Average Error: 15.4 → 0.0
Time: 719.0ms
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 r513562 = x;
        double r513563 = y;
        double r513564 = r513562 + r513563;
        double r513565 = 2.0;
        double r513566 = r513562 * r513565;
        double r513567 = r513566 * r513563;
        double r513568 = r513564 / r513567;
        return r513568;
}
↓
double f(double x, double y) {
        double r513569 = 0.5;
        double r513570 = 1.0;
        double r513571 = y;
        double r513572 = r513570 / r513571;
        double r513573 = x;
        double r513574 = r513570 / r513573;
        double r513575 = r513569 * r513574;
        double r513576 = fma(r513569, r513572, r513575);
        return r513576;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.4
Target0.0
Herbie0.0
\[\frac{0.5}{x} + \frac{0.5}{y}\]

Derivation

  1. Initial program 15.4

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

  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 2020065 +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)))