Average Error: 19.8 → 0.4
Time: 3.0s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{y}{x + y}}{\left(\left(x + y\right) + 1\right) \cdot \frac{x + y}{x}}\]
\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{y}{x + y}}{\left(\left(x + y\right) + 1\right) \cdot \frac{x + y}{x}}
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 / (x + y)) / (((x + y) + 1.0) * ((x + y) / x)));
}

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
Herbie0.4
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

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 associate-*l*19.8

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

  4. Applied *-commutative19.8

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

  5. Applied times-frac4.2

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

  7. Applied associate-/r*0.1

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

  8. Applied associate-*r/0.1

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

  10. Applied associate-/l*0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1))))