LXR linux/include/linux/reciprocal

   1/* SPDX-License-Identifier: GPL-2.0 */
   2#ifndef _LINUX_RECIPROCAL_DIV_H
   3#define _LINUX_RECIPROCAL_DIV_H
   4
   5#include <linux/types.h>
   6
   7/*
   8 * This algorithm is based on the paper "Division by Invariant
   9 * Integers Using Multiplication" by Torbjörn Granlund and Peter
  10 * L. Montgomery.
  11 *
  12 * The assembler implementation from Agner Fog, which this code is
  13 * based on, can be found here:
  14 * http://www.agner.org/optimize/asmlib.zip
  15 *
  16 * This optimization for A/B is helpful if the divisor B is mostly
  17 * runtime invariant. The reciprocal of B is calculated in the
  18 * slow-path with reciprocal_value(). The fast-path can then just use
  19 * a much faster multiplication operation with a variable dividend A
  20 * to calculate the division A/B.
  21 */
  22
  23struct reciprocal_value {
  24        u32 m;
  25        u8 sh1, sh2;
  26};
  27
  28/* "reciprocal_value" and "reciprocal_divide" together implement the basic
  29 * version of the algorithm described in Figure 4.1 of the paper.
  30 */
  31struct reciprocal_value reciprocal_value(u32 d);
  32
  33static inline u32 reciprocal_divide(u32 a, struct reciprocal_value R)
  34{
  35        u32 t = (u32)(((u64)a * R.m) >> 32);
  36        return (t + ((a - t) >> R.sh1)) >> R.sh2;
  37}
  38
  39struct reciprocal_value_adv {
  40        u32 m;
  41        u8 sh, exp;
  42        bool is_wide_m;
  43};
  44
  45/* "reciprocal_value_adv" implements the advanced version of the algorithm
  46 * described in Figure 4.2 of the paper except when "divisor > (1U << 31)" whose
  47 * ceil(log2(d)) result will be 32 which then requires u128 divide on host. The
  48 * exception case could be easily handled before calling "reciprocal_value_adv".
  49 *
  50 * The advanced version requires more complex calculation to get the reciprocal
  51 * multiplier and other control variables, but then could reduce the required
  52 * emulation operations.
  53 *
  54 * It makes no sense to use this advanced version for host divide emulation,
  55 * those extra complexities for calculating multiplier etc could completely
  56 * waive our saving on emulation operations.
  57 *
  58 * However, it makes sense to use it for JIT divide code generation for which
  59 * we are willing to trade performance of JITed code with that of host. As shown
  60 * by the following pseudo code, the required emulation operations could go down
  61 * from 6 (the basic version) to 3 or 4.
  62 *
  63 * To use the result of "reciprocal_value_adv", suppose we want to calculate
  64 * n/d, the pseudo C code will be:
  65 *
  66 *   struct reciprocal_value_adv rvalue;
  67 *   u8 pre_shift, exp;
  68 *
  69 *   // handle exception case.
  70 *   if (d >= (1U << 31)) {
  71 *     result = n >= d;
  72 *     return;
  73 *   }
  74 *
  75 *   rvalue = reciprocal_value_adv(d, 32)
  76 *   exp = rvalue.exp;
  77 *   if (rvalue.is_wide_m && !(d & 1)) {
  78 *     // floor(log2(d & (2^32 -d)))
  79 *     pre_shift = fls(d & -d) - 1;
  80 *     rvalue = reciprocal_value_adv(d >> pre_shift, 32 - pre_shift);
  81 *   } else {
  82 *     pre_shift = 0;
  83 *   }
  84 *
  85 *   // code generation starts.
  86 *   if (imm == 1U << exp) {
  87 *     result = n >> exp;
  88 *   } else if (rvalue.is_wide_m) {
  89 *     // pre_shift must be zero when reached here.
  90 *     t = (n * rvalue.m) >> 32;
  91 *     result = n - t;
  92 *     result >>= 1;
  93 *     result += t;
  94 *     result >>= rvalue.sh - 1;
  95 *   } else {
  96 *     if (pre_shift)
  97 *       result = n >> pre_shift;
  98 *     result = ((u64)result * rvalue.m) >> 32;
  99 *     result >>= rvalue.sh;
 100 *   }
 101 */
 102struct reciprocal_value_adv reciprocal_value_adv(u32 d, u8 prec);
 103
 104#endif /* _LINUX_RECIPROCAL_DIV_H */
 105