Average Error: 19.4 → 0.1
Time: 4.5s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{x \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}}{x + y}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{x \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}}{x + y}
double f(double x, double y) {
        double r475778 = x;
        double r475779 = y;
        double r475780 = r475778 * r475779;
        double r475781 = r475778 + r475779;
        double r475782 = r475781 * r475781;
        double r475783 = 1.0;
        double r475784 = r475781 + r475783;
        double r475785 = r475782 * r475784;
        double r475786 = r475780 / r475785;
        return r475786;
}
↓
double f(double x, double y) {
        double r475787 = x;
        double r475788 = y;
        double r475789 = r475787 + r475788;
        double r475790 = 1.0;
        double r475791 = r475789 + r475790;
        double r475792 = r475788 / r475791;
        double r475793 = r475792 / r475789;
        double r475794 = r475787 * r475793;
        double r475795 = r475794 / r475789;
        return r475795;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.4
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 19.4

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

  3. Applied times-frac7.3

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

  5. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \cdot \frac{y}{\left(x + y\right) + 1}\]
  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 div-inv0.2

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

  10. Applied associate-*l*0.2

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020035 
(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))))