Average Error: 0.2 → 0.3
Time: 3.3s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{0.1111111111111111}{x}\right) - \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\frac{y}{\sqrt[3]{3}}}{\sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{0.1111111111111111}{x}\right) - \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\frac{y}{\sqrt[3]{3}}}{\sqrt{x}}
double code(double x, double y) {
	return ((double) (((double) (1.0 - ((double) (1.0 / ((double) (x * 9.0)))))) - ((double) (y / ((double) (3.0 * ((double) sqrt(x))))))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (1.0 - ((double) (0.1111111111111111 / x)))) - ((double) (((double) (1.0 / ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))))) * ((double) (((double) (y / ((double) cbrt(3.0)))) / ((double) sqrt(x))))))));
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.2
Target0.2
Herbie0.3
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

Derivation

  1. Initial program 0.2

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

  2. Taylor expanded around 0 0.2

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

  4. Applied associate-/r*0.2

    \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{\frac{\frac{y}{3}}{\sqrt{x}}}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity0.2

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

  7. Applied sqrt-prod0.2

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

  8. Applied add-cube-cbrt0.2

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

  9. Applied *-un-lft-identity0.2

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

  10. Applied times-frac0.3

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

  11. Applied times-frac0.3

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020121 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x))))

  (- (- 1 (/ 1 (* x 9))) (/ y (* 3 (sqrt x)))))