Average Error: 10.5 → 0.0
Time: 15.4s
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 r34049856 = x;
        double r34049857 = y;
        double r34049858 = z;
        double r34049859 = r34049858 - r34049856;
        double r34049860 = r34049857 * r34049859;
        double r34049861 = r34049856 + r34049860;
        double r34049862 = r34049861 / r34049858;
        return r34049862;
}
↓
double f(double x, double y, double z) {
        double r34049863 = x;
        double r34049864 = z;
        double r34049865 = r34049863 / r34049864;
        double r34049866 = y;
        double r34049867 = -r34049866;
        double r34049868 = fma(r34049865, r34049867, r34049865);
        double r34049869 = r34049868 + r34049866;
        return r34049869;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.5
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.5

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

  2. Simplified10.5

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

  3. Taylor expanded around 0 3.4

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 +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))