Average Error: 19.8 → 0.1
Time: 4.6s
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 r402868 = x;
        double r402869 = y;
        double r402870 = r402868 * r402869;
        double r402871 = r402868 + r402869;
        double r402872 = r402871 * r402871;
        double r402873 = 1.0;
        double r402874 = r402871 + r402873;
        double r402875 = r402872 * r402874;
        double r402876 = r402870 / r402875;
        return r402876;
}
↓
double f(double x, double y) {
        double r402877 = 1.0;
        double r402878 = x;
        double r402879 = y;
        double r402880 = r402878 + r402879;
        double r402881 = r402880 / r402878;
        double r402882 = r402877 / r402881;
        double r402883 = 1.0;
        double r402884 = r402880 + r402883;
        double r402885 = r402879 / r402884;
        double r402886 = r402882 * r402885;
        double r402887 = r402886 / r402880;
        return r402887;
}

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

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 times-frac7.5

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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 2019353 +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))))