Average Error: 10.5 → 0.0
Time: 3.2s
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 r866613 = x;
        double r866614 = y;
        double r866615 = z;
        double r866616 = r866615 - r866613;
        double r866617 = r866614 * r866616;
        double r866618 = r866613 + r866617;
        double r866619 = r866618 / r866615;
        return r866619;
}
↓
double f(double x, double y, double z) {
        double r866620 = 1.0;
        double r866621 = y;
        double r866622 = r866620 - r866621;
        double r866623 = x;
        double r866624 = z;
        double r866625 = r866623 / r866624;
        double r866626 = fma(r866622, r866625, r866621);
        return r866626;
}

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. 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 2020018 +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))