Average Error: 9.7 → 0.0
Time: 11.9s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\mathsf{fma}\left(\frac{-x}{z}, y, y\right) + \frac{x}{z}\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\mathsf{fma}\left(\frac{-x}{z}, y, y\right) + \frac{x}{z}
double f(double x, double y, double z) {
        double r611389 = x;
        double r611390 = y;
        double r611391 = z;
        double r611392 = r611391 - r611389;
        double r611393 = r611390 * r611392;
        double r611394 = r611389 + r611393;
        double r611395 = r611394 / r611391;
        return r611395;
}
↓
double f(double x, double y, double z) {
        double r611396 = x;
        double r611397 = -r611396;
        double r611398 = z;
        double r611399 = r611397 / r611398;
        double r611400 = y;
        double r611401 = fma(r611399, r611400, r611400);
        double r611402 = r611396 / r611398;
        double r611403 = r611401 + r611402;
        return r611403;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original9.7
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 9.7

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

  2. Simplified9.7

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

  3. Taylor expanded around 0 3.3

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019194 +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))