Average Error: 15.4 → 0.0
Time: 2.5s
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 r490454 = x;
        double r490455 = y;
        double r490456 = r490454 + r490455;
        double r490457 = 2.0;
        double r490458 = r490454 * r490457;
        double r490459 = r490458 * r490455;
        double r490460 = r490456 / r490459;
        return r490460;
}
↓
double f(double x, double y) {
        double r490461 = 0.5;
        double r490462 = 1.0;
        double r490463 = y;
        double r490464 = r490462 / r490463;
        double r490465 = x;
        double r490466 = r490462 / r490465;
        double r490467 = r490461 * r490466;
        double r490468 = fma(r490461, r490464, r490467);
        return r490468;
}

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 2020064 +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)))