Average Error: 20.2 → 0.1
Time: 15.7s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{y \cdot \frac{x}{x + y}}{x + y}}{\left(x + y\right) + 1}\]
\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{y \cdot \frac{x}{x + y}}{x + y}}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r224504 = x;
        double r224505 = y;
        double r224506 = r224504 * r224505;
        double r224507 = r224504 + r224505;
        double r224508 = r224507 * r224507;
        double r224509 = 1.0;
        double r224510 = r224507 + r224509;
        double r224511 = r224508 * r224510;
        double r224512 = r224506 / r224511;
        return r224512;
}
↓
double f(double x, double y) {
        double r224513 = y;
        double r224514 = x;
        double r224515 = r224514 + r224513;
        double r224516 = r224514 / r224515;
        double r224517 = r224513 * r224516;
        double r224518 = r224517 / r224515;
        double r224519 = 1.0;
        double r224520 = r224515 + r224519;
        double r224521 = r224518 / r224520;
        return r224521;
}

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 distribute-lft-in20.2

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

  5. Applied times-frac10.8

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

  6. Simplified8.3

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

  8. Applied *-un-lft-identity8.3

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

  9. Applied times-frac0.2

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

  10. Applied associate-*l*0.2

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

  12. Applied associate-*r/0.2

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

  13. Applied associate-*r/0.2

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2019326 
(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))))