Average Error: 0.1 → 0.0
Time: 8.3s
Precision: 64
\[\frac{x + y}{y + y}\]
↓
\[\mathsf{fma}\left(\frac{1}{2}, \frac{x}{y}, \frac{1}{2}\right)\]
\frac{x + y}{y + y}
↓
\mathsf{fma}\left(\frac{1}{2}, \frac{x}{y}, \frac{1}{2}\right)
double f(double x, double y) {
        double r59894 = x;
        double r59895 = y;
        double r59896 = r59894 + r59895;
        double r59897 = r59895 + r59895;
        double r59898 = r59896 / r59897;
        return r59898;
}
↓
double f(double x, double y) {
        double r59899 = 0.5;
        double r59900 = x;
        double r59901 = y;
        double r59902 = r59900 / r59901;
        double r59903 = fma(r59899, r59902, r59899);
        return r59903;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.1
Target0.0
Herbie0.0
\[0.5 \cdot \frac{x}{y} + 0.5\]

Derivation

  1. Initial program 0.1

    \[\frac{x + y}{y + y}\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x}{y} + \frac{1}{2}}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{x}{y}, \frac{1}{2}\right)}\]

  4. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{x}{y}, \frac{1}{2}\right)\]

Reproduce

herbie shell --seed 2019315 +o rules:numerics
(FPCore (x y)
  :name "Data.Random.Distribution.T:$ccdf from random-fu-0.2.6.2"
  :precision binary64

  :herbie-target
  (+ (* 0.5 (/ x y)) 0.5)

  (/ (+ x y) (+ y y)))