Average Error: 9.2 → 1.0
Time: 4.0s
Precision: binary64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\left(x \cdot \left(\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}} \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\right)\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\right)}^{2}} \cdot \sqrt[3]{\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}}\right)\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\left(x \cdot \left(\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}} \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\right)\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\right)}^{2}} \cdot \sqrt[3]{\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}}\right)
double code(double x, double y) {
	return ((double) (((double) (x * ((double) (((double) (x / y)) + 1.0)))) / ((double) (x + 1.0))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (x * ((double) (((double) cbrt(((double) (((double) (((double) (x / y)) + 1.0)) / ((double) (x + 1.0)))))) * ((double) cbrt(((double) (((double) (((double) (x / y)) + 1.0)) / ((double) (x + 1.0)))))))))) * ((double) (((double) cbrt(((double) pow(((double) cbrt(((double) (((double) (((double) (x / y)) + 1.0)) / ((double) (x + 1.0)))))), 2.0)))) * ((double) cbrt(((double) cbrt(((double) (((double) (((double) (x / y)) + 1.0)) / ((double) (x + 1.0))))))))))));
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original9.2
Target0.1
Herbie1.0
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.2

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

  2. Simplified0.1

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

  4. Applied add-cube-cbrt0.9

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

  5. Applied associate-*r*0.9

    \[\leadsto \color{blue}{\left(x \cdot \left(\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}} \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\right)\right) \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt0.9

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

  8. Applied cbrt-prod1.0

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

  9. Simplified1.0

    \[\leadsto \left(x \cdot \left(\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}} \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\right)\right) \cdot \left(\color{blue}{\sqrt[3]{{\left(\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}}\right)\]

  10. Final simplification1.0

    \[\leadsto \left(x \cdot \left(\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}} \cdot \sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\right)\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}\right)}^{2}} \cdot \sqrt[3]{\sqrt[3]{\frac{\frac{x}{y} + 1}{x + 1}}}\right)\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0)))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))