Average Error: 20.0 → 0.5
Time: 4.6s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}{\frac{x + y}{\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}}} \cdot y}{\left(x + y\right) + 1}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}{\frac{x + y}{\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}}} \cdot y}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r403665 = x;
        double r403666 = y;
        double r403667 = r403665 * r403666;
        double r403668 = r403665 + r403666;
        double r403669 = r403668 * r403668;
        double r403670 = 1.0;
        double r403671 = r403668 + r403670;
        double r403672 = r403669 * r403671;
        double r403673 = r403667 / r403672;
        return r403673;
}
↓
double f(double x, double y) {
        double r403674 = x;
        double r403675 = cbrt(r403674);
        double r403676 = r403675 * r403675;
        double r403677 = y;
        double r403678 = r403674 + r403677;
        double r403679 = cbrt(r403678);
        double r403680 = r403679 * r403679;
        double r403681 = r403676 / r403680;
        double r403682 = r403675 / r403679;
        double r403683 = r403678 / r403682;
        double r403684 = r403681 / r403683;
        double r403685 = r403684 * r403677;
        double r403686 = 1.0;
        double r403687 = r403678 + r403686;
        double r403688 = r403685 / r403687;
        return r403688;
}

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
Herbie0.5
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

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-frac7.8

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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-*r/0.2

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

  9. Applied add-cube-cbrt0.7

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

  10. Applied add-cube-cbrt0.5

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

  11. Applied times-frac0.5

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

  12. Applied associate-/l*0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2020036 +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))))