Average Error: 15.4 → 0.0
Time: 1.8s
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 r462088 = x;
        double r462089 = y;
        double r462090 = r462088 + r462089;
        double r462091 = 2.0;
        double r462092 = r462088 * r462091;
        double r462093 = r462092 * r462089;
        double r462094 = r462090 / r462093;
        return r462094;
}
↓
double f(double x, double y) {
        double r462095 = 0.5;
        double r462096 = 1.0;
        double r462097 = y;
        double r462098 = r462096 / r462097;
        double r462099 = x;
        double r462100 = r462096 / r462099;
        double r462101 = r462095 * r462100;
        double r462102 = fma(r462095, r462098, r462101);
        return r462102;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.4
Target0.0
Herbie0.0
\[\frac{0.5}{x} + \frac{0.5}{y}\]

Derivation

  1. Initial program 15.4

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

  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 2019353 +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)))