Average Error: 20.0 → 0.1
Time: 4.0s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}}{x + y} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\]
\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} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)
double f(double x, double y) {
        double r603985 = x;
        double r603986 = y;
        double r603987 = r603985 * r603986;
        double r603988 = r603985 + r603986;
        double r603989 = r603988 * r603988;
        double r603990 = 1.0;
        double r603991 = r603988 + r603990;
        double r603992 = r603989 * r603991;
        double r603993 = r603987 / r603992;
        return r603993;
}
↓
double f(double x, double y) {
        double r603994 = x;
        double r603995 = y;
        double r603996 = r603994 + r603995;
        double r603997 = r603994 / r603996;
        double r603998 = 1.0;
        double r603999 = r603996 + r603998;
        double r604000 = r603995 / r603999;
        double r604001 = r603997 * r604000;
        double r604002 = r604001 / r603996;
        double r604003 = 1.0;
        double r604004 = cbrt(r604003);
        double r604005 = r604004 * r604004;
        double r604006 = r604002 * r604005;
        return r604006;
}

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 *-un-lft-identity0.2

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

  10. Applied add-cube-cbrt0.2

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

  11. Applied times-frac0.2

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

  12. Applied associate-*l*0.2

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2020046 
(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))))