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| 1 | const std = @import("../std.zig"); |
| 2 | const math = std.math; |
| 3 | const testing = std.testing; |
| 4 | const assert = std.debug.assert; |
| 5 | const Log2Int = math.Log2Int; |
| 6 | |
| 7 | /// Returns the logarithm of `x` for the provided `base`, rounding down to the nearest integer. |
| 8 | /// Asserts that `base > 1` and `x > 0`. |
| 9 | pub fn log_int(comptime T: type, base: T, x: T) Log2Int(T) { |
| 10 | if (@typeInfo(T) != .Int or @typeInfo(T).Int.signedness != .unsigned) |
| 11 | @compileError("log_int requires an unsigned integer, found " ++ @typeName(T)); |
| 12 | |
| 13 | assert(base > 1 and x > 0); |
| 14 | |
| 15 | // Let's denote by [y] the integer part of y. |
| 16 | |
| 17 | // Throughout the iteration the following invariant is preserved: |
| 18 | // power = base ^ exponent |
| 19 | |
| 20 | // Safety and termination. |
| 21 | // |
| 22 | // We never overflow inside the loop because when we enter the loop we have |
| 23 | // power <= [maxInt(T) / base] |
| 24 | // therefore |
| 25 | // power * base <= maxInt(T) |
| 26 | // is a valid multiplication for type `T` and |
| 27 | // exponent + 1 <= log(base, maxInt(T)) <= log2(maxInt(T)) <= maxInt(Log2Int(T)) |
| 28 | // is a valid addition for type `Log2Int(T)`. |
| 29 | // |
| 30 | // This implies also termination because power is strictly increasing, |
| 31 | // hence it must eventually surpass [x / base] < maxInt(T) and we then exit the loop. |
| 32 | |
| 33 | var exponent: Log2Int(T) = 0; |
| 34 | var power: T = 1; |
| 35 | while (power <= x / base) { |
| 36 | power *= base; |
| 37 | exponent += 1; |
| 38 | } |
| 39 | |
| 40 | // If we never entered the loop we must have |
| 41 | // [x / base] < 1 |
| 42 | // hence |
| 43 | // x <= [x / base] * base < base |
| 44 | // thus the result is 0. We can then return exponent, which is still 0. |
| 45 | // |
| 46 | // Otherwise, if we entered the loop at least once, |
| 47 | // when we exit the loop we have that power is exactly divisible by base and |
| 48 | // power / base <= [x / base] < power |
| 49 | // hence |
| 50 | // power <= [x / base] * base <= x < power * base |
| 51 | // This means that |
| 52 | // base^exponent <= x < base^(exponent+1) |
| 53 | // hence the result is exponent. |
| 54 | |
| 55 | return exponent; |
| 56 | } |
| 57 | |
| 58 | test "math.log_int" { |
| 59 | // Test all unsigned integers with 2, 3, ..., 64 bits. |
| 60 | // We cannot test 0 or 1 bits since base must be > 1. |
| 61 | inline for (2..64 + 1) |bits| { |
| 62 | const T = @Type(std.builtin.Type{ |
| 63 | .Int = std.builtin.Type.Int{ .signedness = .unsigned, .bits = @intCast(bits) }, |
| 64 | }); |
| 65 | |
| 66 | // for base = 2, 3, ..., min(maxInt(T),1024) |
| 67 | var base: T = 1; |
| 68 | while (base < math.maxInt(T) and base <= 1024) { |
| 69 | base += 1; |
| 70 | |
| 71 | // test that `log_int(T, base, 1) == 0` |
| 72 | try testing.expectEqual(@as(Log2Int(T), 0), log_int(T, base, 1)); |
| 73 | |
| 74 | // For powers `pow = base^exp > 1` that fit inside T, |
| 75 | // test that `log_int` correctly detects the jump in the logarithm |
| 76 | // from `log(pow-1) == exp-1` to `log(pow) == exp`. |
| 77 | var exp: Log2Int(T) = 0; |
| 78 | var pow: T = 1; |
| 79 | while (pow <= math.maxInt(T) / base) { |
| 80 | exp += 1; |
| 81 | pow *= base; |
| 82 | |
| 83 | try testing.expectEqual(exp - 1, log_int(T, base, pow - 1)); |
| 84 | try testing.expectEqual(exp, log_int(T, base, pow)); |
| 85 | } |
| 86 | } |
| 87 | } |
| 88 | } |
| 89 | |
| 90 | test "math.log_int vs math.log2" { |
| 91 | const types = [_]type{ u2, u3, u4, u8, u16 }; |
| 92 | inline for (types) |T| { |
| 93 | var n: T = 0; |
| 94 | while (n < math.maxInt(T)) { |
| 95 | n += 1; |
| 96 | const special = math.log2_int(T, n); |
| 97 | const general = log_int(T, 2, n); |
| 98 | try testing.expectEqual(special, general); |
| 99 | } |
| 100 | } |
| 101 | } |
| 102 | |
| 103 | test "math.log_int vs math.log10" { |
| 104 | const types = [_]type{ u4, u5, u6, u8, u16 }; |
| 105 | inline for (types) |T| { |
| 106 | var n: T = 0; |
| 107 | while (n < math.maxInt(T)) { |
| 108 | n += 1; |
| 109 | const special = math.log10_int(n); |
| 110 | const general = log_int(T, 10, n); |
| 111 | try testing.expectEqual(special, general); |
| 112 | } |
| 113 | } |
| 114 | } |
Federico Stra <stra.federico@gmail.com> 2023-09-14 21:33:56+02:00
GitHub <noreply@github.com> 2023-09-14 19:33:56+00:00
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