Average Error: 10.1 → 0.0
Time: 4.0s
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 r904952 = x;
        double r904953 = y;
        double r904954 = z;
        double r904955 = r904954 - r904952;
        double r904956 = r904953 * r904955;
        double r904957 = r904952 + r904956;
        double r904958 = r904957 / r904954;
        return r904958;
}
↓
double f(double x, double y, double z) {
        double r904959 = 1.0;
        double r904960 = y;
        double r904961 = r904959 - r904960;
        double r904962 = x;
        double r904963 = z;
        double r904964 = r904962 / r904963;
        double r904965 = fma(r904961, r904964, r904960);
        return r904965;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

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

Derivation

  1. Initial program 10.1

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

  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 2019353 +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))