Average Error: 20.0 → 0.5
Time: 5.1s
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 r396430 = x;
        double r396431 = y;
        double r396432 = r396430 * r396431;
        double r396433 = r396430 + r396431;
        double r396434 = r396433 * r396433;
        double r396435 = 1.0;
        double r396436 = r396433 + r396435;
        double r396437 = r396434 * r396436;
        double r396438 = r396432 / r396437;
        return r396438;
}
↓
double f(double x, double y) {
        double r396439 = x;
        double r396440 = cbrt(r396439);
        double r396441 = r396440 * r396440;
        double r396442 = y;
        double r396443 = r396439 + r396442;
        double r396444 = cbrt(r396443);
        double r396445 = r396444 * r396444;
        double r396446 = r396441 / r396445;
        double r396447 = r396440 / r396444;
        double r396448 = r396443 / r396447;
        double r396449 = r396446 / r396448;
        double r396450 = r396449 * r396442;
        double r396451 = 1.0;
        double r396452 = r396443 + r396451;
        double r396453 = r396450 / r396452;
        return r396453;
}

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))))