Average Error: 9.4 → 0.0
Time: 12.9s
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 r23162566 = x;
        double r23162567 = y;
        double r23162568 = z;
        double r23162569 = r23162568 - r23162566;
        double r23162570 = r23162567 * r23162569;
        double r23162571 = r23162566 + r23162570;
        double r23162572 = r23162571 / r23162568;
        return r23162572;
}
↓
double f(double x, double y, double z) {
        double r23162573 = x;
        double r23162574 = z;
        double r23162575 = r23162573 / r23162574;
        double r23162576 = y;
        double r23162577 = -r23162576;
        double r23162578 = r23162576 + r23162575;
        double r23162579 = fma(r23162575, r23162577, r23162578);
        return r23162579;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original9.4
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 9.4

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

  2. Simplified9.4

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

  3. Taylor expanded around 0 3.3

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 +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))