Average Error: 19.5 → 0.4
Time: 48.6s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{x + y}}{\frac{x + y}{\frac{y}{1 + \left(x + y\right)}}}\]
\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}}{\frac{x + y}{\frac{y}{1 + \left(x + y\right)}}}
double f(double x, double y) {
        double r21763645 = x;
        double r21763646 = y;
        double r21763647 = r21763645 * r21763646;
        double r21763648 = r21763645 + r21763646;
        double r21763649 = r21763648 * r21763648;
        double r21763650 = 1.0;
        double r21763651 = r21763648 + r21763650;
        double r21763652 = r21763649 * r21763651;
        double r21763653 = r21763647 / r21763652;
        return r21763653;
}
↓
double f(double x, double y) {
        double r21763654 = x;
        double r21763655 = y;
        double r21763656 = r21763654 + r21763655;
        double r21763657 = r21763654 / r21763656;
        double r21763658 = 1.0;
        double r21763659 = r21763658 + r21763656;
        double r21763660 = r21763655 / r21763659;
        double r21763661 = r21763656 / r21763660;
        double r21763662 = r21763657 / r21763661;
        return r21763662;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.5
Target0.1
Herbie0.4
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 19.5

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

  3. Applied times-frac8.1

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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-*l/0.1

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

  9. Applied associate-/l*0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))