Average Error: 20.0 → 9.7
Time: 16.3s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{x \cdot x + y \cdot \left(2 \cdot x + y\right)}{\sqrt[3]{x}}} \cdot \frac{y}{\left(x + y\right) + 1}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{x \cdot x + y \cdot \left(2 \cdot x + y\right)}{\sqrt[3]{x}}} \cdot \frac{y}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r334861 = x;
        double r334862 = y;
        double r334863 = r334861 * r334862;
        double r334864 = r334861 + r334862;
        double r334865 = r334864 * r334864;
        double r334866 = 1.0;
        double r334867 = r334864 + r334866;
        double r334868 = r334865 * r334867;
        double r334869 = r334863 / r334868;
        return r334869;
}
↓
double f(double x, double y) {
        double r334870 = x;
        double r334871 = cbrt(r334870);
        double r334872 = r334871 * r334871;
        double r334873 = r334870 * r334870;
        double r334874 = y;
        double r334875 = 2.0;
        double r334876 = r334875 * r334870;
        double r334877 = r334876 + r334874;
        double r334878 = r334874 * r334877;
        double r334879 = r334873 + r334878;
        double r334880 = r334879 / r334871;
        double r334881 = r334872 / r334880;
        double r334882 = r334870 + r334874;
        double r334883 = 1.0;
        double r334884 = r334882 + r334883;
        double r334885 = r334874 / r334884;
        double r334886 = r334881 * r334885;
        return r334886;
}

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
Herbie9.7
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

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. Taylor expanded around 0 20.0

    \[\leadsto \frac{x \cdot y}{\color{blue}{\left({x}^{2} + \left({y}^{2} + 2 \cdot \left(x \cdot y\right)\right)\right)} \cdot \left(\left(x + y\right) + 1\right)}\]

  3. Simplified20.0

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

  5. Applied times-frac9.3

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

  7. Applied add-cube-cbrt9.7

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{x \cdot x + y \cdot \left(2 \cdot x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}\]

  8. Applied associate-/l*9.7

    \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{x \cdot x + y \cdot \left(2 \cdot x + y\right)}{\sqrt[3]{x}}}} \cdot \frac{y}{\left(x + y\right) + 1}\]

  9. Final simplification9.7

    \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{x \cdot x + y \cdot \left(2 \cdot x + y\right)}{\sqrt[3]{x}}} \cdot \frac{y}{\left(x + y\right) + 1}\]

Reproduce

herbie shell --seed 2019323 
(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))))