Average Error: 15.5 → 0.0
Time: 1.3s
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 r554207 = x;
        double r554208 = y;
        double r554209 = r554207 + r554208;
        double r554210 = 2.0;
        double r554211 = r554207 * r554210;
        double r554212 = r554211 * r554208;
        double r554213 = r554209 / r554212;
        return r554213;
}
↓
double f(double x, double y) {
        double r554214 = 0.5;
        double r554215 = 1.0;
        double r554216 = y;
        double r554217 = r554215 / r554216;
        double r554218 = x;
        double r554219 = r554215 / r554218;
        double r554220 = r554214 * r554219;
        double r554221 = fma(r554214, r554217, r554220);
        return r554221;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.5
Target0.0
Herbie0.0
\[\frac{0.5}{x} + \frac{0.5}{y}\]

Derivation

  1. Initial program 15.5

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

  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 2020020 +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)))