Average Error: 10.6 → 0.0
Time: 3.0s
Precision: binary64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[y + \left(1 - y\right) \cdot \frac{x}{z}\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
y + \left(1 - y\right) \cdot \frac{x}{z}
double code(double x, double y, double z) {
	return (((double) (x + ((double) (y * ((double) (z - x)))))) / z);
}
↓
double code(double x, double y, double z) {
	return ((double) (y + ((double) (((double) (1.0 - y)) * (x / z)))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

Enter valid numbers for all inputs

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. Simplified3.2

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

  4. Applied add-cube-cbrt3.7

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

  5. Applied associate-*l*3.7

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

  6. Simplified2.4

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

  7. Taylor expanded around 0 3.6

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2020196 
(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))