Average Error: 0.1 → 0.0
Time: 2.7s
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 r32947997 = x;
        double r32947998 = y;
        double r32947999 = r32947997 + r32947998;
        double r32948000 = r32947998 + r32947998;
        double r32948001 = r32947999 / r32948000;
        return r32948001;
}
↓
double f(double x, double y) {
        double r32948002 = 0.5;
        double r32948003 = x;
        double r32948004 = y;
        double r32948005 = r32948003 / r32948004;
        double r32948006 = fma(r32948002, r32948005, r32948002);
        return r32948006;
}

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 2019164 +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)))