Average Error: 19.8 → 0.1
Time: 3.2s
Precision: binary64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}\]
\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 (((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) ((x / ((double) (x + y))) * ((y / ((double) (((double) (x + y)) + 1.0))) / ((double) (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

Original19.8
Target0.2
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 times-frac_binary6410.8

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

  8. Simplified8.1

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

  10. Applied associate-/r*_binary640.2

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

  12. Applied div-inv_binary640.2

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

  13. Applied associate-*l*_binary640.2

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2020210 
(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))))