Average Error: 15.1 → 0.0
Time: 1.5s
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 r593338 = x;
        double r593339 = y;
        double r593340 = r593338 + r593339;
        double r593341 = 2.0;
        double r593342 = r593338 * r593341;
        double r593343 = r593342 * r593339;
        double r593344 = r593340 / r593343;
        return r593344;
}
↓
double f(double x, double y) {
        double r593345 = 0.5;
        double r593346 = 1.0;
        double r593347 = y;
        double r593348 = r593346 / r593347;
        double r593349 = x;
        double r593350 = r593346 / r593349;
        double r593351 = r593345 * r593350;
        double r593352 = fma(r593345, r593348, r593351);
        return r593352;
}

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 2020035 +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)))