Average Error: 19.8 → 0.2
Time: 13.3s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{y + x}}{y + x} \cdot \frac{1}{\frac{1 + \left(y + x\right)}{y}}\]
\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{x}{y + x}}{y + x} \cdot \frac{1}{\frac{1 + \left(y + x\right)}{y}}
double f(double x, double y) {
        double r303690 = x;
        double r303691 = y;
        double r303692 = r303690 * r303691;
        double r303693 = r303690 + r303691;
        double r303694 = r303693 * r303693;
        double r303695 = 1.0;
        double r303696 = r303693 + r303695;
        double r303697 = r303694 * r303696;
        double r303698 = r303692 / r303697;
        return r303698;
}
↓
double f(double x, double y) {
        double r303699 = x;
        double r303700 = y;
        double r303701 = r303700 + r303699;
        double r303702 = r303699 / r303701;
        double r303703 = r303702 / r303701;
        double r303704 = 1.0;
        double r303705 = 1.0;
        double r303706 = r303705 + r303701;
        double r303707 = r303706 / r303700;
        double r303708 = r303704 / r303707;
        double r303709 = r303703 * r303708;
        return r303709;
}

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-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. Simplified7.8

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

  5. Simplified7.8

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

  7. Applied associate-/r*0.2

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

  8. Simplified0.2

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

  10. Applied clear-num0.2

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

  11. Simplified0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019196 
(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))))