Average Error: 20.0 → 0.2
Time: 13.4s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{y}{x + \left(1 + y\right)} \cdot \frac{\frac{1}{x + y} \cdot x}{x + y}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{y}{x + \left(1 + y\right)} \cdot \frac{\frac{1}{x + y} \cdot x}{x + y}
double f(double x, double y) {
        double r260675 = x;
        double r260676 = y;
        double r260677 = r260675 * r260676;
        double r260678 = r260675 + r260676;
        double r260679 = r260678 * r260678;
        double r260680 = 1.0;
        double r260681 = r260678 + r260680;
        double r260682 = r260679 * r260681;
        double r260683 = r260677 / r260682;
        return r260683;
}
↓
double f(double x, double y) {
        double r260684 = y;
        double r260685 = x;
        double r260686 = 1.0;
        double r260687 = r260686 + r260684;
        double r260688 = r260685 + r260687;
        double r260689 = r260684 / r260688;
        double r260690 = 1.0;
        double r260691 = r260685 + r260684;
        double r260692 = r260690 / r260691;
        double r260693 = r260692 * r260685;
        double r260694 = r260693 / r260691;
        double r260695 = r260689 * r260694;
        return r260695;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
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 20.0

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
  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. Simplified0.2

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

  5. Simplified0.2

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

  7. Applied div-inv0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019179 +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))))