[ANN] XRationals.jl

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From the README.md:

Exact rational arithmetic with IEEE-like special values (NaN, Inf, -Inf), overflow-safe saturation, and lazy normalization for impressive throughput.

Types

Type Alias Backing Overflow Normalization
XRational32 Qx32 Int32 Saturates to Inf/NaN Lazy (Int64 intermediate)

| XRational64 | Qx64 | Int64 | Saturates to Inf/NaN | Lazy (Int128 intermediate) |

Features

  • Exact rational arithmetic with no floating-point rounding
  • IEEE-like NaN, Inf, -Inf encoded in the same struct (0//0, 1//0, -1//0)
  • Overflow saturates to Inf/NaN instead of crashing (Qx types)
  • Lazy GCD normalization: deferred until display, hashing, or conversion
  • 3-13x faster than Rational{Int} for chained arithmetic
  • Fused multiply-add (fma) with exact intermediate computation
  • Cross-width conversion (Qx64 to Qx32) via best rational approximation
  • Zero heap allocation: all arithmetic uses fixed-width integers (Int32/Int64/Int128/Int256)
  • typemin rejection prevents silent negation overflow

Examples

See Use.jl for examples.

Benchmarks

All operations are zero-allocation unless noted. Times are minimum nanoseconds.

Note: the reason that fma for Qx32 and Qx64 takes longer is that it returns the nearest rational to the true value. Rational{Int32} and Rational{Int64} just perform a muladd. This is the reason for slowdown in that function. You can use muladd with Qx32 and Qx64 for improved performance.

32-bit

Operation Rational{Int32} Qx32
construct(7,3) 1 ns 1 ns
a + b 13 ns 2 ns
a - b 13 ns 2 ns
a * b 8 ns 2 ns
a / b 7 ns 2 ns
-a 1 ns 1 ns
a < b 1 ns 2 ns
a == b 1 ns 1 ns
abs(-a) 1 ns 2 ns
inv(a) 1 ns 2 ns
a ^ 3 18 ns 5 ns
a+b+c+d 66 ns 4 ns
ab-cd 37 ns 4 ns
muladd(a,b,a) 23 ns 3 ns
fma(a,b,a) 23 ns 215 ns

64-bit

Operation Rational{Int64} Qx64
construct(7,3) 1 ns 1 ns
a + b 14 ns 3 ns
a - b 15 ns 3 ns
a * b 8 ns 2 ns
a / b 8 ns 3 ns
-a 1 ns 1 ns
a < b 1 ns 2 ns
a == b 1 ns 1 ns
abs(-a) 1 ns 2 ns
inv(a) 1 ns 2 ns
a ^ 3 21 ns 7 ns
a+b+c+d 72 ns 8 ns
ab-cd 41 ns 5 ns
muladd(a,b,a) 27 ns 6 ns
fma(a,b,a) 27 ns 859 ns

Qx64 vs Rational{Int64} Speedup

Qx64 delays GCD normalization until display, hashing, or conversion, giving IEEE-like Inf/NaN semantics and faster arithmetic than stdlib.

Operation Rational{Int64} Qx64 Speedup
construct(7,3) 1 ns 1 ns ~1x
a + b 14 ns 3 ns 5.3x
a - b 15 ns 3 ns 5.5x
a * b 8 ns 2 ns 3.7x
a / b 8 ns 3 ns 3.1x
-a 1 ns 1 ns 1.1x
a < b 1 ns 2 ns 0.84x
a == b 1 ns 1 ns ~1x
abs(-a) 1 ns 1 ns 0.89x
inv(a) 2 ns 2 ns 0.91x
a ^ 3 21 ns 7 ns 3.1x
a+b+c+d 72 ns 8 ns 9.4x
ab-cd 41 ns 5 ns 8.3x
muladd(a,b,a) 27 ns 6 ns 4.5x
fma(a,b,a) 27 ns 859 ns 0.03x

(benchmarking code is available in the repository)

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