Average Error: 0.1 → 0.0
Time: 1.9s
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 r13097613 = x;
        double r13097614 = y;
        double r13097615 = r13097613 + r13097614;
        double r13097616 = r13097614 + r13097614;
        double r13097617 = r13097615 / r13097616;
        return r13097617;
}
↓
double f(double x, double y) {
        double r13097618 = 0.5;
        double r13097619 = x;
        double r13097620 = y;
        double r13097621 = r13097619 / r13097620;
        double r13097622 = fma(r13097618, r13097621, r13097618);
        return r13097622;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.1
Target0.0
Herbie0.0
\[\frac{1}{2} \cdot \frac{x}{y} + \frac{1}{2}\]

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 2019156 +o rules:numerics
(FPCore (x y)
  :name "Data.Random.Distribution.T:$ccdf from random-fu-0.2.6.2"

  :herbie-target
  (+ (* 1/2 (/ x y)) 1/2)

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