Average Error: 20.2 → 0.1
Time: 7.6s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{y \cdot \frac{x}{y + x}}{y + x}}{\left(y + x\right) + 1}\]
\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{y \cdot \frac{x}{y + x}}{y + x}}{\left(y + x\right) + 1}
(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (/ (/ (* y (/ x (+ y x))) (+ y x)) (+ (+ y x) 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 ((y * (x / (y + x))) / (y + x)) / ((y + x) + 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

Original20.2
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 20.2

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

  3. Applied associate-/r*_binary6417.0

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

  4. Simplified8.1

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

  6. Applied *-un-lft-identity_binary648.1

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

  7. Applied times-frac_binary640.2

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

  8. Applied associate-*l*_binary640.2

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

  9. Simplified0.2

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

  11. Applied associate-*l/_binary640.1

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

  12. Simplified0.1

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

  13. Final simplification0.1

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

Alternatives

Reproduce

herbie shell --seed 2021118 
(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))))