Average Error: 10.0 → 0.0
Time: 2.7s
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 r741755 = x;
        double r741756 = y;
        double r741757 = z;
        double r741758 = r741757 - r741755;
        double r741759 = r741756 * r741758;
        double r741760 = r741755 + r741759;
        double r741761 = r741760 / r741757;
        return r741761;
}
↓
double f(double x, double y, double z) {
        double r741762 = 1.0;
        double r741763 = y;
        double r741764 = r741762 - r741763;
        double r741765 = x;
        double r741766 = z;
        double r741767 = r741765 / r741766;
        double r741768 = fma(r741764, r741767, r741763);
        return r741768;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

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

Derivation

  1. Initial program 10.0

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

  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 2020033 +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))