固定点逆sqrt

Question

我正在寻找固定点16.16数字的最佳反平方根算法。下面的代码是我到目前为止（但基本上它取平方根和除以原始数字，我想得到反平方根没有划分）。如果它改变了任何东西，代码将被编译为armv5te。

uint32_t INVSQRT(uint32_t n) 
{ 
    uint64_t op, res, one; 
    op = ((uint64_t)n<<16); 
    res = 0; 
    one = (uint64_t)1 << 46; 
    while (one > op) one >>= 2; 
    while (one != 0) 
    { 
     if (op >= res + one) 
     { 
      op -= (res + one); 
      res += (one<<1); 
     } 
     res >>= 1; 
     one >>= 2; 
    } 
    res<<=16; 
    res /= n; 
    return(res); 
} 

Answer 1

诀窍是采用牛顿法的问题X - 1/Y^2 = 0。因此，给定的x，使用迭代解决方案用于Y。

Y_(n+1) = y_n * (3 - x*y_n^2)/2 

除以2只是一个位移，或者最坏的情况是乘以0.5。这个方案恰好按要求收敛到y = 1/sqrt（x），并且根本没有任何真正的分歧。

唯一的问题是，你需要一个体面的y值。我记得，迭代收敛的估计值y有限制。

Answer 2

ARMv5TE处理器提供了一个快速整数乘法器和一个“计数前导零”指令。他们通常也会有中等大小的缓存。基于此，对于高性能实现而言，最合适的方法似乎是初始近似的表格查找，然后是两个Newton-Raphson迭代，以实现完全准确的结果。我们可以进一步加速这些迭代的第一步，并将其纳入表中，这是Cray计算机四十年前使用的一种技术。

下面的函数fxrsqrt()实现了这种方法。它以一个8位近似值r开始，参数为a的倒数平方根，但不是存储r，而是每个表元素存储3r（在32位条目的低10位中）和（在32位条目的高位22位）。这允许快速计算第一次迭代为 r = 1.5 * r - a * r 。第二次迭代，然后以常规的方式计算为r = 0.5 * R *（3 - R 1 _*（R * A））。

为了能够准确地执行这些计算，无论输入的大小如何，参数a在计算开始时被归一化，本质上表示为2.32定点数乘以比例因子为2 ^scal。在计算结束时，根据公式1/sqrt对结果进行非规范化（2 ²ⁿ）= 2 ^-n。通过舍弃最重要的丢弃位为1的结果，提高了准确性，导致几乎所有结果都被正确舍入。

该代码使用两个辅助函数：__clz()确定非零32位参数中前导零位的数量。 __umulhi()计算两个无符号32位整数的完整64位乘积的32个最高有效位。这两个函数都应该通过编译器内部函数或通过使用一些内联汇编来实现。在下面的代码中，我展示了适用于ARM CPU的便携式实现以及用于x86平台的内联汇编版本。在ARMv5TE平台__clz()应该映射映射到CLZ指令，而__umulhi()应该映射到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, 0x9163067c, 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, 1457 results are too big by one ulp. 
* A total of 2096 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; 
}