Average Error: 10.6 → 0.0
Time: 3.1s
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 r723537 = x;
        double r723538 = y;
        double r723539 = z;
        double r723540 = r723539 - r723537;
        double r723541 = r723538 * r723540;
        double r723542 = r723537 + r723541;
        double r723543 = r723542 / r723539;
        return r723543;
}
↓
double f(double x, double y, double z) {
        double r723544 = 1.0;
        double r723545 = y;
        double r723546 = r723544 - r723545;
        double r723547 = x;
        double r723548 = z;
        double r723549 = r723547 / r723548;
        double r723550 = fma(r723546, r723549, r723545);
        return r723550;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.6
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.6

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

  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 2020036 +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))