Average Error: 15.1 → 0.0
Time: 1.0s
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 r533156 = x;
        double r533157 = y;
        double r533158 = r533156 + r533157;
        double r533159 = 2.0;
        double r533160 = r533156 * r533159;
        double r533161 = r533160 * r533157;
        double r533162 = r533158 / r533161;
        return r533162;
}
↓
double f(double x, double y) {
        double r533163 = 0.5;
        double r533164 = 1.0;
        double r533165 = y;
        double r533166 = r533164 / r533165;
        double r533167 = x;
        double r533168 = r533164 / r533167;
        double r533169 = r533163 * r533168;
        double r533170 = fma(r533163, r533166, r533169);
        return r533170;
}

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 2020062 +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)))