Average Error: 15.3 → 0.0
Time: 1.6s
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 r516956 = x;
        double r516957 = y;
        double r516958 = r516956 + r516957;
        double r516959 = 2.0;
        double r516960 = r516956 * r516959;
        double r516961 = r516960 * r516957;
        double r516962 = r516958 / r516961;
        return r516962;
}
↓
double f(double x, double y) {
        double r516963 = 0.5;
        double r516964 = 1.0;
        double r516965 = y;
        double r516966 = r516964 / r516965;
        double r516967 = x;
        double r516968 = r516964 / r516967;
        double r516969 = r516963 * r516968;
        double r516970 = fma(r516963, r516966, r516969);
        return r516970;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.3
Target0.0
Herbie0.0
\[\frac{0.5}{x} + \frac{0.5}{y}\]

Derivation

  1. Initial program 15.3

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

  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 2020024 +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)))