Average Error: 10.4 → 0.2
Time: 4.2s
Precision: binary64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[y + \left(x \cdot \left(\sqrt[3]{1} \cdot \frac{\sqrt[3]{1}}{z}\right)\right) \cdot \frac{\sqrt[3]{1}}{\frac{1}{1 - y}}\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
y + \left(x \cdot \left(\sqrt[3]{1} \cdot \frac{\sqrt[3]{1}}{z}\right)\right) \cdot \frac{\sqrt[3]{1}}{\frac{1}{1 - y}}
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) (x * ((double) (((double) cbrt(1.0)) * (((double) cbrt(1.0)) / z))))) * (((double) cbrt(1.0)) / (1.0 / ((double) (1.0 - y))))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.4
Target0.0
Herbie0.2
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program Error: 10.4 bits

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

  2. SimplifiedError: 3.3 bits

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

  4. Applied clear-numError: 3.3 bits

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

  6. Applied div-invError: 3.3 bits

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

  7. Applied add-cube-cbrtError: 3.3 bits

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

  8. Applied times-fracError: 3.3 bits

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

  9. Applied associate-*r*Error: 0.2 bits

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

  10. SimplifiedError: 0.2 bits

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

  11. Final simplificationError: 0.2 bits

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

Reproduce

herbie shell --seed 2020200 
(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))