Average Error: 15.2 → 0.0
Time: 1.7s
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 r513808 = x;
        double r513809 = y;
        double r513810 = r513808 + r513809;
        double r513811 = 2.0;
        double r513812 = r513808 * r513811;
        double r513813 = r513812 * r513809;
        double r513814 = r513810 / r513813;
        return r513814;
}
↓
double f(double x, double y) {
        double r513815 = 0.5;
        double r513816 = 1.0;
        double r513817 = y;
        double r513818 = r513816 / r513817;
        double r513819 = x;
        double r513820 = r513816 / r513819;
        double r513821 = r513815 * r513820;
        double r513822 = fma(r513815, r513818, r513821);
        return r513822;
}

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 2020089 +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)))