Average Error: 10.2 → 0.0
Time: 4.4s
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 r4709 = x;
        double r4710 = y;
        double r4711 = z;
        double r4712 = r4711 - r4709;
        double r4713 = r4710 * r4712;
        double r4714 = r4709 + r4713;
        double r4715 = r4714 / r4711;
        return r4715;
}
↓
double f(double x, double y, double z) {
        double r4716 = 1.0;
        double r4717 = y;
        double r4718 = r4716 - r4717;
        double r4719 = x;
        double r4720 = z;
        double r4721 = r4719 / r4720;
        double r4722 = fma(r4718, r4721, r4717);
        return r4722;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.2
Target0.1
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.2

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

  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 2020025 +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))