Average Error: 20.0 → 0.6
Time: 2.8s
Precision: binary64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\sqrt[3]{\frac{y}{\left(x + y\right) + 1}} \cdot \left(\frac{\frac{x}{x + y}}{x + y} \cdot \left(\sqrt[3]{\frac{y}{\left(x + y\right) + 1}} \cdot \sqrt[3]{\frac{y}{\left(x + y\right) + 1}}\right)\right)\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\sqrt[3]{\frac{y}{\left(x + y\right) + 1}} \cdot \left(\frac{\frac{x}{x + y}}{x + y} \cdot \left(\sqrt[3]{\frac{y}{\left(x + y\right) + 1}} \cdot \sqrt[3]{\frac{y}{\left(x + y\right) + 1}}\right)\right)
(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (*
  (cbrt (/ y (+ (+ x y) 1.0)))
  (*
   (/ (/ x (+ x y)) (+ x y))
   (* (cbrt (/ y (+ (+ x y) 1.0))) (cbrt (/ y (+ (+ x y) 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 cbrt(y / ((x + y) + 1.0)) * (((x / (x + y)) / (x + y)) * (cbrt(y / ((x + y) + 1.0)) * cbrt(y / ((x + y) + 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

Original20.0
Target0.1
Herbie0.6
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 20.0

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

  3. Applied add-cbrt-cube_binary6421.1

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

  4. Applied add-cbrt-cube_binary6421.2

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

  5. Applied cbrt-unprod_binary6427.7

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

  6. Simplified27.7

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

  8. Applied times-frac_binary6422.3

    \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{{\left(x + y\right)}^{6}}} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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_binary6439.1

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

  13. Applied *-un-lft-identity_binary6439.1

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

  14. Applied times-frac_binary6439.1

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

  15. Simplified39.1

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

  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 add-cube-cbrt_binary640.6

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

  19. Applied associate-*r*_binary640.6

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

  20. Simplified0.6

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

  21. Final simplification0.6

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

Reproduce

herbie shell --seed 2020231 
(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))))