Average Error: 0.4 → 0.5
Time: 3.8s
Precision: binary64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt[3]{\sqrt{3}} \cdot \left(\sqrt[3]{\sqrt{3}} \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{0.1111111111111111}{x}\right) - 1\right)\right)\right)\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt[3]{\sqrt{3}} \cdot \left(\sqrt[3]{\sqrt{3}} \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{0.1111111111111111}{x}\right) - 1\right)\right)\right)\right)
double code(double x, double y) {
	return ((double) (((double) (3.0 * ((double) sqrt(x)))) * ((double) (((double) (y + ((double) (1.0 / ((double) (x * 9.0)))))) - 1.0))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))) * ((double) (((double) cbrt(((double) sqrt(3.0)))) * ((double) (((double) cbrt(((double) sqrt(3.0)))) * ((double) (((double) sqrt(x)) * ((double) (((double) (y + ((double) (0.1111111111111111 / x)))) - 1.0))))))))));
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.4
Target0.4
Herbie0.5
\[3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right)\]

Derivation

  1. Initial program 0.4

    \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
  2. Using strategy rm

  3. Applied associate-*l*0.4

    \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.4

    \[\leadsto \color{blue}{\left(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}\right)} \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\]

  6. Applied associate-*l*0.6

    \[\leadsto \color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt[3]{3} \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right)}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt0.6

    \[\leadsto \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt[3]{\color{blue}{\sqrt{3} \cdot \sqrt{3}}} \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right)\]

  9. Applied cbrt-prod0.6

    \[\leadsto \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\color{blue}{\left(\sqrt[3]{\sqrt{3}} \cdot \sqrt[3]{\sqrt{3}}\right)} \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right)\]

  10. Applied associate-*l*0.5

    \[\leadsto \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \color{blue}{\left(\sqrt[3]{\sqrt{3}} \cdot \left(\sqrt[3]{\sqrt{3}} \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right)\right)}\]

  11. Taylor expanded around 0 0.5

    \[\leadsto \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt[3]{\sqrt{3}} \cdot \left(\sqrt[3]{\sqrt{3}} \cdot \left(\sqrt{x} \cdot \left(\left(y + \color{blue}{\frac{0.1111111111111111}{x}}\right) - 1\right)\right)\right)\right)\]

  12. Final simplification0.5

    \[\leadsto \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt[3]{\sqrt{3}} \cdot \left(\sqrt[3]{\sqrt{3}} \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{0.1111111111111111}{x}\right) - 1\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2020168 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))