Average Error: 20.2 → 0.1
Time: 3.0s
Precision: binary64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{y \cdot \frac{\frac{x}{y + x}}{y + x}}{\left(y + x\right) + 1}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{y \cdot \frac{\frac{x}{y + x}}{y + x}}{\left(y + x\right) + 1}
(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (/ (* y (/ (/ x (+ y x)) (+ y x))) (+ (+ y x) 1.0)))
double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}
↓
double code(double x, double y) {
	return (y * ((x / (y + x)) / (y + x))) / ((y + x) + 1.0);
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.2
Target0.1
Herbie0.1
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 20.2

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

  3. Applied add-cbrt-cube_binary6429.6

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

  4. Simplified29.6

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

  6. Applied cbrt-prod_binary6428.1

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

  7. Applied times-frac_binary6422.4

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

  8. Simplified22.4

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

  10. Applied add-sqr-sqrt_binary6442.9

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

  11. Applied unpow-prod-down_binary6442.9

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

  12. Applied cbrt-prod_binary6438.9

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

  13. Applied *-un-lft-identity_binary6438.9

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

  14. Applied times-frac_binary6438.9

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

  15. Simplified38.9

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

  16. Simplified0.2

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

  18. Applied associate-*r/_binary640.2

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

  19. Simplified0.1

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

  20. Final simplification0.1

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

Reproduce

herbie shell --seed 2020232 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

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

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