Average Error: 10.3 → 0.0
Time: 3.5s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\mathsf{fma}\left(1 - y, \frac{x}{z}, y\right)\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\mathsf{fma}\left(1 - y, \frac{x}{z}, y\right)
double f(double x, double y, double z) {
        double r830807 = x;
        double r830808 = y;
        double r830809 = z;
        double r830810 = r830809 - r830807;
        double r830811 = r830808 * r830810;
        double r830812 = r830807 + r830811;
        double r830813 = r830812 / r830809;
        return r830813;
}
↓
double f(double x, double y, double z) {
        double r830814 = 1.0;
        double r830815 = y;
        double r830816 = r830814 - r830815;
        double r830817 = x;
        double r830818 = z;
        double r830819 = r830817 / r830818;
        double r830820 = fma(r830816, r830819, r830815);
        return r830820;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.3
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.3

    \[\frac{x + y \cdot \left(z - x\right)}{z}\]

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

  (/ (+ x (* y (- z x))) z))