Average Error: 9.0 → 0.0
Time: 18.1s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\mathsf{fma}\left(\frac{x}{z}, -y, \frac{x}{z}\right) + y\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\mathsf{fma}\left(\frac{x}{z}, -y, \frac{x}{z}\right) + y
double f(double x, double y, double z) {
        double r33004062 = x;
        double r33004063 = y;
        double r33004064 = z;
        double r33004065 = r33004064 - r33004062;
        double r33004066 = r33004063 * r33004065;
        double r33004067 = r33004062 + r33004066;
        double r33004068 = r33004067 / r33004064;
        return r33004068;
}
↓
double f(double x, double y, double z) {
        double r33004069 = x;
        double r33004070 = z;
        double r33004071 = r33004069 / r33004070;
        double r33004072 = y;
        double r33004073 = -r33004072;
        double r33004074 = fma(r33004071, r33004073, r33004071);
        double r33004075 = r33004074 + r33004072;
        return r33004075;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original9.0
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 9.0

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

  2. Taylor expanded around 0 3.1

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"

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

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