Average Error: 15.7 → 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 r459229 = x;
        double r459230 = y;
        double r459231 = r459229 + r459230;
        double r459232 = 2.0;
        double r459233 = r459229 * r459232;
        double r459234 = r459233 * r459230;
        double r459235 = r459231 / r459234;
        return r459235;
}
↓
double f(double x, double y) {
        double r459236 = 0.5;
        double r459237 = 1.0;
        double r459238 = y;
        double r459239 = r459237 / r459238;
        double r459240 = x;
        double r459241 = r459237 / r459240;
        double r459242 = r459236 * r459241;
        double r459243 = fma(r459236, r459239, r459242);
        return r459243;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.7
Target0.0
Herbie0.0
\[\frac{0.5}{x} + \frac{0.5}{y}\]

Derivation

  1. Initial program 15.7

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

  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 2020036 +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)))