80 + 22;
   if(af == 0)
      return LtoF(af);
   while(af & 0xff000000)
   {
      ++ae;
      af >>= 1;
   }
   while((af & 0xff800000) == 0)
   {
      --ae;
      af <<= 1;
   }
   a = (as << 31) | (ae << 23) | (af & 0x007fffff);
   return LtoF(a);
}
 
 
//0 iff a==b; 1 iff a>b; -1 iff a<b
int FP_Cmp(float a_fp, float b_fp)
{
   unsigned long a, b;
   unsigned long as, ae, af, bs, be, bf;
   int gt;
   a = FtoL(a_fp);
   b = FtoL(b_fp);
   if(a == b)
      return 0;
   as = a >> 31;
   bs = b >> 31;
   if(as > bs)
      return -1;
   if(as < bs)
      return 1;
   gt = as ? -1 : 1;
   ae = (a >> 23) & 0xff;
   be = (b >> 23) & 0xff;
   if(ae > be)
      return gt;
   if(ae < be)
      return -gt;
   af = 0x00800000 | (a & 0x007fffff);
   bf = 0x00800000 | (b & 0x007fffff);
   if(af > bf)
      return gt;
   return -gt;
}
 
 
int __ltsf2(float a, float b)
{
   return FP_Cmp(a, b);
}
 
int __lesf2(float a, float b)
{
   return FP_Cmp(a, b);
}
 
int __gtsf2(float a, float b)
{
   return FP_Cmp(a, b);
}
 
int __gesf2(float a, float b)
{
   return FP_Cmp(a, b);
}
 
int __eqsf2(float a, float b)
{
   return FtoL(a) != FtoL(b);
}
 
int __nesf2(float a, float b)
{
   return FtoL(a) != FtoL(b);
}
 
 
float FP_Sqrt(float a)
{
   float x1, y1, x2, y2, x3;
   long i;
   x1 = FP_ToFloat(1);
   y1 = FP_Sub(FP_Mult(x1, x1), a);  //y1=x1*x1-a;
   x2 = FP_ToFloat(100);
   y2 = FP_Sub(FP_Mult(x2, x2), a);
   for(i = 0; i < 10; ++i)
   {
      if(FtoL(y1) == FtoL(y2))
         return x2;
      //x3=x2-(x1-x2)*y2/(y1-y2);
      x3 = FP_Sub(x2, FP_Div(FP_Mult(FP_Sub(x1, x2), y2), FP_Sub(y1, y2)));
      x1 = x2;
      y1 = y2;
      x2 = x3;
      y2 = FP_Sub(FP_Mult(x2, x2), a);
   }
   return x2;
}
 
 
float FP_Cos(float rad)
{
   int n;
   float answer, x2, top, bottom, sign;
   while(FP_Cmp(rad, PI2) > 0)
      rad = FP_Sub(rad, PI2);
   while(FP_Cmp(rad, (float)0.0) < 0)
      rad = FP_Add(rad, PI2);
   answer = 1.0;
   sign = 1.0;
   if(FP_Cmp(rad, PI) >= 0)
   {
      rad = FP_Sub(rad, PI);
      sign = FP_ToFloat(-1);
   }
   if(FP_Cmp(rad, PI_2) >= 0)
   {
      rad = FP_Sub(PI, rad);
      sign = FP_Neg(sign);
   }
   x2 = FP_Mult(rad, rad);
   top = 1.0;
   bottom = 1.0;
   for(n = 2; n < 12; n += 2)
   {
      top = FP_Mult(top, FP_Neg(x2));
      bottom = FP_Mult(bottom, FP_ToFloat((n - 1) * n));
      answer = FP_Add(answer, FP_Div(top, bottom));
   }
   return FP_Mult(answer, sign);
}
 
 
float FP_Sin(float rad)
{
   const float pi_2=(float)(PI/2.0);
   return FP_Cos(FP_Sub(rad, pi_2));
}
 
 
float FP_Atan(float x)
{
   const float b=(float)(PI/8.0);
   const float atan_b=(float)0.37419668; //atan(b);
   int n;
   float answer, x2, top;
   if(FP_Cmp(x, (float)0.0) >= 0)
   {
      if(FP_Cmp(x, (float)1.0) > 0)
         return FP_Sub(PI_2, FP_Atan(FP_Div((float)1.0, x)));
   }
   else
   {
      if(FP_Cmp(x, (float)-1.0) > 0)
         return FP_Sub(-PI_2, FP_Atan(FP_Div((float)1.0, x)));
   }
   if(FP_Cmp(x, (float)0.45) > 0)
   {
      //answer = (x - atan_b) / (1 + x * atan_b);
      answer = FP_Div(FP_Sub(x, atan_b), FP_Add(1.0, FP_Mult(x, atan_b)));
      //answer = b + FP_Atan(answer) - (float)0.034633; /*FIXME fudge?*/
      answer = FP_Sub(FP_Add(b, FP_Atan(answer)), (float)0.034633);
      return answer;
   }
   if(FP_Cmp(x, (float)-0.45) < 0)
   {
      x = FP_Neg(x);
      //answer = (x - atan_b) / (1 + x * atan_b);
      answer = FP_Div(FP_Sub(x, atan_b), FP_Add(1.0, FP_Mult(x, atan_b)));
      //answer = b + FP_Atan(answer) - (float)0.034633; /*FIXME*/
      answer = FP_Sub(FP_Add(b, FP_Atan(answer)), (float)0.034633);
      return FP_Neg(answer);
   }
   answer = x;
   x2 = FP_Mult(FP_Neg(x), x);
   top = x;
   for(n = 3; n < 14; n += 2)
   {
      top = FP_Mult(top, x2);
      answer = FP_Add(answer, FP_Div(top, FP_ToFloat(n)));
   }
   return answer;
}
 
 
float FP_Atan2(float y, float x)
{
   float answer,r;
   r = y / x;
   answer = FP_Atan(r);
   if(FP_Cmp(x, (float)0.0) < 0)
   {
      if(FP_Cmp(y, (float)0.0) > 0)
         answer = FP_Add(answer, PI);
      else
         answer = FP_Sub(answer, PI);
   }
   return answer;
}
 
 
float FP_Exp(float x)
{
   const float e2=(float)7.389056099;
   const float inv_e2=(float)0.135335283;
   float answer, top, bottom, mult;
   int n;
 
   mult = 1.0;
   while(FP_Cmp(x, (float)2.0) > 0)
   {
      mult = FP_Mult(mult, e2);
      x = FP_Add(x, (float)-2.0);
   }
   while(FP_Cmp(x, (float)-2.0) < 0)
   {
      mult = FP_Mult(mult, inv_e2);
      x = FP_Add(x, (float)2.0);
   }
   answer = FP_Add((float)1.0, x);
   top = x;
   bottom = 1.0;
   for(n = 2; n < 15; ++n)
   {
      top = FP_Mult(top, x);
      bottom = FP_Mult(bottom, FP_ToFloat(n));
      answer = FP_Add(answer, FP_Div(top, bottom));
   }
   return FP_Mult(answer, mult);
}
 
 
float FP_Log(float x)
{
   const float log_2=(float)0.69314718; /*log(2.0)*/
   int n;
   float answer, top, add;
   add = 0.0;
   while(FP_Cmp(x, 16.0) > 0)
   {
      x = FP_Mult(x, (float)0.0625);
      add = FP_Add(add, (float)(log_2 * 4));
   }
   while(FP_Cmp(x, 1.5) > 0)
   {
      x = FP_Mult(x, 0.5);
      add = FP_Add(add, log_2);
   }
   while(FP_Cmp(x, 0.5) < 0)
   {
      x = FP_Mult(x, 2.0);
      add = FP_Sub(add, log_2);
   }
   x = FP_Sub(x, 1.0);
   answer = 0.0;
   top = -1.0;
   for(n = 1; n < 14; ++n)
   {
      top = FP_Mult(top, FP_Neg(x));
      answer = FP_Add(answer, FP_Div(top, FP_ToFloat(n)));
   }
   return FP_Add(answer, add);
}
 
 
float FP_Pow(float x, float y)
{
   return FP_Exp(y * FP_Log(x));
}
 
 
/********************************************/
//These five functions will only be used if the flag "-mno-mul" is enabled
#ifdef USE_SW_MULT
unsigned long __mulsi3(unsigned long a, unsigned long b)
{
   unsigned long answer = 0;
   while(b)
   {
      if(b & 1)
         answer += a;
      a <<= 1;
      b >>= 1;
   }
   return answer;
}
 
 
static unsigned long DivideMod(unsigned long a, unsigned long b, int doMod)
{
   unsigned long upper=a, lower=0;
   int i;
   a = b << 31;
   for(i = 0; i < 32; ++i)
   {
      lower = lower << 1;
      if(upper >= a && a && b < 2)
      {
         upper = upper - a;
         lower |= 1;
      }
      a = ((b&2) << 30) | (a >> 1);
      b = b >> 1;
   }
   if(!doMod)
      return lower;
   return upper;
}
 
 
unsigned long __udivsi3(unsigned long a, unsigned long b)
{
   return DivideMod(a, b, 0);
}
 
 
long __divsi3(long a, long b)
{
   long answer, negate=0;
   if(a < 0)
   {
      a = -a;
      negate = !negate;
   }
   if(b < 0)
   {
      b = -b;
      negate = !negate;
   }
   answer = DivideMod(a, b, 0);
   if(negate)
      answer = -answer;
   return answer;
}
 
 
unsigned long __umodsi3(unsigned long a, unsigned long b)
{
   return DivideMod(a, b, 1);
}
#endif
 
 
/*************** Test *****************/
#ifdef WIN32
#undef _LIBC
#include <math.h>
struct {
   char *name;
   float low, high;
   double (*func1)(double);
   float (*func2)(float);
} test_info[]={
   {"cos", -2*PI, 2*PI, cos, FP_Cos},
   {"sin", -2*PI, 2*PI, sin, FP_Sin},
   {"atan", -3.0, 2.0, atan, FP_Atan},
   {"log", (float)0.01, (float)4.0, log, FP_Log},
   {"exp", (float)-5.01, (float)30.0, exp, FP_Exp},
   {"sqrt", (float)0.01, (float)1000.0, sqrt, FP_Sqrt}
};
 
 
void TestMathFull(void)
{
   float a, b, c, d;
   float error1, error2, error3, error4, error5;
   int test;
 
   a = PI * PI;
   b = PI;
   c = FP_Div(a, b);
   printf("%10f %10f %10f %10f %10f\n",
      (double)a, (double)b, (double)(a/b), (double)c, (double)(a/b-c));
   a = a * 200;
   for(b = -(float)2.718281828*100; b < 300; b += (float)23.678)
   {
      c = FP_Div(a, b);
      d = a / b - c;
      printf("%10f %10f %10f %10f %10f\n",
         (double)a, (double)b, (double)(a/b), (double)c, (double)(a/b-c));
   }
   //getch();
 
   for(test = 0; test < 6; ++test)
   {
      printf("\nTesting %s\n", test_info[test].name);
      for(a = test_info[test].low;
          a <= test_info[test].high;
          a += (test_info[test].high-test_info[test].low)/(float)20.0)
      {
         b = (float)test_info[test].func1(a);
         c = test_info[test].func2(a);
         d = b - c;
         printf("%s %10f %10f %10f %10f\n", test_info[test].name, a, b, c, d);
      }
      //getch();
   }
 
   a = FP_ToFloat((long)6.0);
   b = FP_ToFloat((long)2.0);
   printf("%f %f\n", (double)a, (double)b);
   c = FP_Add(a, b);
   printf("add %f %f\n", (double)(a + b), (double)c);
   c = FP_Sub(a, b);
   printf("sub %f %f\n", (double)(a - b), (double)c);
   c = FP_Mult(a, b);
   printf("mult %f %f\n", (double)(a * b), (double)c);
   c = FP_Div(a, b);
   printf("div %f %f\n", (double)(a / b), (double)c);
   //getch();
 
   for(a = (float)-13756.54; a < (float)17400.0; a += (float)64.45)
   {
      for(b = (float)-875.36; b < (float)935.8; b += (float)36.7)
      {
         error1 = (float)1.0 - (a + b) / FP_Add(a, b);
         error2 = (float)1.0 - (a * b) / FP_Mult(a, b);
         error3 = (float)1.0 - (a / b) / FP_Div(a, b);
         error4 = (float)1.0 - a / FP_ToFloat(FP_ToLong(a));
         error5 = error1 + error2 + error3 + error4;
         if(error5 < 0.00005)
            continue;
         printf("ERROR!\n");
         printf("a=%f b=%f\n", (double)a, (double)b);
         printf("  a+b=%f %f\n", (double)(a+b), (double)FP_Add(a, b));
         printf("  a*b=%f %f\n", (double)(a*b), (double)FP_Mult(a, b));
         printf("  a/b=%f %f\n", (double)(a/b), (double)FP_Div(a, b));
         printf("  a=%f %ld %f\n", (double)a, FP_ToLong(a),
                                   (double)FP_ToFloat((long)a));
         printf("  %f %f %f %f\n", (double)error1, (double)error2,
            (double)error3, (double)error4);
         //if(error5 > 0.001) 
         //   getch();
      }
   }
   printf("done.\n");
   //getch();
}
#endif
 
 

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[/] [plasma/] [trunk/] [kernel/] [math.c] - Blame information for rev 421

Line No.	Rev	Author	Line
1	138	rhoads	`/*--------------------------------------------------------------------`
2			`* TITLE: Plasma Floating Point Library`
3			`* AUTHOR: Steve Rhoads (rhoadss@yahoo.com)`
4			`* DATE CREATED: 3/2/06`
5			`* FILENAME: math.c`
6			`* PROJECT: Plasma CPU core`
7			`* COPYRIGHT: Software placed into the public domain by the author.`
8			`* Software 'as is' without warranty. Author liable for nothing.`
9			`* DESCRIPTION:`
10			`* Plasma Floating Point Library`
11			`*--------------------------------------------------------------------`
12			`* IEEE_fp = sign(1) \| exponent(8) \| fraction(23)`
13			`* cos(x)=1-x^2/2!+x^4/4!-x^6/6!+...`
14			`* exp(x)=1+x+x^2/2!+x^3/3!+...`
15			`* ln(1+x)=x-x^2/2+x^3/3-x^4/4+...`
16			`* atan(x)=x-x^3/3+x^5/5-x^7/7+...`
17			`* pow(x,y)=exp(y*ln(x))`
18			`* x=tan(a+b)=(tan(a)+tan(b))/(1-tan(a)*tan(b))`
19			`* atan(x)=b+atan((x-atan(b))/(1+x*atan(b)))`
20			`* ln(ax)=ln(a)+ln(x); ln(x^n)=nln(x)`
21			`--------------------------------------------------------------------/`
22			`#include "rtos.h"`
23
24	195	rhoads	`//#define USE_SW_MULT`
25			`#if !defined(WIN32) && !defined(USE_SW_MULT)`
26			`#define USE_MULT64`
27			`#endif`
28
29	138	rhoads	`#define PI ((float)3.1415926)`
30			`#define PI_2 ((float)(PI/2.0))`
31			`#define PI2 ((float)(PI*2.0))`
32
33			`#define FtoL(X) ((unsigned long)&(X))`
34			`#define LtoF(X) ((float)&(X))`
35
36
37			`float FP_Neg(float a_fp)`
38			`{`
39			`unsigned long a;`
40			`a = FtoL(a_fp);`
41			`a ^= 0x80000000;`
42			`return LtoF(a);`
43			`}`
44
45
46			`float FP_Add(float a_fp, float b_fp)`
47			`{`
48			`unsigned long a, b, c;`
49			`unsigned long as, bs, cs; //sign`
50			`long ae, af, be, bf, ce, cf; //exponent and fraction`
51			`a = FtoL(a_fp);`
52			`b = FtoL(b_fp);`
53			`as = a >> 31; //sign`
54			`ae = (a >> 23) & 0xff; //exponent`
55			`af = 0x00800000 \| (a & 0x007fffff); //fraction`
56			`bs = b >> 31;`
57			`be = (b >> 23) & 0xff;`
58			`bf = 0x00800000 \| (b & 0x007fffff);`
59			`if(ae > be)`
60			`{`
61			`if(ae - be < 30)`
62			`bf >>= ae - be;`
63			`else`
64			`bf = 0;`
65			`ce = ae;`
66			`}`
67			`else`
68			`{`
69			`if(be - ae < 30)`
70			`af >>= be - ae;`
71			`else`
72			`af = 0;`
73			`ce = be;`
74			`}`
75			`cf = (as ? -af : af) + (bs ? -bf : bf);`
76			`cs = cf < 0;`
77			`cf = cf>=0 ? cf : -cf;`
78			`if(cf == 0)`
79			`return LtoF(cf);`
80			`while(cf & 0xff000000)`
81			`{`
82			`++ce;`
83			`cf >>= 1;`
84			`}`
85			`while((cf & 0xff800000) == 0)`
86			`{`
87			`--ce;`
88			`cf <<= 1;`
89			`}`
90			`c = (cs << 31) \| (ce << 23) \| (cf & 0x007fffff);`
91			`if(ce < 1)`
92			`c = 0;`
93			`return LtoF(c);`
94			`}`
95
96
97			`float FP_Sub(float a_fp, float b_fp)`
98			`{`
99			`return FP_Add(a_fp, FP_Neg(b_fp));`
100			`}`
101
102
103			`float FP_Mult(float a_fp, float b_fp)`
104			`{`
105			`unsigned long a, b, c;`
106			`unsigned long as, af, bs, bf, cs, cf;`
107			`long ae, be, ce;`
108	195	rhoads	`#ifndef USE_MULT64`
109	138	rhoads	`unsigned long a2, a1, b2, b1, med1, med2;`
110			`#endif`
111			`unsigned long hi, lo;`
112			`a = FtoL(a_fp);`
113			`b = FtoL(b_fp);`
114			`as = a >> 31;`
115			`ae = (a >> 23) & 0xff;`
116			`af = 0x00800000 \| (a & 0x007fffff);`
117			`bs = b >> 31;`
118			`be = (b >> 23) & 0xff;`
119			`bf = 0x00800000 \| (b & 0x007fffff);`
120			`cs = as ^ bs;`
121	195	rhoads	`#ifndef USE_MULT64`
122	138	rhoads	`a1 = af & 0xffff;`
123			`a2 = af >> 16;`
124			`b1 = bf & 0xffff;`
125			`b2 = bf >> 16;`
126			`lo = a1 * b1;`
127			`med1 = a2 * b1 + (lo >> 16);`
128			`med2 = a1 * b2;`
129			`hi = a2 * b2 + (med1 >> 16) + (med2 >> 16);`
130			`med1 = (med1 & 0xffff) + (med2 & 0xffff);`
131			`hi += (med1 >> 16);`
132			`lo = (med1 << 16) \| (lo & 0xffff);`
133			`#else`
134			`lo = OS_AsmMult(af, bf, &hi);`
135			`#endif`
136			`cf = (hi << 9) \| (lo >> 23);`
137			`ce = ae + be - 0x80 + 1;`
138			`if(cf == 0)`
139			`return LtoF(cf);`
140			`while(cf & 0xff000000)`
141			`{`
142			`++ce;`
143			`cf >>= 1;`
144			`}`
145			`c = (cs << 31) \| (ce << 23) \| (cf & 0x007fffff);`
146			`if(ce < 1)`
147			`c = 0;`
148			`return LtoF(c);`
149			`}`
150
151
152			`float FP_Div(float a_fp, float b_fp)`
153			`{`
154			`unsigned long a, b, c;`
155			`unsigned long as, af, bs, bf, cs, cf;`
156			`unsigned long a1, b1;`
157	195	rhoads	`#ifndef USE_MULT64`
158	138	rhoads	`unsigned long a2, b2, med1, med2;`
159			`#endif`
160			`unsigned long hi, lo;`
161			`long ae, be, ce, d;`
162			`a = FtoL(a_fp);`
163			`b = FtoL(b_fp);`
164			`as = a >> 31;`
165			`ae = (a >> 23) & 0xff;`
166			`af = 0x00800000 \| (a & 0x007fffff);`
167			`bs = b >> 31;`
168			`be = (b >> 23) & 0xff;`
169			`bf = 0x00800000 \| (b & 0x007fffff);`
170			`cs = as ^ bs;`
171			`ce = ae - (be - 0x80) + 6 - 8;`
172			`a1 = af << 4; //8`
173			`b1 = bf >> 8;`
174			`cf = a1 / b1;`
175			`cf <<= 12; //8`
176			`#if 1 /non-quick/`
177	195	rhoads	`#ifndef USE_MULT64`
178	138	rhoads	`a1 = cf & 0xffff;`
179			`a2 = cf >> 16;`
180			`b1 = bf & 0xffff;`
181			`b2 = bf >> 16;`
182			`lo = a1 * b1;`
183			`med1 =a2 * b1 + (lo >> 16);`
184			`med2 = a1 * b2;`
185			`hi = a2 * b2 + (med1 >> 16) + (med2 >> 16);`
186			`med1 = (med1 & 0xffff) + (med2 & 0xffff);`
187			`hi += (med1 >> 16);`
188			`lo = (med1 << 16) \| (lo & 0xffff);`
189			`#else`
190			`lo = OS_AsmMult(cf, bf, &hi);`
191			`#endif`
192			`lo = (hi << 8) \| (lo >> 24);`
193			`d = af - lo; //remainder`
194			`assert(-0xffff < d && d < 0xffff);`
195			`d <<= 16;`
196			`b1 = bf >> 8;`
197			`d = d / (long)b1;`
198			`cf += d;`
199			`#endif`
200			`if(cf == 0)`
201			`return LtoF(cf);`
202			`while(cf & 0xff000000)`
203			`{`
204			`++ce;`
205			`cf >>= 1;`
206			`}`
207			`if(ce < 0)`
208			`ce = 0;`
209			`c = (cs << 31) \| (ce << 23) \| (cf & 0x007fffff);`
210			`if(ce < 1)`
211			`c = 0;`
212			`return LtoF(c);`
213			`}`
214
215
216			`long FP_ToLong(float a_fp)`
217			`{`
218			`unsigned long a;`
219			`unsigned long as;`
220			`long ae;`
221			`long af, shift;`
222			`a = FtoL(a_fp);`
223			`as = a >> 31;`
224			`ae = (a >> 23) & 0xff;`
225			`af = 0x00800000 \| (a & 0x007fffff);`
226			`af <<= 7;`
227			`shift = -(ae - 0x80 - 29);`
228			`if(shift > 0)`
229			`{`
230			`if(shift < 31)`
231			`af >>= shift;`
232			`else`
233			`af = 0;`
234			`}`
235			`af = as ? -af: af;`
236			`return af;`
237			`}`
238
239
240			`float FP_ToFloat(long af)`
241			`{`
242			`unsigned long a;`
243			`unsigned long as, ae;`
244			`as = af>=0 ? 0: 1;`
245			`af = af>=0 ? af: -af;`
246			`ae = 0x80 + 22;`
247			`if(af == 0)`
248			`return LtoF(af);`
249			`while(af & 0xff000000)`
250			`{`
251			`++ae;`
252			`af >>= 1;`
253			`}`
254			`while((af & 0xff800000) == 0)`
255			`{`
256			`--ae;`
257			`af <<= 1;`
258			`}`
259			`a = (as << 31) \| (ae << 23) \| (af & 0x007fffff);`
260			`return LtoF(a);`
261			`}`
262
263
264			`//0 iff a==b; 1 iff a>b; -1 iff a<b`
265			`int FP_Cmp(float a_fp, float b_fp)`
266			`{`
267			`unsigned long a, b;`
268			`unsigned long as, ae, af, bs, be, bf;`
269			`int gt;`
270			`a = FtoL(a_fp);`
271			`b = FtoL(b_fp);`
272			`if(a == b)`
273			`return 0;`
274			`as = a >> 31;`
275			`bs = b >> 31;`
276			`if(as > bs)`
277			`return -1;`
278			`if(as < bs)`
279			`return 1;`
280			`gt = as ? -1 : 1;`
281			`ae = (a >> 23) & 0xff;`
282			`be = (b >> 23) & 0xff;`
283			`if(ae > be)`
284			`return gt;`
285			`if(ae < be)`
286			`return -gt;`
287			`af = 0x00800000 \| (a & 0x007fffff);`
288			`bf = 0x00800000 \| (b & 0x007fffff);`
289			`if(af > bf)`
290			`return gt;`
291			`return -gt;`
292			`}`
293
294
295			`int __ltsf2(float a, float b)`
296			`{`
297			`return FP_Cmp(a, b);`
298			`}`
299
300			`int __lesf2(float a, float b)`
301			`{`
302			`return FP_Cmp(a, b);`
303			`}`
304
305			`int __gtsf2(float a, float b)`
306			`{`
307			`return FP_Cmp(a, b);`
308			`}`
309
310			`int __gesf2(float a, float b)`
311			`{`
312			`return FP_Cmp(a, b);`
313			`}`
314
315			`int __eqsf2(float a, float b)`
316			`{`
317			`return FtoL(a) != FtoL(b);`
318			`}`
319
320			`int __nesf2(float a, float b)`
321			`{`
322			`return FtoL(a) != FtoL(b);`
323			`}`
324
325
326			`float FP_Sqrt(float a)`
327			`{`
328			`float x1, y1, x2, y2, x3;`
329			`long i;`
330			`x1 = FP_ToFloat(1);`
331			`y1 = FP_Sub(FP_Mult(x1, x1), a); //y1=x1*x1-a;`
332			`x2 = FP_ToFloat(100);`
333			`y2 = FP_Sub(FP_Mult(x2, x2), a);`
334			`for(i = 0; i < 10; ++i)`
335			`{`
336			`if(FtoL(y1) == FtoL(y2))`
337			`return x2;`
338			`//x3=x2-(x1-x2)*y2/(y1-y2);`
339			`x3 = FP_Sub(x2, FP_Div(FP_Mult(FP_Sub(x1, x2), y2), FP_Sub(y1, y2)));`
340			`x1 = x2;`
341			`y1 = y2;`
342			`x2 = x3;`
343			`y2 = FP_Sub(FP_Mult(x2, x2), a);`
344			`}`
345			`return x2;`
346			`}`
347
348
349			`float FP_Cos(float rad)`
350			`{`
351			`int n;`
352			`float answer, x2, top, bottom, sign;`
353			`while(FP_Cmp(rad, PI2) > 0)`
354			`rad = FP_Sub(rad, PI2);`
355			`while(FP_Cmp(rad, (float)0.0) < 0)`
356			`rad = FP_Add(rad, PI2);`
357			`answer = 1.0;`
358			`sign = 1.0;`
359			`if(FP_Cmp(rad, PI) >= 0)`
360			`{`
361			`rad = FP_Sub(rad, PI);`
362			`sign = FP_ToFloat(-1);`
363			`}`
364			`if(FP_Cmp(rad, PI_2) >= 0)`
365			`{`
366			`rad = FP_Sub(PI, rad);`
367			`sign = FP_Neg(sign);`
368			`}`
369			`x2 = FP_Mult(rad, rad);`
370			`top = 1.0;`
371			`bottom = 1.0;`
372			`for(n = 2; n < 12; n += 2)`
373			`{`
374			`top = FP_Mult(top, FP_Neg(x2));`
375			`bottom = FP_Mult(bottom, FP_ToFloat((n - 1) * n));`
376			`answer = FP_Add(answer, FP_Div(top, bottom));`
377			`}`
378			`return FP_Mult(answer, sign);`
379			`}`
380
381
382			`float FP_Sin(float rad)`
383			`{`
384			`const float pi_2=(float)(PI/2.0);`
385			`return FP_Cos(FP_Sub(rad, pi_2));`
386			`}`
387
388
389			`float FP_Atan(float x)`
390			`{`
391			`const float b=(float)(PI/8.0);`
392			`const float atan_b=(float)0.37419668; //atan(b);`
393			`int n;`
394			`float answer, x2, top;`
395			`if(FP_Cmp(x, (float)0.0) >= 0)`
396			`{`
397			`if(FP_Cmp(x, (float)1.0) > 0)`
398			`return FP_Sub(PI_2, FP_Atan(FP_Div((float)1.0, x)));`
399			`}`
400			`else`
401			`{`
402			`if(FP_Cmp(x, (float)-1.0) > 0)`
403			`return FP_Sub(-PI_2, FP_Atan(FP_Div((float)1.0, x)));`
404			`}`
405			`if(FP_Cmp(x, (float)0.45) > 0)`
406			`{`
407			`//answer = (x - atan_b) / (1 + x * atan_b);`
408			`answer = FP_Div(FP_Sub(x, atan_b), FP_Add(1.0, FP_Mult(x, atan_b)));`
409			`//answer = b + FP_Atan(answer) - (float)0.034633; /FIXME fudge?/`
410			`answer = FP_Sub(FP_Add(b, FP_Atan(answer)), (float)0.034633);`
411			`return answer;`
412			`}`
413			`if(FP_Cmp(x, (float)-0.45) < 0)`
414			`{`
415			`x = FP_Neg(x);`
416			`//answer = (x - atan_b) / (1 + x * atan_b);`
417			`answer = FP_Div(FP_Sub(x, atan_b), FP_Add(1.0, FP_Mult(x, atan_b)));`
418			`//answer = b + FP_Atan(answer) - (float)0.034633; /FIXME/`
419			`answer = FP_Sub(FP_Add(b, FP_Atan(answer)), (float)0.034633);`
420			`return FP_Neg(answer);`
421			`}`
422			`answer = x;`
423			`x2 = FP_Mult(FP_Neg(x), x);`
424			`top = x;`
425			`for(n = 3; n < 14; n += 2)`
426			`{`
427			`top = FP_Mult(top, x2);`
428			`answer = FP_Add(answer, FP_Div(top, FP_ToFloat(n)));`
429			`}`
430			`return answer;`
431			`}`
432
433
434			`float FP_Atan2(float y, float x)`
435			`{`
436			`float answer,r;`
437			`r = y / x;`
438			`answer = FP_Atan(r);`
439			`if(FP_Cmp(x, (float)0.0) < 0)`
440			`{`
441			`if(FP_Cmp(y, (float)0.0) > 0)`
442			`answer = FP_Add(answer, PI);`
443			`else`
444			`answer = FP_Sub(answer, PI);`
445			`}`
446			`return answer;`
447			`}`
448
449
450			`float FP_Exp(float x)`
451			`{`
452			`const float e2=(float)7.389056099;`
453			`const float inv_e2=(float)0.135335283;`
454			`float answer, top, bottom, mult;`
455			`int n;`
456
457			`mult = 1.0;`
458			`while(FP_Cmp(x, (float)2.0) > 0)`
459			`{`
460			`mult = FP_Mult(mult, e2);`
461			`x = FP_Add(x, (float)-2.0);`
462			`}`
463			`while(FP_Cmp(x, (float)-2.0) < 0)`
464			`{`
465			`mult = FP_Mult(mult, inv_e2);`
466			`x = FP_Add(x, (float)2.0);`
467			`}`
468			`answer = FP_Add((float)1.0, x);`
469			`top = x;`
470			`bottom = 1.0;`
471			`for(n = 2; n < 15; ++n)`
472			`{`
473			`top = FP_Mult(top, x);`
474			`bottom = FP_Mult(bottom, FP_ToFloat(n));`
475			`answer = FP_Add(answer, FP_Div(top, bottom));`
476			`}`
477			`return FP_Mult(answer, mult);`
478			`}`
479
480
481			`float FP_Log(float x)`
482			`{`
483			`const float log_2=(float)0.69314718; /log(2.0)/`
484			`int n;`
485			`float answer, top, add;`
486			`add = 0.0;`
487			`while(FP_Cmp(x, 16.0) > 0)`
488			`{`
489			`x = FP_Mult(x, (float)0.0625);`
490			`add = FP_Add(add, (float)(log_2 * 4));`
491			`}`
492			`while(FP_Cmp(x, 1.5) > 0)`
493			`{`
494			`x = FP_Mult(x, 0.5);`
495			`add = FP_Add(add, log_2);`
496			`}`
497			`while(FP_Cmp(x, 0.5) < 0)`
498			`{`
499			`x = FP_Mult(x, 2.0);`
500			`add = FP_Sub(add, log_2);`
501			`}`
502			`x = FP_Sub(x, 1.0);`
503			`answer = 0.0;`
504			`top = -1.0;`
505			`for(n = 1; n < 14; ++n)`
506			`{`
507			`top = FP_Mult(top, FP_Neg(x));`
508			`answer = FP_Add(answer, FP_Div(top, FP_ToFloat(n)));`
509			`}`
510			`return FP_Add(answer, add);`
511			`}`
512
513
514			`float FP_Pow(float x, float y)`
515			`{`
516			`return FP_Exp(y * FP_Log(x));`
517			`}`
518
519
520	195	rhoads	`/********************************************/`
521			`//These five functions will only be used if the flag "-mno-mul" is enabled`
522			`#ifdef USE_SW_MULT`
523			`unsigned long __mulsi3(unsigned long a, unsigned long b)`
524			`{`
525			`unsigned long answer = 0;`
526	197	rhoads	`while(b)`
527	195	rhoads	`{`
528			`if(b & 1)`
529			`answer += a;`
530			`a <<= 1;`
531			`b >>= 1;`
532			`}`
533			`return answer;`
534			`}`
535
536
537			`static unsigned long DivideMod(unsigned long a, unsigned long b, int doMod)`
538			`{`
539			`unsigned long upper=a, lower=0;`
540			`int i;`
541			`a = b << 31;`
542			`for(i = 0; i < 32; ++i)`
543			`{`
544			`lower = lower << 1;`
545			`if(upper >= a && a && b < 2)`
546			`{`
547			`upper = upper - a;`
548			`lower \|= 1;`
549			`}`
550			`a = ((b&2) << 30) \| (a >> 1);`
551			`b = b >> 1;`
552			`}`
553			`if(!doMod)`
554			`return lower;`
555			`return upper;`
556			`}`
557
558
559			`unsigned long __udivsi3(unsigned long a, unsigned long b)`
560			`{`
561			`return DivideMod(a, b, 0);`
562			`}`
563
564
565			`long __divsi3(long a, long b)`
566			`{`
567			`long answer, negate=0;`
568			`if(a < 0)`
569			`{`
570			`a = -a;`
571			`negate = !negate;`
572			`}`
573			`if(b < 0)`
574			`{`
575			`b = -b;`
576			`negate = !negate;`
577			`}`
578			`answer = DivideMod(a, b, 0);`
579			`if(negate)`
580			`answer = -answer;`
581			`return answer;`
582			`}`
583
584
585			`unsigned long __umodsi3(unsigned long a, unsigned long b)`
586			`{`
587			`return DivideMod(a, b, 1);`
588			`}`
589			`#endif`
590
591
592	138	rhoads	`/************* Test ***************/`
593			`#ifdef WIN32`
594	336	rhoads	`#undef _LIBC`
595	138	rhoads	`#include <math.h>`
596			`struct {`
597			`char *name;`
598			`float low, high;`
599			`double (*func1)(double);`
600			`float (*func2)(float);`
601			`} test_info[]={`
602			`{"cos", -2PI, 2PI, cos, FP_Cos},`
603			`{"sin", -2PI, 2PI, sin, FP_Sin},`
604			`{"atan", -3.0, 2.0, atan, FP_Atan},`
605			`{"log", (float)0.01, (float)4.0, log, FP_Log},`
606			`{"exp", (float)-5.01, (float)30.0, exp, FP_Exp},`
607			`{"sqrt", (float)0.01, (float)1000.0, sqrt, FP_Sqrt}`
608			`};`
609
610
611			`void TestMathFull(void)`
612			`{`
613			`float a, b, c, d;`
614			`float error1, error2, error3, error4, error5;`
615			`int test;`
616
617			`a = PI * PI;`
618			`b = PI;`
619			`c = FP_Div(a, b);`
620			`printf("%10f %10f %10f %10f %10f\n",`
621			`(double)a, (double)b, (double)(a/b), (double)c, (double)(a/b-c));`
622			`a = a * 200;`
623			`for(b = -(float)2.718281828*100; b < 300; b += (float)23.678)`
624			`{`
625			`c = FP_Div(a, b);`
626			`d = a / b - c;`
627			`printf("%10f %10f %10f %10f %10f\n",`
628			`(double)a, (double)b, (double)(a/b), (double)c, (double)(a/b-c));`
629			`}`
630	302	rhoads	`//getch();`
631	138	rhoads
632			`for(test = 0; test < 6; ++test)`
633			`{`
634			`printf("\nTesting %s\n", test_info[test].name);`
635			`for(a = test_info[test].low;`
636			`a <= test_info[test].high;`
637			`a += (test_info[test].high-test_info[test].low)/(float)20.0)`
638			`{`
639			`b = (float)test_info[test].func1(a);`
640			`c = test_info[test].func2(a);`
641			`d = b - c;`
642			`printf("%s %10f %10f %10f %10f\n", test_info[test].name, a, b, c, d);`
643			`}`
644	302	rhoads	`//getch();`
645	138	rhoads	`}`
646
647			`a = FP_ToFloat((long)6.0);`
648			`b = FP_ToFloat((long)2.0);`
649			`printf("%f %f\n", (double)a, (double)b);`
650			`c = FP_Add(a, b);`
651			`printf("add %f %f\n", (double)(a + b), (double)c);`
652			`c = FP_Sub(a, b);`
653			`printf("sub %f %f\n", (double)(a - b), (double)c);`
654			`c = FP_Mult(a, b);`
655			`printf("mult %f %f\n", (double)(a * b), (double)c);`
656			`c = FP_Div(a, b);`
657			`printf("div %f %f\n", (double)(a / b), (double)c);`
658	302	rhoads	`//getch();`
659	138	rhoads
660			`for(a = (float)-13756.54; a < (float)17400.0; a += (float)64.45)`
661			`{`
662			`for(b = (float)-875.36; b < (float)935.8; b += (float)36.7)`
663			`{`
664			`error1 = (float)1.0 - (a + b) / FP_Add(a, b);`
665			`error2 = (float)1.0 - (a * b) / FP_Mult(a, b);`
666			`error3 = (float)1.0 - (a / b) / FP_Div(a, b);`
667			`error4 = (float)1.0 - a / FP_ToFloat(FP_ToLong(a));`
668			`error5 = error1 + error2 + error3 + error4;`
669			`if(error5 < 0.00005)`
670			`continue;`
671			`printf("ERROR!\n");`
672			`printf("a=%f b=%f\n", (double)a, (double)b);`
673			`printf(" a+b=%f %f\n", (double)(a+b), (double)FP_Add(a, b));`
674			`printf(" ab=%f %f\n", (double)(ab), (double)FP_Mult(a, b));`
675			`printf(" a/b=%f %f\n", (double)(a/b), (double)FP_Div(a, b));`
676			`printf(" a=%f %ld %f\n", (double)a, FP_ToLong(a),`
677			`(double)FP_ToFloat((long)a));`
678			`printf(" %f %f %f %f\n", (double)error1, (double)error2,`
679			`(double)error3, (double)error4);`
680	302	rhoads	`//if(error5 > 0.001)`
681			`// getch();`
682	138	rhoads	`}`
683			`}`
684			`printf("done.\n");`
685	302	rhoads	`//getch();`
686	138	rhoads	`}`
687			`#endif`
688
689