Average Error: 15.5 → 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 r520981 = x;
        double r520982 = y;
        double r520983 = r520981 + r520982;
        double r520984 = 2.0;
        double r520985 = r520981 * r520984;
        double r520986 = r520985 * r520982;
        double r520987 = r520983 / r520986;
        return r520987;
}
↓
double f(double x, double y) {
        double r520988 = 0.5;
        double r520989 = 1.0;
        double r520990 = y;
        double r520991 = r520989 / r520990;
        double r520992 = x;
        double r520993 = r520989 / r520992;
        double r520994 = r520988 * r520993;
        double r520995 = fma(r520988, r520991, r520994);
        return r520995;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.5
Target0.0
Herbie0.0
\[\frac{0.5}{x} + \frac{0.5}{y}\]

Derivation

  1. Initial program 15.5

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

  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 2019354 +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)))