Average Error: 19.7 → 0.2
Time: 2.2s
Precision: binary64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}
(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (* (/ (/ x (+ x y)) (+ x y)) (/ 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 ((x / (x + y)) / (x + y)) * (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

Original19.7
Target0.1
Herbie0.2
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 19.7

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

  3. Applied times-frac_binary647.9

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

  4. Simplified36.5

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

  6. Applied add-sqr-sqrt_binary6436.5

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

  7. Applied sqrt-prod_binary6436.6

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

  8. Applied unpow-prod-down_binary6436.6

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

  9. Applied *-un-lft-identity_binary6436.6

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

  10. Applied times-frac_binary6432.9

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

  11. Simplified32.7

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

  12. Simplified0.2

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

  14. Applied *-un-lft-identity_binary640.2

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

  15. Applied *-un-lft-identity_binary640.2

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

  16. Applied times-frac_binary640.2

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

  17. Applied associate-*r*_binary640.2

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

  18. Simplified0.2

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

  19. Final simplification0.2

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

Reproduce

herbie shell --seed 2020224 
(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))))