Average Error: 19.1 → 0.1
Time: 2.7s
Precision: binary64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}
(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (* (/ x (+ x y)) (/ (/ y (+ (+ x y) 1.0)) (+ x y))))
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)) * ((y / ((x + y) + 1.0)) / (x + y));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.1
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 19.1

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

  3. Applied add-cbrt-cube_binary6426.8

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

  4. Simplified26.8

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

  6. Applied sqr-pow_binary6426.8

    \[\leadsto \frac{x \cdot y}{\sqrt[3]{\color{blue}{{\left(x + y\right)}^{\left(\frac{6}{2}\right)} \cdot {\left(x + y\right)}^{\left(\frac{6}{2}\right)}}} \cdot \left(\left(x + y\right) + 1\right)}\]

  7. Applied cbrt-prod_binary6420.2

    \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\sqrt[3]{{\left(x + y\right)}^{\left(\frac{6}{2}\right)}} \cdot \sqrt[3]{{\left(x + y\right)}^{\left(\frac{6}{2}\right)}}\right)} \cdot \left(\left(x + y\right) + 1\right)}\]

  8. Applied associate-*l*_binary6420.2

    \[\leadsto \frac{x \cdot y}{\color{blue}{\sqrt[3]{{\left(x + y\right)}^{\left(\frac{6}{2}\right)}} \cdot \left(\sqrt[3]{{\left(x + y\right)}^{\left(\frac{6}{2}\right)}} \cdot \left(\left(x + y\right) + 1\right)\right)}}\]

  9. Simplified20.1

    \[\leadsto \frac{x \cdot y}{\sqrt[3]{{\left(x + y\right)}^{\left(\frac{6}{2}\right)}} \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}}\]
  10. Using strategy rm

  11. Applied times-frac_binary6413.5

    \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{{\left(x + y\right)}^{\left(\frac{6}{2}\right)}}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}}\]

  12. Simplified4.0

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

  13. Simplified4.0

    \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{y}{\left(x + y\right) + {\left(x + y\right)}^{2}}}\]
  14. Using strategy rm

  15. Applied unpow2_binary644.0

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

  16. Applied distribute-rgt1-in_binary644.0

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

  17. Applied associate-/r*_binary640.1

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

  18. Simplified0.1

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

  19. Final simplification0.1

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

Reproduce

herbie shell --seed 2020233 
(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))))