Average Error: 20.1 → 0.1
Time: 13.5s
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 r482564 = x;
        double r482565 = y;
        double r482566 = r482564 * r482565;
        double r482567 = r482564 + r482565;
        double r482568 = r482567 * r482567;
        double r482569 = 1.0;
        double r482570 = r482567 + r482569;
        double r482571 = r482568 * r482570;
        double r482572 = r482566 / r482571;
        return r482572;
}
↓
double f(double x, double y) {
        double r482573 = x;
        double r482574 = y;
        double r482575 = r482573 + r482574;
        double r482576 = r482573 / r482575;
        double r482577 = 1.0;
        double r482578 = r482575 + r482577;
        double r482579 = r482574 / r482578;
        double r482580 = r482576 * r482579;
        double r482581 = r482580 / r482575;
        return r482581;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
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 20.1

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

  3. Applied add-cube-cbrt20.4

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

  4. Applied associate-*r*20.4

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

  6. Applied times-frac8.3

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

  7. Simplified8.3

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot {\left(\sqrt[3]{x + y}\right)}^{3}}} \cdot \frac{y}{\left(x + y\right) + 1}\]
  8. Using strategy rm

  9. Applied *-un-lft-identity8.3

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

  10. Applied times-frac0.7

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

  11. Simplified0.2

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

  13. Applied associate-*r/0.2

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

  14. Applied associate-*l/0.2

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

  15. Simplified0.1

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

  16. Final simplification0.1

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

Reproduce

herbie shell --seed 2019350 
(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))))