Average Error: 20.0 → 0.1
Time: 4.8s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{1}{\frac{x + y}{x}} \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}\]
\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{1}{\frac{x + y}{x}} \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}
double f(double x, double y) {
        double r520878 = x;
        double r520879 = y;
        double r520880 = r520878 * r520879;
        double r520881 = r520878 + r520879;
        double r520882 = r520881 * r520881;
        double r520883 = 1.0;
        double r520884 = r520881 + r520883;
        double r520885 = r520882 * r520884;
        double r520886 = r520880 / r520885;
        return r520886;
}
↓
double f(double x, double y) {
        double r520887 = 1.0;
        double r520888 = x;
        double r520889 = y;
        double r520890 = r520888 + r520889;
        double r520891 = r520890 / r520888;
        double r520892 = r520887 / r520891;
        double r520893 = 1.0;
        double r520894 = r520890 + r520893;
        double r520895 = r520889 / r520894;
        double r520896 = r520892 * r520895;
        double r520897 = r520896 / r520890;
        return r520897;
}

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-frac8.5

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

  5. Applied associate-/r*0.2

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

  7. Applied associate-*l/0.1

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

  9. Applied clear-num0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019354 +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))))