Add std.math.egcd (Extended GCD). Micro-optimize GCD.

lib/std/math.zig

@@ -238,6 +238,7 @@ pub const sinh = @import("math/sinh.zig").sinh;
 pub const cosh = @import("math/cosh.zig").cosh;
 pub const tanh = @import("math/tanh.zig").tanh;
 pub const gcd = @import("math/gcd.zig").gcd;
+pub const egcd = @import("math/egcd.zig").egcd;
 pub const lcm = @import("math/lcm.zig").lcm;
 pub const gamma = @import("math/gamma.zig").gamma;
 pub const lgamma = @import("math/gamma.zig").lgamma;

lib/std/math/egcd.zig

@@ -0,0 +1,240 @@
+//! Extended Greatest Common Divisor (https://mathworld.wolfram.com/ExtendedGreatestCommonDivisor.html)
+const std = @import("../std.zig");
+
+/// Result type of `egcd`.
+pub fn Result(S: anytype) type {
+    const N = switch (S) {
+        comptime_int => comptime_int,
+        else => |T| std.meta.Int(.unsigned, @bitSizeOf(T)),
+    };
+
+    return struct {
+        gcd: N,
+        bezout_coeff_1: S,
+        bezout_coeff_2: S,
+    };
+}
+
+/// Returns the Extended Greatest Common Divisor (EGCD) of two signed integers (`a` and `b`) which are not both zero.
+pub fn egcd(a: anytype, b: anytype) Result(@TypeOf(a, b)) {
+    const S = switch (@TypeOf(a, b)) {
+        comptime_int => b: {
+            const n = @max(@abs(a), @abs(b));
+            break :b std.math.IntFittingRange(-n, n);
+        },
+        else => |T| T,
+    };
+
+    if (@typeInfo(S) != .int or @typeInfo(S).int.signedness != .signed)
+        @compileError("`a` and `b` must be signed integers");
+
+    std.debug.assert(a != 0 or b != 0);
+
+    if (a == 0) return .{ .gcd = @abs(b), .bezout_coeff_1 = 0, .bezout_coeff_2 = std.math.sign(b) };
+    if (b == 0) return .{ .gcd = @abs(a), .bezout_coeff_1 = std.math.sign(a), .bezout_coeff_2 = 0 };
+
+    const other: S, const odd: S, const shift, const switch_coeff = b: {
+        const xz = @ctz(@as(S, a));
+        const yz = @ctz(@as(S, b));
+        break :b if (xz < yz) .{ b, a, xz, true } else .{ a, b, yz, false };
+    };
+    const toinv = @shrExact(other, @intCast(shift));
+    const ctrl = @shrExact(odd, @intCast(shift)); // Invariant: |s|, |t|, |ctrl| < |MIN_OF(S)|
+    const abs_ctrl = @abs(ctrl);
+    const half_ctrl: S = @intCast(1 + abs_ctrl >> 1);
+
+    var s: S = std.math.sign(toinv);
+    var t: S = 0;
+
+    var x = @abs(toinv);
+    var y = abs_ctrl;
+
+    {
+        const xz = @ctz(x);
+        x = @shrExact(x, @intCast(xz));
+        for (0..xz) |_| {
+            const half_s = s >> 1;
+            if (s & 1 == 0)
+                s = half_s
+            else
+                s = half_s + half_ctrl;
+        }
+    }
+
+    var y_minus_x = y -% x;
+    while (y_minus_x != 0) : (y_minus_x = y -% x) {
+        const t_minus_s = t - s;
+        const copy_x = x;
+        const copy_s = s;
+        const xz = @ctz(y_minus_x);
+
+        s -= t;
+        const carry = x < y;
+        x -%= y;
+        if (carry) {
+            x = y_minus_x;
+            y = copy_x;
+            s = t_minus_s;
+            t = copy_s;
+        }
+        x = @shrExact(x, @intCast(xz));
+        for (0..xz) |_| {
+            const half_s = s >> 1;
+            if (s & 1 == 0)
+                s = half_s
+            else
+                s = half_s + half_ctrl;
+        }
+
+        if (s < 0) s = @intCast(abs_ctrl - @abs(s));
+    }
+
+    // Using integer widening is only a temporary solution.
+    const W = std.meta.Int(.signed, @bitSizeOf(S) * 2);
+    t = @intCast(@divExact(y - @as(W, s) * toinv, ctrl));
+    const final_s, const final_t = if (switch_coeff) .{ t, s } else .{ s, t };
+    return .{
+        .gcd = @shlExact(y, @intCast(shift)),
+        .bezout_coeff_1 = final_s,
+        .bezout_coeff_2 = final_t,
+    };
+}
+
+test {
+    {
+        const a: i2 = 0;
+        const b: i2 = 1;
+        const r = egcd(a, b);
+        const g = r.gcd;
+        const s: i2 = r.bezout_coeff_1;
+        const t: i2 = r.bezout_coeff_2;
+        try std.testing.expect(s * a + t * b == g);
+    }
+    {
+        const a: i8 = -128;
+        const b: i8 = 127;
+        const r = egcd(a, b);
+        const g = r.gcd;
+        const s: i16 = r.bezout_coeff_1;
+        const t: i16 = r.bezout_coeff_2;
+        try std.testing.expect(s * a + t * b == g);
+    }
+    {
+        const a: i16 = -32768;
+        const b: i16 = -32768;
+        const r = egcd(a, b);
+        const g = r.gcd;
+        const s: i32 = r.bezout_coeff_1;
+        const t: i32 = r.bezout_coeff_2;
+        try std.testing.expect(s * a + t * b == g);
+    }
+    {
+        const a: i32 = 128;
+        const b: i32 = 112;
+        const r = egcd(a, b);
+        const g = r.gcd;
+        const s: i64 = r.bezout_coeff_1;
+        const t: i64 = r.bezout_coeff_2;
+        try std.testing.expect(s * a + t * b == g);
+    }
+    {
+        const a: i32 = 4 * 89;
+        const b: i32 = 2 * 17;
+        const r = egcd(a, b);
+        const g = r.gcd;
+        const s: i64 = r.bezout_coeff_1;
+        const t: i64 = r.bezout_coeff_2;
+        try std.testing.expect(s * a + t * b == g);
+    }
+    {
+        const a: i8 = 127;
+        const b: i8 = 126;
+        const r = egcd(a, b);
+        const g = r.gcd;
+        const s: i16 = r.bezout_coeff_1;
+        const t: i16 = r.bezout_coeff_2;
+        try std.testing.expect(s * a + t * b == g);
+    }
+    {
+        const a: i4 = -8;
+        const b: i4 = 1;
+        const r = egcd(a, b);
+        const g = r.gcd;
+        const s = r.bezout_coeff_1;
+        const t = r.bezout_coeff_2;
+        try std.testing.expect(s * a + t * b == g);
+    }
+    {
+        const a: i4 = -8;
+        const b: i4 = 5;
+        const r = egcd(a, b);
+        const g = r.gcd;
+        const s: i8 = r.bezout_coeff_1;
+        const t: i8 = r.bezout_coeff_2;
+        try std.testing.expect(s * a + t * b == g);
+    }
+    {
+        const a: i32 = 0;
+        const b: i32 = 5;
+        const r = egcd(a, b);
+        const g = r.gcd;
+        const s = r.bezout_coeff_1;
+        const t = r.bezout_coeff_2;
+        try std.testing.expect(s * a + t * b == g);
+    }
+    {
+        const a: i32 = 5;
+        const b: i32 = 0;
+        const r = egcd(a, b);
+        const g = r.gcd;
+        const s = r.bezout_coeff_1;
+        const t = r.bezout_coeff_2;
+        try std.testing.expect(s * a + t * b == g);
+    }
+
+    {
+        const a: i32 = 21;
+        const b: i32 = 15;
+        const r = egcd(a, b);
+        const g = r.gcd;
+        const s = r.bezout_coeff_1;
+        const t = r.bezout_coeff_2;
+        try std.testing.expect(s * a + t * b == g);
+    }
+    {
+        const a: i32 = -21;
+        const b: i32 = 15;
+        const r = egcd(a, b);
+        const g = r.gcd;
+        const s = r.bezout_coeff_1;
+        const t = r.bezout_coeff_2;
+        try std.testing.expect(s * a + t * b == g);
+    }
+    {
+        const a = -21;
+        const b = 15;
+        const r = egcd(a, b);
+        const g = r.gcd;
+        const s = r.bezout_coeff_1;
+        const t = r.bezout_coeff_2;
+        try std.testing.expect(s * a + t * b == g);
+    }
+    {
+        const a = 927372692193078999176;
+        const b = 573147844013817084101;
+        const r = egcd(a, b);
+        const g = r.gcd;
+        const s = r.bezout_coeff_1;
+        const t = r.bezout_coeff_2;
+        try std.testing.expect(s * a + t * b == g);
+    }
+    {
+        const a = 453973694165307953197296969697410619233826;
+        const b = 280571172992510140037611932413038677189525;
+        const r = egcd(a, b);
+        const g = r.gcd;
+        const s = r.bezout_coeff_1;
+        const t = r.bezout_coeff_2;
+        try std.testing.expect(s * a + t * b == g);
+    }
+}

lib/std/math/gcd.zig

@@ -1,20 +1,22 @@
-//! Greatest common divisor (https://mathworld.wolfram.com/GreatestCommonDivisor.html)
-const std = @import("std");
+//! Greatest Common Divisor (https://mathworld.wolfram.com/GreatestCommonDivisor.html)
+const std = @import("../std.zig");
 
-/// Returns the greatest common divisor (GCD) of two unsigned integers (`a` and `b`) which are not both zero.
+/// Returns the Greatest Common Divisor (GCD) of two unsigned integers (`a` and `b`) which are not both zero.
 /// For example, the GCD of `8` and `12` is `4`, that is, `gcd(8, 12) == 4`.
 pub fn gcd(a: anytype, b: anytype) @TypeOf(a, b) {
     const N = switch (@TypeOf(a, b)) {
         // convert comptime_int to some sized int type for @ctz
         comptime_int => std.math.IntFittingRange(@min(a, b), @max(a, b)),
         else => |T| T,
     };
+
     if (@typeInfo(N) != .int or @typeInfo(N).int.signedness != .unsigned) {
-        @compileError("`a` and `b` must be usigned integers");
+        @compileError("`a` and `b` must be unsigned integers");
     }
 
     // using an optimised form of Stein's algorithm:
     // https://en.wikipedia.org/wiki/Binary_GCD_algorithm
+
     std.debug.assert(a != 0 or b != 0);
 
     if (a == 0) return b;
@@ -26,25 +28,27 @@ pub fn gcd(a: anytype, b: anytype) @TypeOf(a, b) {
     const xz = @ctz(x);
     const yz = @ctz(y);
     const shift = @min(xz, yz);
-    x >>= @intCast(xz);
-    y >>= @intCast(yz);
-
-    var diff = y -% x;
-    while (diff != 0) : (diff = y -% x) {
-        // ctz is invariant under negation, we
-        // put it here to ease data dependencies,
-        // makes the CPU happy.
-        const zeros = @ctz(diff);
-        if (x > y) diff = -%diff;
-        y = @min(x, y);
-        x = diff >> @intCast(zeros);
+    x = @shrExact(x, @intCast(xz));
+    y = @shrExact(y, @intCast(yz));
+
+    var y_minus_x = y -% x;
+    while (y_minus_x != 0) : (y_minus_x = y -% x) {
+        const copy_x = x;
+        const zeros = @ctz(y_minus_x);
+        const carry = x < y;
+        x -%= y;
+        if (carry) {
+            x = y_minus_x;
+            y = copy_x;
+        }
+        x = @shrExact(x, @intCast(zeros));
     }
-    return y << @intCast(shift);
+
+    return @shlExact(y, @intCast(shift));
 }
 
 test gcd {
     const expectEqual = std.testing.expectEqual;
-
     try expectEqual(gcd(0, 5), 5);
     try expectEqual(gcd(5, 0), 5);
     try expectEqual(gcd(8, 12), 4);