|
| 1 | +/// Creates unsigned and signed division functions optimized for dividing integers with the same |
| 2 | +/// bitwidth as the largest operand in an asymmetrically sized division. For example, x86-64 has an |
| 3 | +/// assembly instruction that can divide a 128 bit integer by a 64 bit integer if the quotient fits |
| 4 | +/// in 64 bits. The 128 bit version of this algorithm would use that fast hardware division to |
| 5 | +/// construct a full 128 bit by 128 bit division. |
| 6 | +#[macro_export] |
| 7 | +macro_rules! impl_asymmetric { |
| 8 | + ( |
| 9 | + $unsigned_name:ident, // name of the unsigned division function |
| 10 | + $signed_name:ident, // name of the signed division function |
| 11 | + $zero_div_fn:ident, // function called when division by zero is attempted |
| 12 | + $half_division:ident, // function for division of a $uX by a $uX |
| 13 | + $asymmetric_division:ident, // function for division of a $uD by a $uX |
| 14 | + $n_h:expr, // the number of bits in a $iH or $uH |
| 15 | + $uH:ident, // unsigned integer with half the bit width of $uX |
| 16 | + $uX:ident, // unsigned integer with half the bit width of $uD |
| 17 | + $uD:ident, // unsigned integer type for the inputs and outputs of `$unsigned_name` |
| 18 | + $iD:ident, // signed integer type for the inputs and outputs of `$signed_name` |
| 19 | + $($unsigned_attr:meta),*; // attributes for the unsigned function |
| 20 | + $($signed_attr:meta),* // attributes for the signed function |
| 21 | + ) => { |
| 22 | + /// Computes the quotient and remainder of `duo` divided by `div` and returns them as a |
| 23 | + /// tuple. |
| 24 | + $( |
| 25 | + #[$unsigned_attr] |
| 26 | + )* |
| 27 | + pub fn $unsigned_name(duo: $uD, div: $uD) -> ($uD,$uD) { |
| 28 | + fn carrying_mul(lhs: $uX, rhs: $uX) -> ($uX, $uX) { |
| 29 | + let tmp = (lhs as $uD).wrapping_mul(rhs as $uD); |
| 30 | + (tmp as $uX, (tmp >> ($n_h * 2)) as $uX) |
| 31 | + } |
| 32 | + fn carrying_mul_add(lhs: $uX, mul: $uX, add: $uX) -> ($uX, $uX) { |
| 33 | + let tmp = (lhs as $uD).wrapping_mul(mul as $uD).wrapping_add(add as $uD); |
| 34 | + (tmp as $uX, (tmp >> ($n_h * 2)) as $uX) |
| 35 | + } |
| 36 | + |
| 37 | + let n: u32 = $n_h * 2; |
| 38 | + |
| 39 | + // Many of these subalgorithms are taken from trifecta.rs, see that for better |
| 40 | + // documentation. |
| 41 | + |
| 42 | + let duo_lo = duo as $uX; |
| 43 | + let duo_hi = (duo >> n) as $uX; |
| 44 | + let div_lo = div as $uX; |
| 45 | + let div_hi = (div >> n) as $uX; |
| 46 | + if div_hi == 0 { |
| 47 | + if div_lo == 0 { |
| 48 | + $zero_div_fn() |
| 49 | + } |
| 50 | + if duo_hi < div_lo { |
| 51 | + // `$uD` by `$uX` division with a quotient that will fit into a `$uX` |
| 52 | + let (quo, rem) = unsafe { $asymmetric_division(duo, div_lo) }; |
| 53 | + return (quo as $uD, rem as $uD) |
| 54 | + } else if (div_lo >> $n_h) == 0 { |
| 55 | + // Short division of $uD by a $uH. |
| 56 | + |
| 57 | + // Some x86_64 CPUs have bad division implementations that make specializing |
| 58 | + // this case faster. |
| 59 | + let div_0 = div_lo as $uH as $uX; |
| 60 | + let (quo_hi, rem_3) = $half_division(duo_hi, div_0); |
| 61 | + |
| 62 | + let duo_mid = |
| 63 | + ((duo >> $n_h) as $uH as $uX) |
| 64 | + | (rem_3 << $n_h); |
| 65 | + let (quo_1, rem_2) = $half_division(duo_mid, div_0); |
| 66 | + |
| 67 | + let duo_lo = |
| 68 | + (duo as $uH as $uX) |
| 69 | + | (rem_2 << $n_h); |
| 70 | + let (quo_0, rem_1) = $half_division(duo_lo, div_0); |
| 71 | + |
| 72 | + return ( |
| 73 | + (quo_0 as $uD) |
| 74 | + | ((quo_1 as $uD) << $n_h) |
| 75 | + | ((quo_hi as $uD) << n), |
| 76 | + rem_1 as $uD |
| 77 | + ) |
| 78 | + } else { |
| 79 | + // Short division using the $uD by $uX division |
| 80 | + let (quo_hi, rem_hi) = $half_division(duo_hi, div_lo); |
| 81 | + let tmp = unsafe { |
| 82 | + $asymmetric_division((duo_lo as $uD) | ((rem_hi as $uD) << n), div_lo) |
| 83 | + }; |
| 84 | + return ((tmp.0 as $uD) | ((quo_hi as $uD) << n), tmp.1 as $uD) |
| 85 | + } |
| 86 | + } |
| 87 | + |
| 88 | + let duo_lz = duo_hi.leading_zeros(); |
| 89 | + let div_lz = div_hi.leading_zeros(); |
| 90 | + let rel_leading_sb = div_lz.wrapping_sub(duo_lz); |
| 91 | + if rel_leading_sb < $n_h { |
| 92 | + // Some x86_64 CPUs have bad hardware division implementations that make putting |
| 93 | + // a two possibility algorithm here beneficial. We also avoid a full `$uD` |
| 94 | + // multiplication. |
| 95 | + let shift = n - duo_lz; |
| 96 | + let duo_sig_n = (duo >> shift) as $uX; |
| 97 | + let div_sig_n = (div >> shift) as $uX; |
| 98 | + let quo = $half_division(duo_sig_n, div_sig_n).0; |
| 99 | + let div_lo = div as $uX; |
| 100 | + let div_hi = (div >> n) as $uX; |
| 101 | + let (tmp_lo, carry) = carrying_mul(quo, div_lo); |
| 102 | + let (tmp_hi, overflow) = carrying_mul_add(quo, div_hi, carry); |
| 103 | + let tmp = (tmp_lo as $uD) | ((tmp_hi as $uD) << n); |
| 104 | + if (overflow != 0) || (duo < tmp) { |
| 105 | + return ( |
| 106 | + (quo - 1) as $uD, |
| 107 | + duo.wrapping_add(div).wrapping_sub(tmp) |
| 108 | + ) |
| 109 | + } else { |
| 110 | + return ( |
| 111 | + quo as $uD, |
| 112 | + duo - tmp |
| 113 | + ) |
| 114 | + } |
| 115 | + } else { |
| 116 | + // This has been adapted from |
| 117 | + // https://www.codeproject.com/tips/785014/uint-division-modulus which was in turn |
| 118 | + // adapted from Hacker's Delight. This is similar to the two possibility algorithm |
| 119 | + // in that it uses only more significant parts of `duo` and `div` to divide a large |
| 120 | + // integer with a smaller division instruction. |
| 121 | + |
| 122 | + let div_extra = n - div_lz; |
| 123 | + let div_sig_n = (div >> div_extra) as $uX; |
| 124 | + let tmp = unsafe { |
| 125 | + $asymmetric_division(duo >> 1, div_sig_n) |
| 126 | + }; |
| 127 | + |
| 128 | + let mut quo = tmp.0 >> ((n - 1) - div_lz); |
| 129 | + if quo != 0 { |
| 130 | + quo -= 1; |
| 131 | + } |
| 132 | + |
| 133 | + // Note that this is a full `$uD` multiplication being used here |
| 134 | + let mut rem = duo - (quo as $uD).wrapping_mul(div); |
| 135 | + if div <= rem { |
| 136 | + quo += 1; |
| 137 | + rem -= div; |
| 138 | + } |
| 139 | + return (quo as $uD, rem) |
| 140 | + } |
| 141 | + } |
| 142 | + |
| 143 | + /// Computes the quotient and remainder of `duo` divided by `div` and returns them as a |
| 144 | + /// tuple. |
| 145 | + $( |
| 146 | + #[$signed_attr] |
| 147 | + )* |
| 148 | + pub fn $signed_name(duo: $iD, div: $iD) -> ($iD, $iD) { |
| 149 | + match (duo < 0, div < 0) { |
| 150 | + (false, false) => { |
| 151 | + let t = $unsigned_name(duo as $uD, div as $uD); |
| 152 | + (t.0 as $iD, t.1 as $iD) |
| 153 | + }, |
| 154 | + (true, false) => { |
| 155 | + let t = $unsigned_name(duo.wrapping_neg() as $uD, div as $uD); |
| 156 | + ((t.0 as $iD).wrapping_neg(), (t.1 as $iD).wrapping_neg()) |
| 157 | + }, |
| 158 | + (false, true) => { |
| 159 | + let t = $unsigned_name(duo as $uD, div.wrapping_neg() as $uD); |
| 160 | + ((t.0 as $iD).wrapping_neg(), t.1 as $iD) |
| 161 | + }, |
| 162 | + (true, true) => { |
| 163 | + let t = $unsigned_name(duo.wrapping_neg() as $uD, div.wrapping_neg() as $uD); |
| 164 | + (t.0 as $iD, (t.1 as $iD).wrapping_neg()) |
| 165 | + }, |
| 166 | + } |
| 167 | + } |
| 168 | + } |
| 169 | +} |
0 commit comments