Average Error: 10.4 → 0.0
Time: 4.3s
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 r924768 = x;
        double r924769 = y;
        double r924770 = z;
        double r924771 = r924770 - r924768;
        double r924772 = r924769 * r924771;
        double r924773 = r924768 + r924772;
        double r924774 = r924773 / r924770;
        return r924774;
}
↓
double f(double x, double y, double z) {
        double r924775 = 1.0;
        double r924776 = y;
        double r924777 = r924775 - r924776;
        double r924778 = x;
        double r924779 = z;
        double r924780 = r924778 / r924779;
        double r924781 = fma(r924777, r924780, r924776);
        return r924781;
}

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 2019354 +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))