Average Error: 15.6 → 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 r517834 = x;
        double r517835 = y;
        double r517836 = r517834 + r517835;
        double r517837 = 2.0;
        double r517838 = r517834 * r517837;
        double r517839 = r517838 * r517835;
        double r517840 = r517836 / r517839;
        return r517840;
}
↓
double f(double x, double y) {
        double r517841 = 0.5;
        double r517842 = 1.0;
        double r517843 = y;
        double r517844 = r517842 / r517843;
        double r517845 = x;
        double r517846 = r517842 / r517845;
        double r517847 = r517841 * r517846;
        double r517848 = fma(r517841, r517844, r517847);
        return r517848;
}

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 2020033 +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)))