Average Error: 19.2 → 0.1
Time: 14.0s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) + 1}}{x + y}\]
\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} \cdot y}{\left(x + y\right) + 1}}{x + y}
double f(double x, double y) {
        double r268630 = x;
        double r268631 = y;
        double r268632 = r268630 * r268631;
        double r268633 = r268630 + r268631;
        double r268634 = r268633 * r268633;
        double r268635 = 1.0;
        double r268636 = r268633 + r268635;
        double r268637 = r268634 * r268636;
        double r268638 = r268632 / r268637;
        return r268638;
}
↓
double f(double x, double y) {
        double r268639 = x;
        double r268640 = y;
        double r268641 = r268639 + r268640;
        double r268642 = r268639 / r268641;
        double r268643 = r268642 * r268640;
        double r268644 = 1.0;
        double r268645 = r268641 + r268644;
        double r268646 = r268643 / r268645;
        double r268647 = r268646 / r268641;
        return r268647;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 19.2

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

  3. Applied times-frac7.6

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

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019305 
(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))))