Average Error: 15.3 → 0.0
Time: 1.4s
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 r445319 = x;
        double r445320 = y;
        double r445321 = r445319 + r445320;
        double r445322 = 2.0;
        double r445323 = r445319 * r445322;
        double r445324 = r445323 * r445320;
        double r445325 = r445321 / r445324;
        return r445325;
}
↓
double f(double x, double y) {
        double r445326 = 0.5;
        double r445327 = 1.0;
        double r445328 = y;
        double r445329 = r445327 / r445328;
        double r445330 = x;
        double r445331 = r445327 / r445330;
        double r445332 = r445326 * r445331;
        double r445333 = fma(r445326, r445329, r445332);
        return r445333;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.3
Target0.0
Herbie0.0
\[\frac{0.5}{x} + \frac{0.5}{y}\]

Derivation

  1. Initial program 15.3

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

  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 2020021 +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)))