// Fast integer multiplication using FFT over the complex numbers.
// [Donald Ervin Knuth: The Art of Computer Programming, Vol. II:
// Seminumerical Algorithms, second edition. Section 4.3.3, p. 290-294.]
// Bruno Haible 6.5.1996, 24.-25.8.1996
// FFT in the complex domain has the drawback that it needs careful round-off
// error analysis. But for CPUs with good floating-point performance it might
// nevertheless be better than FFT mod Z/mZ.
// It is important to have precise round-off error estimates. Although
// (by laws of statistics) in the average the actual round-off error makes up
// only half of the bits provided for round-off protection, we cannot rely
// on this average behaviour, but have to produce correct results.
//
// Knuth's formula (42), p. 294, says:
// If we want to multiply l-bit words using an FFT(2^k), our floating point
// numbers shall have m >= 2(k+l) + k + log_2 k + 3.5 mantissa bits.
//
// Here is a more careful analysis, using absolute error estimates.
//
// 1. We assume floating point numbers with radix 2, with the properties:
// (i) Multiplication with 2^n and 2^-n is exact.
// (ii) Negation x -> -x is exact.
// (iii) Addition: When adding x and y, with |x| <= 2^a, |y| <= 2^a,
// the result |x+y| <= 2^(a+1) has an error <= e*2^(a+1-m).
// (iv) Multiplication: When multiplying x and y, with |x| <= 2^a,
// |y| <= 2^b, the result |x*y| <= 2^(a+b) has an error <= e*2^(a+b-m).
// Here e = 1 for a truncating arithmetic, but e = 1/2 for a rounding
// arithmetic like IEEE single and double floats.
// 2. Let's introduce some notation: err(x) means |x'-x| where x is the
// exact mathematical value and x' is its representation in the machine.
// 3. From 1. we get for real numbers x,y:
// (i) err(2^n*x) = 2^n * err(x),
// (ii) err(-x) = err(x),
// (iii) |x| <= 2^a, |y| <= 2^a, then
// err(x+y) <= err(x) + err(y) + e*2^(a+1-m),
// [or .... ............... + e*2^(a-m) if |x+y| <= 2^a].
// (iv) |x| <= 2^a, |y| <= 2^b, then
// err(x*y) <= 2^a * err(y) + 2^b * err(x) + e*2^(a+b-m).
// 4. Our complex arithmetic will be based on the formulas:
// (i) 2^n*(x+iy) = (2^n*x)+i(2^n*y)
// (ii) -(x+iy) = (-x)+i(-y)
// (iii) (x+iy)+(u+iv) = (x+u)+i(y+v)
// (iv) (x+iy)*(u+iv) = (x*u-y*v)+i(x*v+y*u)
// The notation err(z) means |z'-z|, as above, with |.| being the usual
// absolute value on complex numbers (_not_ the L^1 norm).
// 5. From 3. and 4. we get for complex numbers x,y:
// (i) err(2^n*x) = 2^n * err(x),
// (ii) err(-x) = err(x),
// (iii) |x| <= 2^a, |y| <= 2^a, then
// err(x+y) <= err(x) + err(y) + e*2^(a+3/2-m),
// (iv) |x| <= 2^a, |y| <= 2^b, then
// err(x*y) <= 2^a * err(y) + 2^b * err(x) + 3*e*2^(a+b+1/2-m).
// 6. We start out with precomputed roots of unity:
// |exp(2 pi i/2^n)| <= 1,
// err(exp(2 pi i/2^n)) <= e*2^(1/2-m), (even err(..)=0 for n=0,1,2),
// and compute exp(2 pi i * j/2^k) according to the binary digits of j.
// This way, each root of unity will be a product of at most k precomputed
// roots. If (j mod 2^(k-2)) has n bits, then exp(2 pi i * j/2^k) will
// be computed using n factors, i.e. n-1 complex multiplications, and by
// 5.iv. we'll have
// err(exp(2 pi i * j/2^k)) <= n*e*2^(1/2-m) + max(n-1,0)*3*e*2^(1/2-m)
// = max(4*n-3,0)*e*2^(1/2-m).
// Hence the maximum roots-of-unity error is (set n=k-2)
// err(w^j) <= (4*k-11)*e*2^(1/2-m),
// and the average roots-of-unity error is (set n=(k-2)/2)
// < 2*(k-2)*e*2^(1/2-m).
// 7. Now we start the FFT.
// Before the first step, x_i are integral, |x_i| < 2^l and err(x_i) = 0.
// After the first butterfly, which replaces (x(i1),x(i2)) by
// (x(i1) + x(i2), x(i1) - x(i2)), we have |x_i| < 2^(l+1) and err(x_i) = 0.
// Then, for each of the remaining k-1 steps, a butterfly replaces
// (x(i1),x(i2)) by (x(i1) + w^exp*x(i2), x(i1) - w^exp*x(i2)). Thus,
// after n steps we have |x_i| < 2^(l+n) and err(x_i) <= E_i where
// E_0 = 0,
// E_1 = 0,
// E_(n+1) = (E_n + 2^(l+n)*(4*k-11)*e*2^(1/2-m) + 3*e*2^(l+n+1/2-m))
// + E_n + e*2^(l+n+3/2-m)
// = 2*E_n + (4*k-6)*e*2^(l+n+1/2-m)
// hence E_n = (2^n-2)*(4*k-6)*e*2^(l+1/2-m).
// Setting n = k, we have proved that after the FFT ends, we have
// |x_i| < 2^(l+k) and err(x_i) <= (4*k-6)*e*2^(l+k+1/2-m).
// 8. The same error analysis holds for the y_i and their FFT. After we
// multiply z_i := x_i * y_i, we have
// |z_i| < 2^(2*l+2*k) and err(z_i) <= (8*k-9)*e*2^(2*l+2*k+1/2-m).
// 9. Then an inverse FFT on z_i is done, which is the same as an FFT
// followed by a permutation and a division by 2^k. After n steps of
// the FFT, we have |z_i| < 2^(2*l+2*k+n) and err(z_i) <= E_i where
// E_0 = (8*k-9)*e*2^(2*l+2*k+1/2-m),
// E_(n+1) = (E_n + 2^(2*l+2*k+n)*(4*k-11)*e*2^(1/2-m)
// + 3*e*2^(2*l+2*k+n+1/2-m))
// + E_n + e*2^(2*l+2*k+n+3/2-m)
// = 2*E_n + (4*k-6)*e*2^(2*l+2*k+n+1/2-m)
// hence E_n = 2^n*(8*k-9)*e*2^(2*l+2*k+1/2-m)
// + (2^n-1)*(4*k-6)*e*2^(2*l+2*k+1/2-m).
// So, after the FFT, we have (set n=k) |z_i| < 2^(2*l+3*k) and
// err(z_i) <= (12*k-15)*e*2^(2*l+3*k+1/2-m).
// Permutation doesn't change the estimates. After division by 2^k, we get
// |z_i| < 2^(2*l+2*k) and
// err(z_i) <= (12*k-15)*e*2^(2*l+2*k+1/2-m).
// 10. When converting the z_i back to integers, we know that z_i should be
// real, integral, and |z_i| < 2^(2*l+k). We can only guarantee that we
// can find the integral z_i from the floating-point computation if
// (12*k-15)*e*2^(2*l+2*k+1/2-m) < 1/2.
// 11. Assuming e = 1/2 and m = 53 (typical values for IEEE double arithmetic),
// we get the constraint 2*l < m - 1/2 - 2*k - log_2(12*k-15).
// k = 2 l <= 22
// k = 3 l <= 21
// k = 4 l <= 19
// k = 5 l <= 18
// k = 6 l <= 17
// k = 7 l <= 16
// k = 8 l <= 15
// k = 9 l <= 13
// k = 10 l <= 12
// k = 11 l <= 11
// k = 12 l <= 10
// k = 13 l <= 9
// k = 14 l <= 8
// k = 15 l <= 7
// k = 16 l <= 6
// k = 17 l <= 5
// k = 18 l <= 4
// k = 19 l <= 3
// k = 20 l <= 2
// Assuming e = 1/2 and m = 64 ("long double" arithmetic on i387/i486/i586),
// we get the constraint 2*l < m - 1/2 - 2*k - log_2(12*k-15).
// k = 2 l <= 28
// k = 3 l <= 26
// k = 4 l <= 25
// k = 5 l <= 24
// k = 6 l <= 22
// k = 7 l <= 21
// k = 8 l <= 20
// k = 9 l <= 19
// k = 10 l <= 18
// k = 11 l <= 17
// k = 12 l <= 16
// k = 13 l <= 15
// k = 14 l <= 14
// k = 15 l <= 13
// k = 16 l <= 12
// k = 17 l <= 10
// k = 18 l <= 9
// k = 19 l <= 8
// k = 20 l <= 7
// k = 21 l <= 6
// k = 22 l <= 5
// k = 23 l <= 4
// k = 24 l <= 3
// k = 25 l <= 2
#if !(intDsize==32)
#error "complex fft implemented only for intDsize==32"
#endif
#include "cln/floatparam.h"
#include "cln/io.h"
#include "cln/abort.h"
#if defined(HAVE_LONGDOUBLE) && (long_double_mant_bits > double_mant_bits) && (defined(__i386__) || defined(__m68k__) || (defined(__sparc__) && 0))
// Only these CPUs have fast "long double"s in hardware.
// On SPARC, "long double"s are emulated in software and don't work.
typedef long double fftc_real;
#define fftc_real_mant_bits long_double_mant_bits
#define fftc_real_rounds long_double_rounds
#else
typedef double fftc_real;
#define fftc_real_mant_bits double_mant_bits
#define fftc_real_rounds double_rounds
#endif
typedef struct fftc_complex
{
fftc_real re;
fftc_real im;
}
fftc_complex;
static const fftc_complex fftc_roots_of_1 [32+1] =
// roots_of_1[n] is a (2^n)th root of unity in C.
// Also roots_of_1[n-1] = roots_of_1[n]^2.
// For simplicity we choose roots_of_1[n] = exp(2 pi i/2^n).
{
#if (fftc_real_mant_bits == double_mant_bits)
// These values have 64 bit precision.
{ 1.0, 0.0 },
{ -1.0, 0.0 },
{ 0.0, 1.0 },
{ 0.7071067811865475244, 0.7071067811865475244 },
{ 0.9238795325112867561, 0.38268343236508977172 },
{ 0.9807852804032304491, 0.19509032201612826784 },
{ 0.99518472667219688623, 0.098017140329560601996 },
{ 0.9987954562051723927, 0.049067674327418014254 },
{ 0.9996988186962042201, 0.024541228522912288032 },
{ 0.99992470183914454094, 0.0122715382857199260795 },
{ 0.99998117528260114264, 0.0061358846491544753597 },
{ 0.9999952938095761715, 0.00306795676296597627 },
{ 0.99999882345170190993, 0.0015339801862847656123 },
{ 0.99999970586288221914, 7.6699031874270452695e-4 },
{ 0.99999992646571785114, 3.8349518757139558907e-4 },
{ 0.9999999816164292938, 1.9174759731070330744e-4 },
{ 0.9999999954041073129, 9.5873799095977345874e-5 },
{ 0.99999999885102682754, 4.793689960306688455e-5 },
{ 0.99999999971275670683, 2.3968449808418218729e-5 },
{ 0.9999999999281891767, 1.1984224905069706422e-5 },
{ 0.99999999998204729416, 5.9921124526424278428e-6 },
{ 0.99999999999551182357, 2.9960562263346607504e-6 },
{ 0.99999999999887795586, 1.4980281131690112288e-6 },
{ 0.999999999999719489, 7.4901405658471572114e-7 },
{ 0.99999999999992987223, 3.7450702829238412391e-7 },
{ 0.99999999999998246807, 1.8725351414619534487e-7 },
{ 0.999999999999995617, 9.36267570730980828e-8 },
{ 0.99999999999999890425, 4.6813378536549092695e-8 },
{ 0.9999999999999997261, 2.340668926827455276e-8 },
{ 0.99999999999999993153, 1.1703344634137277181e-8 },
{ 0.99999999999999998287, 5.8516723170686386908e-9 },
{ 0.9999999999999999957, 2.925836158534319358e-9 },
{ 0.9999999999999999989, 1.4629180792671596806e-9 }
#else (fftc_real_mant_bits > double_mant_bits)
// These values have 128 bit precision.
{ 1.0L, 0.0L },
{ -1.0L, 0.0L },
{ 0.0L, 1.0L },
{ 0.707106781186547524400844362104849039284L, 0.707106781186547524400844362104849039284L },
{ 0.923879532511286756128183189396788286823L, 0.38268343236508977172845998403039886676L },
{ 0.980785280403230449126182236134239036975L, 0.195090322016128267848284868477022240928L },
{ 0.995184726672196886244836953109479921574L, 0.098017140329560601994195563888641845861L },
{ 0.998795456205172392714771604759100694444L, 0.0490676743274180142549549769426826583147L },
{ 0.99969881869620422011576564966617219685L, 0.0245412285229122880317345294592829250654L },
{ 0.99992470183914454092164649119638322435L, 0.01227153828571992607940826195100321214037L },
{ 0.999981175282601142656990437728567716173L, 0.00613588464915447535964023459037258091705L },
{ 0.999995293809576171511580125700119899554L, 0.00306795676296597627014536549091984251894L },
{ 0.99999882345170190992902571017152601905L, 0.001533980186284765612303697150264079079954L },
{ 0.999999705862882219160228217738765677117L, 7.66990318742704526938568357948576643142e-4L },
{ 0.99999992646571785114473148070738785695L, 3.83495187571395589072461681181381263396e-4L },
{ 0.999999981616429293808346915402909714504L, 1.91747597310703307439909561989000933469e-4L },
{ 0.99999999540410731289097193313960614896L, 9.58737990959773458705172109764763511872e-5L },
{ 0.9999999988510268275626733077945541084L, 4.79368996030668845490039904946588727468e-5L },
{ 0.99999999971275670684941397221864177609L, 2.39684498084182187291865771650218200947e-5L },
{ 0.999999999928189176709775095883850490262L, 1.198422490506970642152156159698898480473e-5L },
{ 0.99999999998204729417728262414778410738L, 5.99211245264242784287971180889086172999e-6L },
{ 0.99999999999551182354431058417299732444L, 2.99605622633466075045481280835705981183e-6L },
{ 0.999999999998877955886077016551752536504L, 1.49802811316901122885427884615536112069e-6L },
{ 0.999999999999719488971519214794719584451L, 7.49014056584715721130498566730655637157e-7L },
{ 0.99999999999992987224287980123972873676L, 3.74507028292384123903169179084633177398e-7L },
{ 0.99999999999998246806071995015624773673L, 1.8725351414619534486882457659356361712e-7L },
{ 0.999999999999995617015179987529456656217L, 9.3626757073098082799067286680885620193e-8L },
{ 0.999999999999998904253794996881763834182L, 4.68133785365490926951155181385400969594e-8L },
{ 0.99999999999999972606344874922040343793L, 2.34066892682745527595054934190348440379e-8L },
{ 0.999999999999999931515862187305098514444L, 1.170334463413727718124621350323810379807e-8L },
{ 0.999999999999999982878965546826274482047L, 5.8516723170686386908097901008341396944e-9L },
{ 0.999999999999999995719741386706568611352L, 2.92583615853431935792823046906895590202e-9L },
{ 0.999999999999999998929935346676642152265L, 1.46291807926715968052953216186596371037e-9L }
#endif
};
// Define this for (cheap) consistency checks.
//#define DEBUG_FFTC
// Define the algorithm of the backward FFT:
// Either FORWARD (a normal FFT followed by a permutation)
// or RECIPROOT (an FFT with reciprocal root of unity)
// or CLEVER (an FFT with reciprocal root of unity but clever computation
// of the reciprocals).
// Drawback of FORWARD: the permutation pass.
// Drawback of RECIPROOT: need all the powers of the root, not only half of them.
#define FORWARD 42
#define RECIPROOT 43
#define CLEVER 44
#define FFTC_BACKWARD CLEVER
static fftc_real fftc_pow2_table[64] = // table of powers of 2
{
1.0,
2.0,
4.0,
8.0,
16.0,
32.0,
64.0,
128.0,
256.0,
512.0,
1024.0,
2048.0,
4096.0,
8192.0,
16384.0,
32768.0,
65536.0,
131072.0,
262144.0,
524288.0,
1048576.0,
2097152.0,
4194304.0,
8388608.0,
16777216.0,
33554432.0,
67108864.0,
134217728.0,
268435456.0,
536870912.0,
1073741824.0,
2147483648.0,
4294967296.0,
8589934592.0,
17179869184.0,
34359738368.0,
68719476736.0,
137438953472.0,
274877906944.0,
549755813888.0,
1099511627776.0,
2199023255552.0,
4398046511104.0,
8796093022208.0,
17592186044416.0,
35184372088832.0,
70368744177664.0,
140737488355328.0,
281474976710656.0,
562949953421312.0,
1125899906842624.0,
2251799813685248.0,
4503599627370496.0,
9007199254740992.0,
18014398509481984.0,
36028797018963968.0,
72057594037927936.0,
144115188075855872.0,
288230376151711744.0,
576460752303423488.0,
1152921504606846976.0,
2305843009213693952.0,
4611686018427387904.0,
9223372036854775808.0
};
// For a constant expression n (0 <= n < 128), returns 2^n of type fftc_real.
#define fftc_pow2(n) \
(((n) & 64 ? (fftc_real)18446744073709551616.0 : (fftc_real)1.0) \
* ((n) & 32 ? (fftc_real)4294967296.0 : (fftc_real)1.0) \
* ((n) & 16 ? (fftc_real)65536.0 : (fftc_real)1.0) \
* ((n) & 8 ? (fftc_real)256.0 : (fftc_real)1.0) \
* ((n) & 4 ? (fftc_real)16.0 : (fftc_real)1.0) \
* ((n) & 2 ? (fftc_real)4.0 : (fftc_real)1.0) \
* ((n) & 1 ? (fftc_real)2.0 : (fftc_real)1.0) \
)
// r := a + b
static inline void add (const fftc_complex& a, const fftc_complex& b, fftc_complex& r)
{
r.re = a.re + b.re;
r.im = a.im + b.im;
}
// r := a - b
static inline void sub (const fftc_complex& a, const fftc_complex& b, fftc_complex& r)
{
r.re = a.re - b.re;
r.im = a.im - b.im;
}
// r := a * b
static inline void mul (fftc_real a, const fftc_complex& b, fftc_complex& r)
{
r.re = a * b.re;
r.im = a * b.im;
}
// r := a * b
static inline void mul (const fftc_complex& a, const fftc_complex& b, fftc_complex& r)
{
var fftc_real r_re = a.re * b.re - a.im * b.im;
var fftc_real r_im = a.re * b.im + a.im * b.re;
r.re = r_re;
r.im = r_im;
}
#ifndef _BIT_REVERSE
#define _BIT_REVERSE
// Reverse an n-bit number x. n>0.
static uintL bit_reverse (uintL n, uintL x)
{
var uintL y = 0;
do {
y <<= 1;
y |= (x & 1);
x >>= 1;
} while (!(--n == 0));
return y;
}
#endif
// Compute a complex convolution using FFT: z[0..N-1] := x[0..N-1] * y[0..N-1].
static void fftc_convolution (const uintL n, const uintL N, // N = 2^n
fftc_complex * x, // N numbers
fftc_complex * y, // N numbers
fftc_complex * z // N numbers result
)
{
CL_ALLOCA_STACK;
#if (FFTC_BACKWARD == RECIPROOT) || defined(DEBUG_FFTC)
var fftc_complex* const w = cl_alloc_array(fftc_complex,N);
#else
var fftc_complex* const w = cl_alloc_array(fftc_complex,(N>>1)+1);
#endif
var uintL i;
// Initialize w[i] to w^i, w a primitive N-th root of unity.
w[0] = fftc_roots_of_1[0];
#if (FFTC_BACKWARD == RECIPROOT) || defined(DEBUG_FFTC)
{
var int j;
for (j = n-1; j>=0; j--) {
var fftc_complex r_j = fftc_roots_of_1[n-j];
w[1<<j] = r_j;
for (i = (2<<j); i < N; i += (2<<j))
mul(w[i],r_j, w[i+(1<<j)]);
}
}
#else
{
var int j;
for (j = n-1; j>=0; j--) {
var fftc_complex r_j = fftc_roots_of_1[n-j];
w[1<<j] = r_j;
for (i = (2<<j); i < N>>1; i += (2<<j))
mul(w[i],r_j, w[i+(1<<j)]);
}
}
#endif
#ifdef DEBUG_FFTC
// Check that w is really a primitive N-th root of unity.
{
var fftc_real epsilon;
epsilon = (fftc_real)(n>=3 ? 4*n-11 : 0);
epsilon = epsilon / fftc_pow2(fftc_real_mant_bits);
if (fftc_real_rounds == rounds_to_nearest)
epsilon = 0.5*epsilon;
epsilon = 1.414*epsilon;
// epsilon = (4*k-11)*e*2^(1/2-m).
var fftc_real part;
var fftc_complex& w_N = w[N>>1];
part = w_N.re - (fftc_real)(-1.0);
if (part > epsilon || part < -epsilon)
cl_abort();
part = w_N.im;
if (part > epsilon || part < -epsilon)
cl_abort();
}
{
var fftc_complex w_N;
mul(w[N-1],fftc_roots_of_1[n], w_N);
// Since there was one more multiplication,
// we have to replace n by (n+1) in the epsilon above.
var fftc_real epsilon;
epsilon = (fftc_real)(4*n-7);
epsilon = epsilon / fftc_pow2(fftc_real_mant_bits);
if (fftc_real_rounds == rounds_to_nearest)
epsilon = 0.5*epsilon;
epsilon = 1.414*epsilon;
// epsilon = (4*k-7)*e*2^(1/2-m).
var fftc_real part;
part = w_N.re - (fftc_real)1.0;
if (part > epsilon || part < -epsilon)
cl_abort();
part = w_N.im;
if (part > epsilon || part < -epsilon)
cl_abort();
}
#endif
var bool squaring = (x == y);
// Do an FFT of length N on x.
{
var sintL l;
/* l = n-1 */ {
var const uintL tmax = N>>1; // tmax = 2^(n-1)
for (var uintL t = 0; t < tmax; t++) {
var uintL i1 = t;
var uintL i2 = i1 + tmax;
// Butterfly: replace (x(i1),x(i2)) by
// (x(i1) + x(i2), x(i1) - x(i2)).
var fftc_complex tmp;
tmp = x[i2];
sub(x[i1],tmp, x[i2]);
add(x[i1],tmp, x[i1]);
}
}
for (l = n-2; l>=0; l--) {
var const uintL smax = (uintL)1 << (n-1-l);
var const uintL tmax = (uintL)1 << l;
for (var uintL s = 0; s < smax; s++) {
var uintL exp = bit_reverse(n-1-l,s) << l;
for (var uintL t = 0; t < tmax; t++) {
var uintL i1 = (s << (l+1)) + t;
var uintL i2 = i1 + tmax;
// Butterfly: replace (x(i1),x(i2)) by
// (x(i1) + w^exp*x(i2), x(i1) - w^exp*x(i2)).
var fftc_complex tmp;
mul(x[i2],w[exp], tmp);
sub(x[i1],tmp, x[i2]);
add(x[i1],tmp, x[i1]);
}
}
}
}
// Do an FFT of length N on y.
if (!squaring) {
var sintL l;
/* l = n-1 */ {
var uintL const tmax = N>>1; // tmax = 2^(n-1)
for (var uintL t = 0; t < tmax; t++) {
var uintL i1 = t;
var uintL i2 = i1 + tmax;
// Butterfly: replace (y(i1),y(i2)) by
// (y(i1) + y(i2), y(i1) - y(i2)).
var fftc_complex tmp;
tmp = y[i2];
sub(y[i1],tmp, y[i2]);
add(y[i1],tmp, y[i1]);
}
}
for (l = n-2; l>=0; l--) {
var const uintL smax = (uintL)1 << (n-1-l);
var const uintL tmax = (uintL)1 << l;
for (var uintL s = 0; s < smax; s++) {
var uintL exp = bit_reverse(n-1-l,s) << l;
for (var uintL t = 0; t < tmax; t++) {
var uintL i1 = (s << (l+1)) + t;
var uintL i2 = i1 + tmax;
// Butterfly: replace (y(i1),y(i2)) by
// (y(i1) + w^exp*y(i2), y(i1) - w^exp*y(i2)).
var fftc_complex tmp;
mul(y[i2],w[exp], tmp);
sub(y[i1],tmp, y[i2]);
add(y[i1],tmp, y[i1]);
}
}
}
}
// Multiply the transformed vectors into z.
for (i = 0; i < N; i++)
mul(x[i],y[i], z[i]);
// Undo an FFT of length N on z.
{
var uintL l;
for (l = 0; l < n-1; l++) {
var const uintL smax = (uintL)1 << (n-1-l);
var const uintL tmax = (uintL)1 << l;
#if FFTC_BACKWARD != CLEVER
for (var uintL s = 0; s < smax; s++) {
var uintL exp = bit_reverse(n-1-l,s) << l;
#if FFTC_BACKWARD == RECIPROOT
if (exp > 0)
exp = N - exp; // negate exp (use w^-1 instead of w)
#endif
for (var uintL t = 0; t < tmax; t++) {
var uintL i1 = (s << (l+1)) + t;
var uintL i2 = i1 + tmax;
// Inverse Butterfly: replace (z(i1),z(i2)) by
// ((z(i1)+z(i2))/2, (z(i1)-z(i2))/(2*w^exp)).
// Do the division by 2 later.
var fftc_complex sum;
var fftc_complex diff;
add(z[i1],z[i2], sum);
sub(z[i1],z[i2], diff);
z[i1] = sum;
mul(diff,w[exp], z[i2]);
}
}
#else // FFTC_BACKWARD == CLEVER: clever handling of negative exponents
/* s = 0, exp = 0 */ {
for (var uintL t = 0; t < tmax; t++) {
var uintL i1 = t;
var uintL i2 = i1 + tmax;
// Inverse Butterfly: replace (z(i1),z(i2)) by
// ((z(i1)+z(i2))/2, (z(i1)-z(i2))/(2*w^exp)),
// with exp <-- 0.
// Do the division by 2 later.
var fftc_complex sum;
var fftc_complex diff;
add(z[i1],z[i2], sum);
sub(z[i1],z[i2], diff);
z[i1] = sum;
z[i2] = diff;
}
}
for (var uintL s = 1; s < smax; s++) {
var uintL exp = bit_reverse(n-1-l,s) << l;
exp = (N>>1) - exp; // negate exp (use w^-1 instead of w)
for (var uintL t = 0; t < tmax; t++) {
var uintL i1 = (s << (l+1)) + t;
var uintL i2 = i1 + tmax;
// Inverse Butterfly: replace (z(i1),z(i2)) by
// ((z(i1)+z(i2))/2, (z(i1)-z(i2))/(2*w^exp)),
// with exp <-- (N/2 - exp).
// Do the division by 2 later.
var fftc_complex sum;
var fftc_complex diff;
add(z[i1],z[i2], sum);
sub(z[i2],z[i1], diff); // note that w^(N/2) = -1
z[i1] = sum;
mul(diff,w[exp], z[i2]);
}
}
#endif
}
/* l = n-1 */ {
var const fftc_real pow2 = (fftc_real)1 / (fftc_real)N;
var const uintL tmax = N>>1; // tmax = 2^(n-1)
for (var uintL t = 0; t < tmax; t++) {
var uintL i1 = t;
var uintL i2 = i1 + tmax;
// Inverse Butterfly: replace (z(i1),z(i2)) by
// ((z(i1)+z(i2))/2, (z(i1)-z(i2))/2).
// Do all the divisions by 2 now.
var fftc_complex sum;
var fftc_complex diff;
add(z[i1],z[i2], sum);
sub(z[i1],z[i2], diff);
mul(pow2,sum, z[i1]);
mul(pow2,diff, z[i2]);
}
}
}
#if FFTC_BACKWARD == FORWARD
// Swap z[i] and z[N-i] for 0 < i < N/2.
for (i = (N>>1)-1; i > 0; i--) {
var fftc_complex tmp = z[i];
z[i] = z[N-i];
z[N-i] = tmp;
}
#endif
}
// For a given k >= 2, the maximum l is determined by
// 2*l < m - 1/2 - 2*k - log_2(12*k-15) - (1 if e=1.0, 0 if e=0.5).
// This is a decreasing function of k.
#define max_l(k) \
(int)((fftc_real_mant_bits \
- 2*(k) \
- ((k)<=2 ? 4 : (k)<=3 ? 5 : (k)<=5 ? 6 : (k)<=8 ? 7 : (k)<=16 ? 8 : (k)<=31 ? 9 : 10) \
- (fftc_real_rounds == rounds_to_nearest ? 0 : 1)) \
/ 2)
static int max_l_table[32+1] =
{ 0, 0, max_l(2), max_l(3), max_l(4), max_l(5), max_l(6),
max_l(7), max_l(8), max_l(9), max_l(10), max_l(11), max_l(12), max_l(13),
max_l(14), max_l(15), max_l(16), max_l(17), max_l(18), max_l(19), max_l(20),
max_l(21), max_l(22), max_l(23), max_l(24), max_l(25), max_l(26), max_l(27),
max_l(28), max_l(29), max_l(30), max_l(31), max_l(32)
};
// Split len uintD's below sourceptr into chunks of l bits, thus filling
// N complex numbers at x.
static void fill_factor (uintL N, fftc_complex* x, uintL l,
const uintD* sourceptr, uintL len)
{
var uintL i;
if (max_l(2) > intDsize && l > intDsize) {
// l > intDsize
if (max_l(2) > 64 && l > 64) {
fprint(std::cerr, "FFT problem: l > 64 not supported by pow2_table\n");
cl_abort();
}
var fftc_real carry = 0;
var sintL carrybits = 0; // number of bits in carry (>=0, <l)
i = 0;
while (len > 0) {
var uintD digit = lsprefnext(sourceptr);
if (carrybits+intDsize >= l) {
x[i].re = carry + (fftc_real)(digit & bitm(l-carrybits)) * fftc_pow2_table[carrybits];
x[i].im = (fftc_real)0;
i++;
carry = (l-carrybits == intDsize ? (fftc_real)0 : (fftc_real)(digit >> (l-carrybits)));
carrybits = carrybits+intDsize-l;
} else {
carry = carry + (fftc_real)digit * fftc_pow2_table[carrybits];
carrybits = carrybits+intDsize;
}
len--;
}
if (carrybits > 0) {
x[i].re = carry;
x[i].im = (fftc_real)0;
i++;
}
if (i > N)
cl_abort();
} else if (max_l(2) >= intDsize && l == intDsize) {
// l = intDsize
if (len > N)
cl_abort();
for (i = 0; i < len; i++) {
var uintD digit = lsprefnext(sourceptr);
x[i].re = (fftc_real)digit;
x[i].im = (fftc_real)0;
}
} else {
// l < intDsize
var const uintD l_mask = bit(l)-1;
var uintD carry = 0;
var sintL carrybits = 0; // number of bits in carry (>=0, <intDsize)
for (i = 0; i < N; i++) {
if (carrybits >= l) {
x[i].re = (fftc_real)(carry & l_mask);
x[i].im = (fftc_real)0;
carry >>= l;
carrybits -= l;
} else {
if (len == 0)
break;
len--;
var uintD digit = lsprefnext(sourceptr);
x[i].re = (fftc_real)((carry | (digit << carrybits)) & l_mask);
x[i].im = (fftc_real)0;
carry = digit >> (l-carrybits);
carrybits = intDsize - (l-carrybits);
}
}
while (carrybits > 0) {
if (!(i < N))
cl_abort();
x[i].re = (fftc_real)(carry & l_mask);
x[i].im = (fftc_real)0;
carry >>= l;
carrybits -= l;
i++;
}
if (len > 0)
cl_abort();
}
for ( ; i < N; i++) {
x[i].re = (fftc_real)0;
x[i].im = (fftc_real)0;
}
}
// Given a not too large floating point number, round it to the nearest integer.
static inline fftc_real fftc_fround (fftc_real x)
{
return
#if (fftc_real_rounds == rounds_to_nearest)
(x + (fftc_pow2(fftc_real_mant_bits-1)+fftc_pow2(fftc_real_mant_bits-2)))
- (fftc_pow2(fftc_real_mant_bits-1)+fftc_pow2(fftc_real_mant_bits-2));
#elif (fftc_real_rounds == rounds_to_infinity)
(fftc_pow2(fftc_real_mant_bits-1)+fftc_pow2(fftc_real_mant_bits-2))
- ((fftc_pow2(fftc_real_mant_bits-1)+fftc_pow2(fftc_real_mant_bits-2))
- (x + (fftc_real)0.5));
#else // rounds_to_zero, rounds_to_minus_infinity
((x + (fftc_real)0.5)
+ (fftc_pow2(fftc_real_mant_bits-1)+fftc_pow2(fftc_real_mant_bits-2)))
- (fftc_pow2(fftc_real_mant_bits-1)+fftc_pow2(fftc_real_mant_bits-2));
#endif
}
// Given a not too large floating point number, round it down.
static inline fftc_real fftc_ffloor (fftc_real x)
{
#if (fftc_real_rounds == rounds_to_nearest)
var fftc_real y =
(x + (fftc_pow2(fftc_real_mant_bits-1)+fftc_pow2(fftc_real_mant_bits-2)))
- (fftc_pow2(fftc_real_mant_bits-1)+fftc_pow2(fftc_real_mant_bits-2));
if (y <= x)
return y;
else
return y - (fftc_real)1.0;
#elif (fftc_real_rounds == rounds_to_infinity)
return
(fftc_pow2(fftc_real_mant_bits-1)+fftc_pow2(fftc_real_mant_bits-2))
- ((fftc_pow2(fftc_real_mant_bits-1)+fftc_pow2(fftc_real_mant_bits-2))
- x);
#else // rounds_to_zero, rounds_to_minus_infinity
return
(x + (fftc_pow2(fftc_real_mant_bits-1)+fftc_pow2(fftc_real_mant_bits-2)))
- (fftc_pow2(fftc_real_mant_bits-1)+fftc_pow2(fftc_real_mant_bits-2));
#endif
}
// Combine the N complex numbers at z into uintD's below destptr.
// The z[i] are known to be approximately integers >= 0, < N*2^(2*l).
// Assumes room for floor(N*l/intDsize)+(1+ceiling((n+2*l)/intDsize)) uintD's
// below destptr. Fills len digits and returns (destptr lspop len).
static uintD* unfill_product (uintL n, uintL N, // N = 2^n
const fftc_complex * z, uintL l,
uintD* destptr)
{
var uintL i;
if (n + 2*l <= intDsize) {
// 2-digit carry is sufficient, l < intDsize
var uintD carry0 = 0;
var uintD carry1 = 0;
var uintL shift = 0; // shift next digit before adding it to the carry, >=0, <intDsize
for (i = 0; i < N; i++) {
// Fetch z[i].re and round it to the nearest uintD.
var uintD digit = (uintD)(z[i].re + (fftc_real)0.5);
if (shift > 0) {
carry1 += digit >> (intDsize-shift);
digit = digit << shift;
}
if ((carry0 += digit) < digit)
carry1 += 1;
shift += l;
if (shift >= intDsize) {
lsprefnext(destptr) = carry0;
carry0 = carry1;
carry1 = 0;
shift -= intDsize;
}
}
lsprefnext(destptr) = carry0;
lsprefnext(destptr) = carry1;
} else if (n + 2*l <= 2*intDsize) {
// 3-digit carry is sufficient, l < intDsize
#if HAVE_DD
var uintDD carry0 = 0;
var uintD carry1 = 0;
var uintL shift = 0; // shift next digit before adding it to the carry, >=0, <intDsize
for (i = 0; i < N; i++) {
// Fetch z[i].re and round it to the nearest uintDD.
var uintDD digit = (uintDD)(z[i].re + (fftc_real)0.5);
if (shift > 0) {
carry1 += (uintD)(digit >> (2*intDsize-shift));
digit = digit << shift;
}
if ((carry0 += digit) < digit)
carry1 += 1;
shift += l;
if (shift >= intDsize) {
lsprefnext(destptr) = lowD(carry0);
carry0 = highlowDD(carry1,highD(carry0));
carry1 = 0;
shift -= intDsize;
}
}
lsprefnext(destptr) = lowD(carry0);
lsprefnext(destptr) = highD(carry0);
lsprefnext(destptr) = carry1;
#else
var uintD carry0 = 0;
var uintD carry1 = 0;
var uintD carry2 = 0;
var uintL shift = 0; // shift next digit before adding it to the carry, >=0, <intDsize
for (i = 0; i < N; i++) {
// Fetch z[i].re and round it to the nearest uintDD.
var fftc_real zi = z[i].re + (fftc_real)0.5;
var const fftc_real pow2_intDsize = fftc_pow2(intDsize);
var const fftc_real invpow2_intDsize = (fftc_real)1.0/pow2_intDsize;
var uintD digit1 = (uintD)(zi * invpow2_intDsize);
var uintD digit0 = (uintD)(zi - (fftc_real)digit1 * pow2_intDsize);
if (shift > 0) {
carry2 += digit1 >> (intDsize-shift);
digit1 = (digit1 << shift) | (digit0 >> (intDsize-shift));
digit0 = digit0 << shift;
}
if ((carry0 += digit0) < digit0)
if ((carry1 += 1) == 0)
carry2 += 1;
if ((carry1 += digit1) < digit1)
carry2 += 1;
shift += l;
if (shift >= intDsize) {
lsprefnext(destptr) = carry0;
carry0 = carry1;
carry1 = carry2;
carry2 = 0;
shift -= intDsize;
}
}
lsprefnext(destptr) = carry0;
lsprefnext(destptr) = carry1;
lsprefnext(destptr) = carry2;
#endif
} else {
// 1-digit+1-float carry is sufficient
var uintD carry0 = 0;
var fftc_real carry1 = 0;
var uintL shift = 0; // shift next digit before adding it to the carry, >=0, <intDsize
for (i = 0; i < N; i++) {
// Fetch z[i].re and round it to the nearest integer.
var fftc_real digit = fftc_fround(z[i].re);
#ifdef DEBUG_FFTC
if (!(digit >= (fftc_real)0
&& z[i].re > digit - (fftc_real)0.5
&& z[i].re < digit + (fftc_real)0.5))
cl_abort();
#endif
if (shift > 0)
digit = digit * fftc_pow2_table[shift];
var fftc_real digit1 = fftc_ffloor(digit*((fftc_real)1.0/fftc_pow2(intDsize)));
var uintD digit0 = (uintD)(digit - digit1*fftc_pow2(intDsize));
carry1 += digit1;
if ((carry0 += digit0) < digit0)
carry1 += (fftc_real)1.0;
shift += l;
while (shift >= intDsize) {
lsprefnext(destptr) = carry0;
var fftc_real tmp = fftc_ffloor(carry1*((fftc_real)1.0/fftc_pow2(intDsize)));
carry0 = (uintD)(carry1 - tmp*fftc_pow2(intDsize));
carry1 = tmp;
shift -= intDsize;
}
}
if (carry0 > 0 || carry1 > (fftc_real)0.0) {
lsprefnext(destptr) = carry0;
while (carry1 > (fftc_real)0.0) {
var fftc_real tmp = fftc_ffloor(carry1*((fftc_real)1.0/fftc_pow2(intDsize)));
lsprefnext(destptr) = (uintD)(carry1 - tmp*fftc_pow2(intDsize));
carry1 = tmp;
}
}
}
return destptr;
}
static inline void mulu_fftcomplex_nocheck (const uintD* sourceptr1, uintC len1,
const uintD* sourceptr2, uintC len2,
uintD* destptr)
// Es ist 2 <= len1 <= len2.
{
// We have to find parameters l and k such that
// ceiling(len1*intDsize/l) + ceiling(len2*intDsize/l) - 1 <= 2^k,
// and (12*k-15)*e*2^(2*l+2*k+1/2-m) < 1/2.
// Try primarily to minimize k. Minimizing l buys you nothing.
var uintL k;
// Computing k: If len1 and len2 differ much, we'll split source2 -
// hence for the moment just substitute len1 for len2.
//
// First approximation of k: A necessary condition for
// 2*ceiling(len1*intDsize/l) - 1 <= 2^k
// is 2*len1*intDsize/l_max - 1 <= 2^k.
{
var const int l = max_l(2);
var uintL lhs = 2*ceiling((uintL)len1*intDsize,l) - 1; // >=1
if (lhs < 3)
k = 2;
else
integerlength32(lhs-1, k=); // k>=2
}
// Try whether this k is ok or whether we have to increase k.
for ( ; ; k++) {
if (k >= sizeof(max_l_table)/sizeof(max_l_table[0])
|| max_l_table[k] <= 0) {
fprint(std::cerr, "FFT problem: numbers too big, floating point precision not sufficient\n");
cl_abort();
}
if (2*ceiling((uintL)len1*intDsize,max_l_table[k])-1 <= ((uintL)1 << k))
break;
}
// We could try to reduce l, keeping the same k. But why should we?
// Calculate the number of pieces in which source2 will have to be
// split. Each of the pieces must satisfy
// ceiling(len1*intDsize/l) + ceiling(len2*intDsize/l) - 1 <= 2^k,
var uintL len2p;
// Try once with k, once with k+1. Compare them.
{
var uintL remaining_k = ((uintL)1 << k) + 1 - ceiling((uintL)len1*intDsize,max_l_table[k]);
var uintL max_piecelen_k = floor(remaining_k*max_l_table[k],intDsize);
var uintL numpieces_k = ceiling(len2,max_piecelen_k);
var uintL remaining_k1 = ((uintL)1 << (k+1)) + 1 - ceiling((uintL)len1*intDsize,max_l_table[k+1]);
var uintL max_piecelen_k1 = floor(remaining_k1*max_l_table[k+1],intDsize);
var uintL numpieces_k1 = ceiling(len2,max_piecelen_k1);
if (numpieces_k <= 2*numpieces_k1) {
// keep k
len2p = max_piecelen_k;
} else {
// choose k+1
k = k+1;
len2p = max_piecelen_k1;
}
}
var const uintL l = max_l_table[k];
var const uintL n = k;
var const uintL N = (uintL)1 << n;
CL_ALLOCA_STACK;
var fftc_complex* const x = cl_alloc_array(fftc_complex,N);
var fftc_complex* const y = cl_alloc_array(fftc_complex,N);
#ifdef DEBUG_FFTC
var fftc_complex* const z = cl_alloc_array(fftc_complex,N);
#else
var fftc_complex* const z = x; // put z in place of x - saves memory
#endif
var uintD* const tmpprod1 = cl_alloc_array(uintD,len1+1);
var uintL tmpprod_len = floor(l<<n,intDsize)+6;
var uintD* const tmpprod = cl_alloc_array(uintD,tmpprod_len);
var uintL destlen = len1+len2;
clear_loop_lsp(destptr,destlen);
do {
if (len2p > len2)
len2p = len2;
if (len2p == 1) {
// cheap case
var uintD* tmpptr = arrayLSDptr(tmpprod1,len1+1);
mulu_loop_lsp(lspref(sourceptr2,0),sourceptr1,tmpptr,len1);
if (addto_loop_lsp(tmpptr,destptr,len1+1))
if (inc_loop_lsp(destptr lspop (len1+1),destlen-(len1+1)))
cl_abort();
} else {
var bool squaring = ((sourceptr1 == sourceptr2) && (len1 == len2p));
// Fill factor x.
fill_factor(N,x,l,sourceptr1,len1);
// Fill factor y.
if (!squaring)
fill_factor(N,y,l,sourceptr2,len2p);
// Multiply.
if (!squaring)
fftc_convolution(n,N, &x[0], &y[0], &z[0]);
else
fftc_convolution(n,N, &x[0], &x[0], &z[0]);
#ifdef DEBUG_FFTC
// Check result.
{
var fftc_real re_lo_limit = (fftc_real)(-0.5);
var fftc_real re_hi_limit = (fftc_real)N * fftc_pow2_table[l] * fftc_pow2_table[l] + (fftc_real)0.5;
var fftc_real im_lo_limit = (fftc_real)(-0.5);
var fftc_real im_hi_limit = (fftc_real)0.5;
for (var uintL i = 0; i < N; i++) {
if (!(z[i].im > im_lo_limit
&& z[i].im < im_hi_limit))
cl_abort();
if (!(z[i].re > re_lo_limit
&& z[i].re < re_hi_limit))
cl_abort();
}
}
#endif
var uintD* tmpLSDptr = arrayLSDptr(tmpprod,tmpprod_len);
var uintD* tmpMSDptr = unfill_product(n,N,z,l,tmpLSDptr);
var uintL tmplen =
#if CL_DS_BIG_ENDIAN_P
tmpLSDptr - tmpMSDptr;
#else
tmpMSDptr - tmpLSDptr;
#endif
if (tmplen > tmpprod_len)
cl_abort();
// Add result to destptr[-destlen..-1]:
if (tmplen > destlen) {
if (test_loop_msp(tmpMSDptr,tmplen-destlen))
cl_abort();
tmplen = destlen;
}
if (addto_loop_lsp(tmpLSDptr,destptr,tmplen))
if (inc_loop_lsp(destptr lspop tmplen,destlen-tmplen))
cl_abort();
}
// Decrement len2.
destptr = destptr lspop len2p;
destlen -= len2p;
sourceptr2 = sourceptr2 lspop len2p;
len2 -= len2p;
} while (len2 > 0);
}
#ifndef _CHECKSUM
#define _CHECKSUM
// Compute a checksum: number mod (2^intDsize-1).
static uintD compute_checksum (const uintD* sourceptr, uintC len)
{
var uintD tmp = ~(uintD)0; // -1-(sum mod 2^intDsize-1), always >0
do {
var uintD digit = lsprefnext(sourceptr);
if (digit < tmp)
tmp -= digit; // subtract digit
else
tmp -= digit+1; // subtract digit-(2^intDsize-1)
} while (--len > 0);
return ~tmp;
}
// Multiply two checksums modulo (2^intDsize-1).
static inline uintD multiply_checksum (uintD checksum1, uintD checksum2)
{
var uintD checksum;
var uintD cksum_hi;
#if HAVE_DD
var uintDD cksum = muluD(checksum1,checksum2);
cksum_hi = highD(cksum); checksum = lowD(cksum);
#else
muluD(checksum1,checksum2, cksum_hi =, checksum =);
#endif
if ((checksum += cksum_hi) + 1 <= cksum_hi)
checksum += 1;
return checksum;
}
#endif // _CHECKSUM
static void mulu_fftcomplex (const uintD* sourceptr1, uintC len1,
const uintD* sourceptr2, uintC len2,
uintD* destptr)
{
// Compute checksums of the arguments and multiply them.
var uintD checksum1 = compute_checksum(sourceptr1,len1);
var uintD checksum2 = compute_checksum(sourceptr2,len2);
var uintD checksum = multiply_checksum(checksum1,checksum2);
mulu_fftcomplex_nocheck(sourceptr1,len1,sourceptr2,len2,destptr);
if (!(checksum == compute_checksum(destptr,len1+len2))) {
fprint(std::cerr, "FFT problem: checksum error\n");
cl_abort();
}
}
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