Inverse sqrt for fixed point
The trick is to apply Newton's method to the problem x - 1/y^2 = 0. So, given x, solve for y using an iterative scheme.
Y_(n+1) = y_n * (3 - x*y_n^2)/2
The divide by 2 is just a bit shift, or at worst, a multiply by 0.5. This scheme converges to y=1/sqrt(x), exactly as requested, and without any true divides at all.
The only problem is that you need a decent starting value for y. As I recall there are limits on the estimate y for the iterations to converge.
ARMv5TE processors provide a fast integer multiplier, and a "count leading zeros" instruction. They also typically come with moderately sized caches. Based on this, the most suitable approach for a high-performance implementation appears to be a table lookup for an initial approximation, followed by two Newton-Raphson iterations to achieve fully accurate results. We can speed up the first of these iterations further with additional pre-computation that is incorporated into the table, a technique used by Cray computers forty years ago.
The function fxrsqrt() below implements this approach. It starts out with an 8-bit approximation r to the reciprocal square root of the argument a, but instead of storing r, each table element stores 3r (in the lower ten bits of the 32-bit entry) and r3 (in the upper 22 bits of the 32-bit entry). This allows the quick computation of the first iteration as
r1 = 0.5 * (3 * r - a * r3). The second iteration is then computed in the conventional way as r2 = 0.5 * r1 * (3 - r1 * (r1 * a)).
To be able to perform these computations accurately, regardless of the magnitude of the input, the argument a is normalized at the start of the computation, in essence representing it as a 2.32 fixed-point number multiplied with a scale factor of 2scal. At the end of the computation the result is denormalized according to formula 1/sqrt(22n) = 2-n. By rounding up results whose most significant discarded bit is 1, accuracy is improved, resulting in almost all results being correctly rounded. The exhaustive test reports: results too low: 639 too high: 1454 not correctly rounded: 2093
The code makes use of two helper functions: __clz() determines the number of leading zero bits in a non-zero 32-bit argument. __umulhi() computes the 32 most significant bits of a full 64-bit product of two unsigned 32-bit integers. Both functions should be implemented either via compiler intrinsics, or by using a bit of inline assembly. In the code below I am showing portable implementations well suited to ARM CPUs along with inline assembly versions for x86 platforms. On ARMv5TE platforms __clz() should be mapped map to the CLZ instruction, and __umulhi() should be mapped to UMULL.
#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <math.h>
#define USE_OWN_INTRINSICS 1
#if USE_OWN_INTRINSICS
__forceinline int __clz (uint32_t a)
{
int r;
__asm__ ("bsrl %1,%0\n\t" : "=r"(r): "r"(a));
return 31 - r;
}
uint32_t __umulhi (uint32_t a, uint32_t b)
{
uint32_t r;
__asm__ ("movl %1,%%eax\n\tmull %2\n\tmovl %%edx,%0\n\t"
: "=r"(r) : "r"(a), "r"(b) : "eax", "edx");
return r;
}
#else // USE_OWN_INTRINSICS
int __clz (uint32_t a)
{
uint32_t r = 32;
if (a >= 0x00010000) { a >>= 16; r -= 16; }
if (a >= 0x00000100) { a >>= 8; r -= 8; }
if (a >= 0x00000010) { a >>= 4; r -= 4; }
if (a >= 0x00000004) { a >>= 2; r -= 2; }
r -= a - (a & (a >> 1));
return r;
}
uint32_t __umulhi (uint32_t a, uint32_t b)
{
return (uint32_t)(((uint64_t)a * b) >> 32);
}
#endif // USE_OWN_INTRINSICS
/*
* For each sub-interval in [1, 4), use an 8-bit approximation r to reciprocal
* square root. To speed up subsequent Newton-Raphson iterations, each entry in
* the table combines two pieces of information: The least-significant 10 bits
* store 3*r, the most-significant 22 bits store r**3, rounded from 24 down to
* 22 bits such that accuracy is optimized.
*/
uint32_t rsqrt_tab [96] =
{
0xfa0bdefa, 0xee6af6ee, 0xe5effae5, 0xdaf27ad9,
0xd2eff6d0, 0xc890aec4, 0xc10366bb, 0xb9a71ab2,
0xb4da2eac, 0xadce7ea3, 0xa6f2b29a, 0xa279a694,
0x9beb568b, 0x97a5c685, 0x9163027c, 0x8d4fd276,
0x89501e70, 0x8563da6a, 0x818ac664, 0x7dc4fe5e,
0x7a122258, 0x7671be52, 0x72e44a4c, 0x6f68fa46,
0x6db22a43, 0x6a52623d, 0x67041a37, 0x65639634,
0x622ffe2e, 0x609cba2b, 0x5d837e25, 0x5bfcfe22,
0x58fd461c, 0x57838619, 0x560e1216, 0x53300a10,
0x51c72e0d, 0x50621a0a, 0x4da48204, 0x4c4c2e01,
0x4af789fe, 0x49a689fb, 0x485a11f8, 0x4710f9f5,
0x45cc2df2, 0x448b4def, 0x421505e9, 0x40df5de6,
0x3fadc5e3, 0x3e7fe1e0, 0x3d55c9dd, 0x3d55d9dd,
0x3c2f41da, 0x39edd9d4, 0x39edc1d4, 0x38d281d1,
0x37bae1ce, 0x36a6c1cb, 0x3595d5c8, 0x3488f1c5,
0x3488fdc5, 0x337fbdc2, 0x3279ddbf, 0x317749bc,
0x307831b9, 0x307879b9, 0x2f7d01b6, 0x2e84ddb3,
0x2d9005b0, 0x2d9015b0, 0x2c9ec1ad, 0x2bb0a1aa,
0x2bb0f5aa, 0x2ac615a7, 0x29ded1a4, 0x29dec9a4,
0x28fabda1, 0x2819e99e, 0x2819ed9e, 0x273c3d9b,
0x273c359b, 0x2661dd98, 0x258ad195, 0x258af195,
0x24b71192, 0x24b6b192, 0x23e6058f, 0x2318118c,
0x2318718c, 0x224da189, 0x224dd989, 0x21860d86,
0x21862586, 0x20c19183, 0x20c1b183, 0x20001580
};
/* This function computes the reciprocal square root of its 16.16 fixed-point
* argument. After normalization of the argument if uses the most significant
* bits of the argument for a table lookup to obtain an initial approximation
* accurate to 8 bits. This is followed by two Newton-Raphson iterations with
* quadratic convergence. Finally, the result is denormalized and some simple
* rounding is applied to maximize accuracy.
*
* To speed up the first NR iteration, for the initial 8-bit approximation r0
* the lookup table supplies 3*r0 along with r0**3. A first iteration computes
* a refined estimate r1 = 1.5 * r0 - x * r0**3. The second iteration computes
* the final result as r2 = 0.5 * r1 * (3 - r1 * (r1 * x)).
*
* The accuracy for all arguments in [0x00000001, 0xffffffff] is as follows:
* 639 results are too small by one ulp, 1454 results are too big by one ulp.
* A total of 2093 results deviate from the correctly rounded result.
*/
uint32_t fxrsqrt (uint32_t a)
{
uint32_t s, r, t, scal;
/* handle special case of zero input */
if (a == 0) return ~a;
/* normalize argument */
scal = __clz (a) & 0xfffffffe;
a = a << scal;
/* initial approximation */
t = rsqrt_tab [(a >> 25) - 32];
/* first NR iteration */
r = (t << 22) - __umulhi (t, a);
/* second NR iteration */
s = __umulhi (r, a);
s = 0x30000000 - __umulhi (r, s);
r = __umulhi (r, s);
/* denormalize and round result */
r = ((r >> (18 - (scal >> 1))) + 1) >> 1;
return r;
}
/* reference implementation, 16.16 reciprocal square root of non-zero argment */
uint32_t ref_fxrsqrt (uint32_t a)
{
double arg = a / 65536.0;
double rsq = sqrt (1.0 / arg);
uint32_t r = (uint32_t)(rsq * 65536.0 + 0.5);
return r;
}
int main (void)
{
uint32_t arg = 0x00000001;
uint32_t res, ref;
uint32_t err, lo = 0, hi = 0;
do {
res = fxrsqrt (arg);
ref = ref_fxrsqrt (arg);
err = 0;
if (res < ref) {
err = ref - res;
lo++;
}
if (res > ref) {
err = res - ref;
hi++;
}
if (err > 1) {
printf ("!!!! arg=%08x res=%08x ref=%08x\n", arg, res, ref);
return EXIT_FAILURE;
}
arg++;
} while (arg);
printf ("results too low: %u too high: %u not correctly rounded: %u\n",
lo, hi, lo + hi);
return EXIT_SUCCESS;
}