Average Error: 19.8 → 0.2
Time: 16.3s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{\frac{x}{x + y}}{x + y}}{\frac{\left(x + y\right) + 1}{y}}\]
\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{x}{x + y}}{x + y}}{\frac{\left(x + y\right) + 1}{y}}
double f(double x, double y) {
        double r245687 = x;
        double r245688 = y;
        double r245689 = r245687 * r245688;
        double r245690 = r245687 + r245688;
        double r245691 = r245690 * r245690;
        double r245692 = 1.0;
        double r245693 = r245690 + r245692;
        double r245694 = r245691 * r245693;
        double r245695 = r245689 / r245694;
        return r245695;
}
↓
double f(double x, double y) {
        double r245696 = x;
        double r245697 = y;
        double r245698 = r245696 + r245697;
        double r245699 = r245696 / r245698;
        double r245700 = r245699 / r245698;
        double r245701 = 1.0;
        double r245702 = r245698 + r245701;
        double r245703 = r245702 / r245697;
        double r245704 = r245700 / r245703;
        return r245704;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.1
Herbie0.2
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 19.8

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

  3. Applied times-frac8.0

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

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019306 +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))))