Average Error: 15.2 → 0.0
Time: 3.3s
Precision: 64
\[\frac{x + y}{\left(x \cdot 2\right) \cdot y}\]
↓
\[\mathsf{fma}\left(0.5, \frac{1}{y}, 0.5 \cdot \frac{1}{x}\right)\]
\frac{x + y}{\left(x \cdot 2\right) \cdot y}
↓
\mathsf{fma}\left(0.5, \frac{1}{y}, 0.5 \cdot \frac{1}{x}\right)
double f(double x, double y) {
        double r807947 = x;
        double r807948 = y;
        double r807949 = r807947 + r807948;
        double r807950 = 2.0;
        double r807951 = r807947 * r807950;
        double r807952 = r807951 * r807948;
        double r807953 = r807949 / r807952;
        return r807953;
}
↓
double f(double x, double y) {
        double r807954 = 0.5;
        double r807955 = 1.0;
        double r807956 = y;
        double r807957 = r807955 / r807956;
        double r807958 = x;
        double r807959 = r807955 / r807958;
        double r807960 = r807954 * r807959;
        double r807961 = fma(r807954, r807957, r807960);
        return r807961;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.2
Target0.0
Herbie0.0
\[\frac{0.5}{x} + \frac{0.5}{y}\]

Derivation

  1. Initial program 15.2

    \[\frac{x + y}{\left(x \cdot 2\right) \cdot y}\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{0.5 \cdot \frac{1}{y} + 0.5 \cdot \frac{1}{x}}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{1}{y}, 0.5 \cdot \frac{1}{x}\right)}\]

  4. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(0.5, \frac{1}{y}, 0.5 \cdot \frac{1}{x}\right)\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, C"
  :precision binary64

  :herbie-target
  (+ (/ 0.5 x) (/ 0.5 y))

  (/ (+ x y) (* (* x 2) y)))