Average Error: 10.3 → 0.0
Time: 17.6s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\mathsf{fma}\left(\frac{x}{z}, -y, y + \frac{x}{z}\right)\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\mathsf{fma}\left(\frac{x}{z}, -y, y + \frac{x}{z}\right)
double f(double x, double y, double z) {
        double r36683840 = x;
        double r36683841 = y;
        double r36683842 = z;
        double r36683843 = r36683842 - r36683840;
        double r36683844 = r36683841 * r36683843;
        double r36683845 = r36683840 + r36683844;
        double r36683846 = r36683845 / r36683842;
        return r36683846;
}
↓
double f(double x, double y, double z) {
        double r36683847 = x;
        double r36683848 = z;
        double r36683849 = r36683847 / r36683848;
        double r36683850 = y;
        double r36683851 = -r36683850;
        double r36683852 = r36683850 + r36683849;
        double r36683853 = fma(r36683849, r36683851, r36683852);
        return r36683853;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.3
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.3

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

  2. Simplified10.3

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

  3. Taylor expanded around 0 3.6

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019170 +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))