Average Error: 15.7 → 0.0
Time: 993.0ms
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 r603043 = x;
        double r603044 = y;
        double r603045 = r603043 + r603044;
        double r603046 = 2.0;
        double r603047 = r603043 * r603046;
        double r603048 = r603047 * r603044;
        double r603049 = r603045 / r603048;
        return r603049;
}
↓
double f(double x, double y) {
        double r603050 = 0.5;
        double r603051 = 1.0;
        double r603052 = y;
        double r603053 = r603051 / r603052;
        double r603054 = x;
        double r603055 = r603051 / r603054;
        double r603056 = r603050 * r603055;
        double r603057 = fma(r603050, r603053, r603056);
        return r603057;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.7
Target0.0
Herbie0.0
\[\frac{0.5}{x} + \frac{0.5}{y}\]

Derivation

  1. Initial program 15.7

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

  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 2020083 +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)))