Average Error: 0.1 → 0.0
Time: 2.9s
Precision: 64
\[\frac{x + y}{y + y}\]
↓
\[\mathsf{fma}\left(\frac{x}{y}, \frac{1}{2}, \frac{1}{2}\right)\]
\frac{x + y}{y + y}
↓
\mathsf{fma}\left(\frac{x}{y}, \frac{1}{2}, \frac{1}{2}\right)
double f(double x, double y) {
        double r21697673 = x;
        double r21697674 = y;
        double r21697675 = r21697673 + r21697674;
        double r21697676 = r21697674 + r21697674;
        double r21697677 = r21697675 / r21697676;
        return r21697677;
}
↓
double f(double x, double y) {
        double r21697678 = x;
        double r21697679 = y;
        double r21697680 = r21697678 / r21697679;
        double r21697681 = 0.5;
        double r21697682 = fma(r21697680, r21697681, r21697681);
        return r21697682;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 +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)))