Average Error: 15.1 → 0.0
Time: 1.9s
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 r652984 = x;
        double r652985 = y;
        double r652986 = r652984 + r652985;
        double r652987 = 2.0;
        double r652988 = r652984 * r652987;
        double r652989 = r652988 * r652985;
        double r652990 = r652986 / r652989;
        return r652990;
}
↓
double f(double x, double y) {
        double r652991 = 0.5;
        double r652992 = 1.0;
        double r652993 = y;
        double r652994 = r652992 / r652993;
        double r652995 = x;
        double r652996 = r652992 / r652995;
        double r652997 = r652991 * r652996;
        double r652998 = fma(r652991, r652994, r652997);
        return r652998;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.1
Target0.0
Herbie0.0
\[\frac{0.5}{x} + \frac{0.5}{y}\]

Derivation

  1. Initial program 15.1

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

  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 2020034 +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)))