Average Error: 15.2 → 0.0
Time: 2.0s
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 r521291 = x;
        double r521292 = y;
        double r521293 = r521291 + r521292;
        double r521294 = 2.0;
        double r521295 = r521291 * r521294;
        double r521296 = r521295 * r521292;
        double r521297 = r521293 / r521296;
        return r521297;
}
↓
double f(double x, double y) {
        double r521298 = 0.5;
        double r521299 = 1.0;
        double r521300 = y;
        double r521301 = r521299 / r521300;
        double r521302 = x;
        double r521303 = r521299 / r521302;
        double r521304 = r521298 * r521303;
        double r521305 = fma(r521298, r521301, r521304);
        return r521305;
}

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 2020001 +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)))