Average Error: 15.2 → 0.0
Time: 1.1s
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 r493475 = x;
        double r493476 = y;
        double r493477 = r493475 + r493476;
        double r493478 = 2.0;
        double r493479 = r493475 * r493478;
        double r493480 = r493479 * r493476;
        double r493481 = r493477 / r493480;
        return r493481;
}
↓
double f(double x, double y) {
        double r493482 = 0.5;
        double r493483 = 1.0;
        double r493484 = y;
        double r493485 = r493483 / r493484;
        double r493486 = x;
        double r493487 = r493483 / r493486;
        double r493488 = r493482 * r493487;
        double r493489 = fma(r493482, r493485, r493488);
        return r493489;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.2
Target0.0
Herbie0.0
\[\frac{0.5}{x} + \frac{0.5}{y}\]

Derivation

  1. Initial program 15.2

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

  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 2020018 +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)))