Average Error: 0.1 → 0.0
Time: 740.0ms
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 r738080 = x;
        double r738081 = y;
        double r738082 = r738080 + r738081;
        double r738083 = r738081 + r738081;
        double r738084 = r738082 / r738083;
        return r738084;
}
↓
double f(double x, double y) {
        double r738085 = 0.5;
        double r738086 = x;
        double r738087 = y;
        double r738088 = r738086 / r738087;
        double r738089 = fma(r738085, r738088, r738085);
        return r738089;
}

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 2020001 +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)))