reciprocal_div.h

  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 */