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//**************************************
// Name: Calculate PI to 10's of thousands of digits
// Description:This method implements the Salamin / Brent / Gauss Arithmetic-
Geometric Mean pi formula.
Let A[0] = 1, B[0] = 1/Sqrt(2)
Then iterate from 1 to 'n'.
A[n] = (A[n-1] + B[n-1])/2
B[n] = Sqrt(A[n-1]*B[n-1])
C[n]^2 = A[n]^2 - B[n]^2 (or) C[n] = (A[n-1]-B[n-1])/2
n
PI[n] = 4A[n+1]^2 / (1-(Sum (2^(j+1))*C[j]^2))
j = 1
There is an actual error calculation, but it comes out to slightly
more than double on each iteration. I think it results in about 17
million correct digits, instead of 16 million if it actually
doubled. PI16 generates 178,000 digits. PI19 to over a million.
PI22 is 10 million, and PI26 to 200 million.
For what little it's worth, this program is placed into the public
domain by its author, Carey Bloodworth, on September 21, 1996.
The math routines in this program are not general purpose routines.
They have been optimized and written for this specific use. The
concepts are of course portable, but not the implementation.
The program run time is about O(n^1.7). That's fairly good growth,
compared to things such as the classic arctangents. But not as good
as it should be, if I used a FFT based multiplication. Also, the 'O'
is fairly large. In fact, I'd guess that this program could compute
one million digits of pi in about the same time as my previously
posted arctangent method, in spite of this one having less than n^2
growth.
The program has not been cleaned up. It's written rather crudely
and dirty. The RSqrt() in particular is rather dirty, having gone
through numerous changes and attempts at speeding it up.
But I'm not planning on doing any more with it. The growth isn't as
low as I'd hoped, and until I find a faster multiplication, the
method isn't any better than simpler arctangents.
I currently use a base of 10,000 simply because it made debugging
easier. A base of 65,536 and modifying the FastMul() to handle sizes
that aren't powers of two would allow a bit better efficiency.
// By: Bob Stout (republished under Open Content License)
//
//
// Inputs:None
//
// Returns:None
//
//Assumes:None
//
//Side Effects:None
//This code is copyrighted and has limited warranties.
//Please see http://www.Planet-Source-Code.com/xq/ASP/txtCodeId.657/lngWId.3/qx/vb/scripts/ShowCode.htm
//for details.
//**************************************
/* +++Date last modified: 05-Jul-1997 */
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <time.h>
#include "snipmath.h"
#ifdef __TURBOC__
typedef short int Indexer;
#else
typedef long int Indexer;
#endif
typedef short int Short;
typedef long int Long;
Short *a, *b, *c, *Work1, *MulWork, *Work2, *Work3;
int SIZE;
/*
** Work1 is explicitly used in Reciprocal() and RSqrt()
** Work1 is implicitly used in Divide() and Sqrt()
** Work2 is explicitly used in Divide() and Sqrt()
** Work3 is used only in the AGM and holds the previous reciprocal
** square root, allowing us to save some time in RSqrt()
*/
void Copy(Short *a, Short *b, Indexer Len)
{
while (Len--) a[Len] = b[Len];
}
/*
** This rounds and copies a 2x mul result into a normal result
** Our number format will never have more than one unit of integer,
** and after a mul, we have two, so we need to fix that.
*/
void Round2x(Short *a, Short *b, Indexer Len)
{
Indexer x;
Short carry;
carry = 0;
if (b[Len+1] >= 5000)
carry = 1;
for (x = Len; x > 0; x--)
{
carry += b[x];
a[x-1] = carry % 10000;
carry /= 10000;
}
}
void DivBy2(Short *n, Indexer Len)
{
Indexer x;
Long temp;
temp = 0;
for (x = 0; x < Len; x++)
{
temp = (Long)n[x]+temp*10000;
n[x] = (Short)(temp/2);
temp = temp%2;
}
}
void DoCarries(Short *limit, Short *cur, Short carry)
{
Long temp;
while ((cur >= limit) && (carry != 0))
{
temp = *cur+carry;
carry = 0;
if (temp >= 10000)
{
carry = 1;
temp -= 10000;
}
*cur = temp;
--cur;
}
}
void DoBorrows(Short *limit, Short *cur, Short borrow)
{
Long temp;
while ((cur >= limit) && (borrow != 0))
{
temp = *cur-borrow;
borrow = 0;
if (temp < 0)
{
borrow = 1;
temp += 10000;
}
*cur = temp;
--cur;
};
}
void PrintShort2(char *str, Short *num, Indexer Len)
{
Indexer x;
printf("%s ", str);
printf("%u.", num[0]);
for (x = 1; x < Len; x++)
printf("%04u", num[x]);
printf("\n");
}
void PrintShort(char *str, Short *num, Indexer Len)
{
Indexer x;
int printed = 0;
printf("%s ", str);
printf("%u.\n", num[0]);
for (x = 1; x < Len; x++)
{
printf("%02d", num[x]/100);
printed += 2;
if ((printed % 1000) == 0)
{
printf("\n\n\n\n");
printed = 0;
}
else if ((printed % 50) == 0)
{
printf("\n");
}
else if ((printed % 10) == 0)
{
printf(" ");
}
printf("%02d", num[x] % 100);
printed += 2;
if ((printed % 1000) == 0)
{
printf("\n\n\n\n");
printed = 0;
}
else if ((printed % 50) == 0)
{
printf("\n");
}
else if ((printed % 10) == 0)
{
printf(" ");
}
}
printf("\n");
}
/* sum = a + b */
Short Add(Short *sum, Short *a, Short *b, Indexer Len)
{
Long s, carry;
Indexer x;
carry = 0;
for (x = Len - 1; x >= 0; x--)
{
s = (Long)a[x] + (Long)b[x] + carry;
sum[x] = (Short)(s%10000);
carry = s/10000;
}
return carry;
}
/* dif = a-b */
Short Sub(Short *dif, Short *a, Short *b, Indexer Len)
{
Long d, borrow;
Indexer x;
borrow = 0;
for (x = Len - 1; x >= 0; x--)
{
d = (Long)a[x] - (Long)b[x] - borrow;
borrow = 0;
if (d < 0)
{
borrow = 1;
d += 10000;
}
dif[x] = (Short)d;
}
return borrow;
}
void Negate(Short *num, Indexer Len)
{
Indexer x;Long d, borrow;
borrow = 0;
for (x = Len - 1; x >= 0; x--)
{
d = 0 - num[x] - borrow;
borrow = 0;
if (d < 0)
{
borrow = 1;
d += 10000;
}
num[x] = (Short)d;
}
}
/* prod = a*b. prod should be twice normal length */
void Mul(Short *prod, Short *a, Short *b, Indexer Len)
{
Long p;
Indexer ia, ib, ip;
if ((prod == a) || (b == prod))
{
printf("MUL product can't be one of the other arguments\n");
exit(1);
}
for (ip = 0; ip < Len * 2; ip++)
prod[ip] = 0;
for (ib = Len - 1; ib >= 0; ib--)
{
if (b[ib] == 0)
continue;
ip = ib + Len;
p = 0;
for (ia = Len - 1; ia >= 0; ia--)
{
p = (Long)a[ia]*(Long)b[ib] + p + prod[ip];
prod[ip] = p%10000;
p = p/10000;
ip--;
}
while ((p) && (ip >= 0))
{
p += prod[ip];
prod[ip] = p%10000;
p = p/10000;
ip--;
}
}
}
/*
** This is based on the simple O(n^1.585) method, although my
** growth seems to be closer to O(n^1.7)
**
** It's fairly simple. a*b is: a2b2(B^2+B)+(a2-a1)(b1-b2)B+a1b1(B+1)
**
** For a = 4711 and b = 6397, a2 = 47 a1 = 11 b2 = 63 b1 = 97 Base = 100
**
** If we did that the normal way, we'd do
** a2b2 = 47*63 = 2961
** a2b1 = 47*97 = 4559
** a1b2 = 11*63 = 693
** a1b1 = 11*97 = 1067
**
** 29 61
**45 59
** 6 93
**10 67
** -----------
** 30 13 62 67
**
** Or, we'd need N*N multiplications.
**
** With the 'fractal' method, we compute:
** a2b2 = 47*63 = 2961
** (a2-b1)(b1-b2) = (47-11)(97-63) = 36*34 = 1224
** a1b1 = 11*97 = 1067
**
** 29 61
**29 61
**12 24
**10 67
**10 67
** -----------
** 30 13 62 67
**
** We need only 3 multiplications, plus a few additions. And of course,
** additions are a lot simpler and faster than multiplications, so we
** end up ahead.
**
** The way it is written requires that the size to be a power of two.
** That's why the program itself requires the size to be a power of two.
** There is no actual requirement in the method, it's just how I did it
** because I would be working close to powers of two anyway, and it was
** easier.
*/
void FastMul(Short *prod, Short *a, Short *b, Indexer Len)
{
Indexer x, HLen;
int SignA, SignB;
Short *offset;
Short *NextProd;
/*
** This is the threshold where it's faster to do it the ordinary way
** On my computer, it's 16. Yours may be different.
*/
if (Len <= 16 )
{
Mul(prod, a, b, Len);
return;
}
HLen = Len/2;
NextProd = prod + 2*Len;
SignA = Sub(prod, a, a + HLen, HLen);
if (SignA)
{
SignA = 1;
Negate(prod, HLen);
}
SignB = Sub(prod + HLen, b + HLen, b, HLen);
if (SignB)
{
SignB = 1;
Negate(prod + HLen, HLen);
}
FastMul(NextProd, prod, prod + HLen, HLen);
for (x = 0; x < 2 * Len; x++)
prod[x] = 0;
offset = prod + HLen;
if (SignA == SignB)
DoCarries(prod, offset - 1, Add(offset, offset, NextProd, Len));
else DoBorrows(prod, offset - 1, Sub(offset, offset, NextProd, Len));
FastMul(NextProd, a, b, HLen);
offset = prod + HLen;
DoCarries(prod, offset - 1, Add(offset, offset, NextProd, Len));
Add(prod, prod, NextProd, Len);
FastMul(NextProd, a + HLen, b + HLen, HLen);
offset = prod + HLen;
DoCarries(prod, offset - 1, Add(offset, offset, NextProd, Len));
offset = prod + Len;
DoCarries(prod, offset - 1, Add(offset, offset, NextProd, Len));
}
/*
** Compute the reciprocal of 'a'.
** x[k+1] = x[k]*(2-a*x[k])
*/
void Reciprocal(Short *r, Short *n, Indexer Len)
{
Indexer x, SubLen;
int iter;
double t;
/* Estimate our first reciprocal */
for (x = 0; x < Len; x++)
r[x] = 0;
t = n[0] + n[1]*1.0e-4 + n[2]*1.0e-8;
if (t == 0.0)
t = 1;
t = 1.0/t;
r[0] = t;
t = (t - floor(t))*10000.0;
r[1] = t;
t = (t - floor(t))*10000.0;
r[2] = t;
iter = log(SIZE)/log(2.0) + 1;
SubLen = 1;
while (iter--)
{
SubLen *= 2;
if (SubLen > SIZE)
SubLen = SIZE;
FastMul(MulWork, n, r, SubLen);
Round2x(Work1, MulWork, SubLen);
MulWork[0] = 2;
for (x = 1; x < Len; x++)
MulWork[x] = 0;
Sub(Work1, MulWork, Work1, SubLen);
FastMul(MulWork, r, Work1, SubLen);
Round2x(r, MulWork, SubLen);
}
}
/*
** Computes the reciprocal of the square root of 'a'
** y[k+1] = y[k]*(3-a*y[k]^2)/2
**
**
** We can save quite a bit of time by using part of our previous
** results! Since the number is converging onto a specific value,
** at least part of our previous result will be correct, so we
** can skip a bit of work.
*/
void RSqrt(Short *r, Short *n, Indexer Len, Indexer SubLen)
{
Indexer x;
int iter;
double t;
/* Estimate our first reciprocal square root */
if (SubLen < 2 )
{
for (x = 0; x < Len; x++)
r[x] = 0;
t = n[0] + n[1]*1.0e-4 + n[2]*1.0e-8;
if (t == 0.0)
t = 1;
t = 1.0/sqrt(t);
r[0] = t;
t = (t - floor(t))*10000.0;
r[1] = t;
t = (t - floor(t))*10000.0;
r[2] = t;
}
/*
** PrintShort2("\n ~R: ", r, SIZE);
*/
if (SubLen > SIZE)
SubLen = SIZE;
iter = SubLen;
while (iter <= SIZE*2)
{
FastMul(MulWork, r, r, SubLen);
Round2x(Work1, MulWork, SubLen);
FastMul(MulWork, n, Work1, SubLen);
Round2x(Work1, MulWork, SubLen);
MulWork[0] = 3;
for (x = 1; x < SubLen; x++)
MulWork[x] = 0;
Sub(Work1, MulWork, Work1, SubLen);
FastMul(MulWork, r, Work1, SubLen);
Round2x(r, MulWork, SubLen);
DivBy2(r, SubLen);
/*
printf("%3d", SubLen);
PrintShort2("R: ", r, SubLen);
printf("%3d", SubLen);
PrintShort2("R: ", r, Len);
*/
SubLen *= 2;
if (SubLen > SIZE)
SubLen = SIZE;
iter *= 2;
}
}
/*
** d = a/b by computing the reciprocal of b and then multiplying
** that by a. It's faster this way because the normal digit by
** digit method has N^2 growth, and this method will have the same
** growth as our multiplication method.
*/
void Divide(Short *d, Short *a, Short *b, Indexer Len)
{
Reciprocal(Work2, b, Len);
FastMul(MulWork, a, Work2, Len);
Round2x(d, MulWork, Len);
}
/*
** r = sqrt(n) by computing the reciprocal of the square root of
** n and then multiplying that by n
*/
void Sqrt(Short *r, Short *n, Indexer Len, Indexer SubLen)
{
RSqrt(Work2, n, Len, SubLen);
FastMul(MulWork, n, Work2, Len);
Round2x(r, MulWork, Len);
}
void usage(void)
{
puts("This program requires the number of digits/4 to calculate");
puts("This number must be a power of two.");
exit(-1);
}
int main(int argc,char *argv[])
{
Indexer x;
Indexer SubLen;
double Pow2, tempd, carryd;
int AGMIter;
int NeededIter;
clock_t T0, T1, T2, T3;
if (argc < 2)
usage();
SIZE = atoi(argv[1]);
if (!ispow2(SIZE))
usage();
a = (Short *)malloc(sizeof(Short)*SIZE);
b = (Short *)malloc(sizeof(Short)*SIZE);
c = (Short *)malloc(sizeof(Short)*SIZE);
Work1 = (Short *)malloc(sizeof(Short)*SIZE);
Work2 = (Short *)malloc(sizeof(Short)*SIZE);
Work3 = (Short *)malloc(sizeof(Short)*SIZE);
MulWork = (Short *)malloc(sizeof(Short)*SIZE*4);
if ((a ==NULL) || (b == NULL) || (c == NULL) || (Work1 == NULL) ||
(Work2 == NULL) || (MulWork == NULL))
{
printf("Unable to allocate memory\n");exit(1);
}
NeededIter = log(SIZE)/log(2.0) + 1;
Pow2 = 4.0;
AGMIter = NeededIter + 1;
SubLen = 1;
T0 = clock();
for (x = 0; x < SIZE; x++)
a[x] = b[x] = c[x] = Work3[x] = 0;
a[0] = 1;
Work3[0] = 2;
RSqrt(b, Work3, SIZE, 1);
c[0] = 1;
T1 = clock();
/*
printf("AGMIter %d\n", AGMIter);
PrintShort2("a AGM: ", a, SIZE);
PrintShort2("b AGM: ", b, SIZE);
PrintShort2("C sum: ", c, SIZE);
*/
while (AGMIter--)
{
Sub(Work1, a, b, SIZE);/* w = (a-b)/2 */
DivBy2(Work1, SIZE);
FastMul(MulWork, Work1, Work1, SIZE); /* m = w*w */
Round2x(Work1, MulWork, SIZE);
carryd = 0.0; /* m = m* w^(J+1)*/
for (x = SIZE - 1; x >= 0; x--)
{
tempd = Pow2*Work1[x] + carryd;
carryd = floor(tempd / 10000.0);
Work1[x] = tempd - carryd*10000.0;
}
Pow2 *= 2.0;
Sub(c, c, Work1, SIZE);/* c = c - m*/
/* Save some time on last iter */
if (AGMIter != 0)
FastMul(MulWork, a, b, SIZE);/* m = a*b */
Add(a, a, b, SIZE);/* a = (a+b)/2 */
DivBy2(a, SIZE);
if (AGMIter != 0)
{
Round2x(Work3, MulWork, SIZE);
Sqrt(b, Work3, SIZE, SubLen); /* b = sqrt(a*b) */
}
/*
printf("AGMIter %d\n", AGMIter);
PrintShort2("a AGM: ", a, SIZE);
PrintShort2("b AGM: ", b, SIZE);
PrintShort2("C sum: ", c, SIZE);
*/
SubLen *= 2;if (SubLen > SIZE) SubLen = SIZE;
}
T2 = clock();
FastMul(MulWork, a, a, SIZE);
Round2x(a, MulWork, SIZE);
carryd = 0.0;
for (x = SIZE - 1; x >= 0; x--)
{
tempd = 4.0*a[x] + carryd;
carryd = floor(tempd / 10000.0);
a[x] = tempd - carryd*10000.0;
}
Divide(b, a, c, SIZE);
T3 = clock();
PrintShort("\nPI = ", b, SIZE);
printf("\nTotal Execution time: %12.4lf\n",
(double)(T3 - T0) / CLOCKS_PER_SEC);
printf("Setup time : %12.4lf\n",
(double)(T1 - T0) / CLOCKS_PER_SEC);
printf("AGM Calculation time: %12.4lf\n",
(double)(T2 - T1) / CLOCKS_PER_SEC);
printf("Post calc time : %12.4lf\n",
(double)(T3 - T2) / CLOCKS_PER_SEC);
return 0;
}
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