Average Error: 20.2 → 0.4
Time: 17.0s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{y + x}}{\frac{y + x}{\frac{y}{\left(y + x\right) + 1}}}\]
\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{x}{y + x}}{\frac{y + x}{\frac{y}{\left(y + x\right) + 1}}}
double f(double x, double y) {
        double r15461652 = x;
        double r15461653 = y;
        double r15461654 = r15461652 * r15461653;
        double r15461655 = r15461652 + r15461653;
        double r15461656 = r15461655 * r15461655;
        double r15461657 = 1.0;
        double r15461658 = r15461655 + r15461657;
        double r15461659 = r15461656 * r15461658;
        double r15461660 = r15461654 / r15461659;
        return r15461660;
}
↓
double f(double x, double y) {
        double r15461661 = x;
        double r15461662 = y;
        double r15461663 = r15461662 + r15461661;
        double r15461664 = r15461661 / r15461663;
        double r15461665 = 1.0;
        double r15461666 = r15461663 + r15461665;
        double r15461667 = r15461662 / r15461666;
        double r15461668 = r15461663 / r15461667;
        double r15461669 = r15461664 / r15461668;
        return r15461669;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.2
Target0.2
Herbie0.4
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 20.2

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

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

Reproduce

herbie shell --seed 2019172 +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))))