Average Error: 19.9 → 0.5
Time: 4.9s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\left(\sqrt[3]{\frac{x}{x + y}} \cdot \sqrt[3]{\frac{x}{x + y}}\right) \cdot \sqrt[3]{\frac{x}{x + y}}}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\left(\sqrt[3]{\frac{x}{x + y}} \cdot \sqrt[3]{\frac{x}{x + y}}\right) \cdot \sqrt[3]{\frac{x}{x + y}}}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r371424 = x;
        double r371425 = y;
        double r371426 = r371424 * r371425;
        double r371427 = r371424 + r371425;
        double r371428 = r371427 * r371427;
        double r371429 = 1.0;
        double r371430 = r371427 + r371429;
        double r371431 = r371428 * r371430;
        double r371432 = r371426 / r371431;
        return r371432;
}
↓
double f(double x, double y) {
        double r371433 = x;
        double r371434 = y;
        double r371435 = r371433 + r371434;
        double r371436 = r371433 / r371435;
        double r371437 = cbrt(r371436);
        double r371438 = r371437 * r371437;
        double r371439 = r371438 * r371437;
        double r371440 = r371439 / r371435;
        double r371441 = 1.0;
        double r371442 = r371435 + r371441;
        double r371443 = r371434 / r371442;
        double r371444 = r371440 * r371443;
        return r371444;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
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 19.9

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

  3. Applied times-frac7.9

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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 add-cube-cbrt0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2020021 +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))))