Average Error: 19.8 → 0.1
Time: 2.8s
Precision: binary64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\left(\frac{y}{y + x} \cdot \frac{x}{y + x}\right) \cdot \frac{1}{\left(y + x\right) + 1}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\left(\frac{y}{y + x} \cdot \frac{x}{y + x}\right) \cdot \frac{1}{\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 (+ y x)) (/ x (+ y x))) (/ 1.0 (+ (+ 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 / (y + x)) * (x / (y + x))) * (1.0 / ((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 19.8

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

  3. Applied distribute-lft-in_binary6419.8

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

  4. Simplified19.8

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

  5. Simplified19.8

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

  7. Applied associate-/r*_binary6416.9

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

  8. Simplified7.9

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

  10. Applied *-un-lft-identity_binary647.9

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

  11. Applied times-frac_binary640.2

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

  12. Applied associate-*r*_binary640.2

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

  13. Simplified0.1

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

  15. Applied div-inv_binary640.1

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

  16. Simplified0.1

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

  17. Final simplification0.1

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

Reproduce

herbie shell --seed 2020219 
(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))))