Average Error: 19.8 → 0.2
Time: 9.5s
Precision: binary64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{1}{x + y} \cdot \left(\frac{x}{x + y} \cdot \left(y \cdot \frac{1}{\left(x + y\right) + 1}\right)\right)\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{1}{x + y} \cdot \left(\frac{x}{x + y} \cdot \left(y \cdot \frac{1}{\left(x + y\right) + 1}\right)\right)
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) ((1.0 / ((double) (x + y))) * ((double) ((x / ((double) (x + y))) * ((double) (y * (1.0 / ((double) (((double) (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

Original19.8
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 19.8

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

  3. Applied times-frac7.7

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

  5. Applied *-un-lft-identity7.7

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

  6. Applied times-frac0.2

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

  7. Applied associate-*l*0.2

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

  9. Applied div-inv0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2020182 
(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))))