Average Error: 19.8 → 0.1
Time: 4.0s
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 (((double) (x * y)) / ((double) (((double) (((double) (x + y)) * ((double) (x + y)))) * ((double) (((double) (x + y)) + 1.0)))));
}
↓
double code(double x, double y) {
	return (((double) (y * ((x / ((double) (y + x))) / ((double) (y + x))))) / ((double) (((double) (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

Original19.8
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 Error: 19.8 bits

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

  3. Applied add-cube-cbrtError: 20.1 bits

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

  4. Applied add-cube-cbrtError: 20.2 bits

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

  5. Applied swap-sqrError: 20.3 bits

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

  6. SimplifiedError: 20.2 bits

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

  8. Applied times-fracError: 8.4 bits

    \[\leadsto \color{blue}{\frac{x}{{\left(\sqrt[3]{x + y}\right)}^{4} \cdot \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]

  9. SimplifiedError: 8.4 bits

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

  11. Applied add-sqr-sqrtError: 35.9 bits

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

  12. Applied cbrt-prodError: 36.0 bits

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

  13. Applied unpow-prod-downError: 36.0 bits

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

  14. Applied *-un-lft-identityError: 36.0 bits

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

  15. Applied times-fracError: 32.3 bits

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

  16. SimplifiedError: 32.1 bits

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

  17. SimplifiedError: 0.2 bits

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

  19. Applied associate-*r/Error: 0.2 bits

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

  20. SimplifiedError: 0.1 bits

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

  21. Final simplificationError: 0.1 bits

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

Reproduce

herbie shell --seed 2020205 
(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))))