Average Error: 20.0 → 0.1
Time: 4.3s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}\]
\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{x}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}
double f(double x, double y) {
        double r407906 = x;
        double r407907 = y;
        double r407908 = r407906 * r407907;
        double r407909 = r407906 + r407907;
        double r407910 = r407909 * r407909;
        double r407911 = 1.0;
        double r407912 = r407909 + r407911;
        double r407913 = r407910 * r407912;
        double r407914 = r407908 / r407913;
        return r407914;
}
↓
double f(double x, double y) {
        double r407915 = x;
        double r407916 = y;
        double r407917 = r407915 + r407916;
        double r407918 = r407915 / r407917;
        double r407919 = 1.0;
        double r407920 = r407917 + r407919;
        double r407921 = r407916 / r407920;
        double r407922 = r407918 * r407921;
        double r407923 = r407922 / r407917;
        return r407923;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
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 20.0

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

  3. Applied times-frac8.2

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

  5. Applied *-un-lft-identity8.2

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

  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 associate-*l/7.7

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

  10. Applied associate-*r/7.7

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020046 +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))))