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 r446108 = x;
        double r446109 = y;
        double r446110 = r446108 + r446109;
        double r446111 = 2.0;
        double r446112 = r446108 * r446111;
        double r446113 = r446112 * r446109;
        double r446114 = r446110 / r446113;
        return r446114;
}
↓
double f(double x, double y) {
        double r446115 = 0.5;
        double r446116 = 1.0;
        double r446117 = y;
        double r446118 = r446116 / r446117;
        double r446119 = x;
        double r446120 = r446116 / r446119;
        double r446121 = r446115 * r446120;
        double r446122 = fma(r446115, r446118, r446121);
        return r446122;
}

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)))