Average Error: 20.4 → 0.2
Time: 4.1s
Precision: binary64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{x \cdot \frac{1}{x + y}}{x + y} \cdot \frac{y}{x + \left(y + 1\right)}\]
\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 \cdot \frac{1}{x + y}}{x + y} \cdot \frac{y}{x + \left(y + 1\right)}
double code(double x, double y) {
	return ((double) (((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) (((double) (((double) (x * ((double) (1.0 / ((double) (x + y)))))) / ((double) (x + y)))) * ((double) (y / ((double) (x + ((double) (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

Original20.4
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 20.4

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

  3. Applied times-frac8.0

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

  4. Simplified8.0

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

  6. Applied associate-/r*0.2

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

  8. Applied div-inv0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020163 
(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))))