Average Error: 10.2 → 1.1
Time: 2.7s
Precision: binary64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[y + \left(x \cdot \left(\sqrt[3]{1 - y} \cdot \frac{\sqrt[3]{1 - y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)\right) \cdot \frac{\sqrt[3]{1 - y}}{\sqrt[3]{z}}\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
y + \left(x \cdot \left(\sqrt[3]{1 - y} \cdot \frac{\sqrt[3]{1 - y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)\right) \cdot \frac{\sqrt[3]{1 - y}}{\sqrt[3]{z}}
double code(double x, double y, double z) {
	return ((double) (((double) (x + ((double) (y * ((double) (z - x)))))) / z));
}
↓
double code(double x, double y, double z) {
	return ((double) (y + ((double) (((double) (x * ((double) (((double) cbrt(((double) (1.0 - y)))) * ((double) (((double) cbrt(((double) (1.0 - y)))) / ((double) (((double) cbrt(z)) * ((double) cbrt(z)))))))))) * ((double) (((double) cbrt(((double) (1.0 - y)))) / ((double) cbrt(z))))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original10.2
Target0.0
Herbie1.1
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.2

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

  2. Simplified3.2

    \[\leadsto \color{blue}{y + x \cdot \frac{1 - y}{z}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt3.8

    \[\leadsto y + x \cdot \frac{1 - y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

  5. Applied add-cube-cbrt3.8

    \[\leadsto y + x \cdot \frac{\color{blue}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\]

  6. Applied times-frac3.8

    \[\leadsto y + x \cdot \color{blue}{\left(\frac{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{1 - y}}{\sqrt[3]{z}}\right)}\]

  7. Applied associate-*r*1.1

    \[\leadsto y + \color{blue}{\left(x \cdot \frac{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{1 - y}}{\sqrt[3]{z}}}\]

  8. Simplified1.1

    \[\leadsto y + \color{blue}{\left(x \cdot \left(\sqrt[3]{1 - y} \cdot \frac{\sqrt[3]{1 - y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)\right)} \cdot \frac{\sqrt[3]{1 - y}}{\sqrt[3]{z}}\]

  9. Final simplification1.1

    \[\leadsto y + \left(x \cdot \left(\sqrt[3]{1 - y} \cdot \frac{\sqrt[3]{1 - y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)\right) \cdot \frac{\sqrt[3]{1 - y}}{\sqrt[3]{z}}\]

Reproduce

herbie shell --seed 2020184 
(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))