Average Error: 10.4 → 0.0
Time: 18.1s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\left(\frac{x}{z} + y\right) + \frac{x}{z} \cdot \left(-y\right)\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\left(\frac{x}{z} + y\right) + \frac{x}{z} \cdot \left(-y\right)
double f(double x, double y, double z) {
        double r485179 = x;
        double r485180 = y;
        double r485181 = z;
        double r485182 = r485181 - r485179;
        double r485183 = r485180 * r485182;
        double r485184 = r485179 + r485183;
        double r485185 = r485184 / r485181;
        return r485185;
}
↓
double f(double x, double y, double z) {
        double r485186 = x;
        double r485187 = z;
        double r485188 = r485186 / r485187;
        double r485189 = y;
        double r485190 = r485188 + r485189;
        double r485191 = -r485189;
        double r485192 = r485188 * r485191;
        double r485193 = r485190 + r485192;
        return r485193;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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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. Simplified10.4

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

  3. Taylor expanded around 0 3.4

    \[\leadsto \color{blue}{\left(\frac{x}{z} + y\right) - \frac{x \cdot y}{z}}\]
  4. Using strategy rm

  5. Applied sub-neg3.4

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019347 +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))