Average Error: 10.4 → 0.0
Time: 3.2s
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 r793896 = x;
        double r793897 = y;
        double r793898 = z;
        double r793899 = r793898 - r793896;
        double r793900 = r793897 * r793899;
        double r793901 = r793896 + r793900;
        double r793902 = r793901 / r793898;
        return r793902;
}
↓
double f(double x, double y, double z) {
        double r793903 = 1.0;
        double r793904 = y;
        double r793905 = r793903 - r793904;
        double r793906 = x;
        double r793907 = z;
        double r793908 = r793906 / r793907;
        double r793909 = fma(r793905, r793908, r793904);
        return r793909;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.4
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.4

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

  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 2020027 +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))