Average Error: 20.0 → 0.1
Time: 13.6s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{1 + \left(x + y\right)} \cdot \frac{y}{x + y}}{x + y}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\frac{x}{1 + \left(x + y\right)} \cdot \frac{y}{x + y}}{x + y}
double f(double x, double y) {
        double r427726 = x;
        double r427727 = y;
        double r427728 = r427726 * r427727;
        double r427729 = r427726 + r427727;
        double r427730 = r427729 * r427729;
        double r427731 = 1.0;
        double r427732 = r427729 + r427731;
        double r427733 = r427730 * r427732;
        double r427734 = r427728 / r427733;
        return r427734;
}
↓
double f(double x, double y) {
        double r427735 = x;
        double r427736 = 1.0;
        double r427737 = y;
        double r427738 = r427735 + r427737;
        double r427739 = r427736 + r427738;
        double r427740 = r427735 / r427739;
        double r427741 = r427737 / r427738;
        double r427742 = r427740 * r427741;
        double r427743 = r427742 / r427738;
        return r427743;
}

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

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. Simplified20.0

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

  4. Applied times-frac7.8

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

  5. Simplified7.8

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

  6. Simplified0.2

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

  8. Applied associate-*r/0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019194 
(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))))