Average Error: 15.6 → 0.0
Time: 1.4s
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 r686490 = x;
        double r686491 = y;
        double r686492 = r686490 + r686491;
        double r686493 = 2.0;
        double r686494 = r686490 * r686493;
        double r686495 = r686494 * r686491;
        double r686496 = r686492 / r686495;
        return r686496;
}
↓
double f(double x, double y) {
        double r686497 = 0.5;
        double r686498 = 1.0;
        double r686499 = y;
        double r686500 = r686498 / r686499;
        double r686501 = x;
        double r686502 = r686498 / r686501;
        double r686503 = r686497 * r686502;
        double r686504 = fma(r686497, r686500, r686503);
        return r686504;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.6
Target0.0
Herbie0.0
\[\frac{0.5}{x} + \frac{0.5}{y}\]

Derivation

  1. Initial program 15.6

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

  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 2020100 +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)))