Average Error: 15.1 → 0.0
Time: 1.0s
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 r512408 = x;
        double r512409 = y;
        double r512410 = r512408 + r512409;
        double r512411 = 2.0;
        double r512412 = r512408 * r512411;
        double r512413 = r512412 * r512409;
        double r512414 = r512410 / r512413;
        return r512414;
}
↓
double f(double x, double y) {
        double r512415 = 0.5;
        double r512416 = 1.0;
        double r512417 = y;
        double r512418 = r512416 / r512417;
        double r512419 = x;
        double r512420 = r512416 / r512419;
        double r512421 = r512415 * r512420;
        double r512422 = fma(r512415, r512418, r512421);
        return r512422;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.1
Target0.0
Herbie0.0
\[\frac{0.5}{x} + \frac{0.5}{y}\]

Derivation

  1. Initial program 15.1

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

  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 2020062 +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)))