Average Error: 19.5 → 0.1
Time: 7.7s
Precision: binary64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
↓
\[\frac{\frac{x}{x + y} \cdot \frac{y}{x + y}}{\left(x + y\right) + 1} \]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\frac{x}{x + y} \cdot \frac{y}{x + y}}{\left(x + y\right) + 1}

(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))) (+ (+ x y) 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 ((x / (x + y)) * (y / (x + y))) / ((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

Original19.5
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 19.5

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

  2. Applied associate-/r*_binary6416.6

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

  3. Simplified7.6

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

  4. Applied associate-/r*_binary640.1

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

  5. Applied div-inv_binary640.2

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

  6. Applied associate-*l*_binary640.2

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2021280 
(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))))