Average Error: 19.2 → 0.2
Time: 16.1s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
↓
\[\frac{\frac{x}{y + x}}{y + x} \cdot \frac{1}{\frac{\left(y + x\right) + 1.0}{y}}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}
↓
\frac{\frac{x}{y + x}}{y + x} \cdot \frac{1}{\frac{\left(y + x\right) + 1.0}{y}}
double f(double x, double y) {
        double r20274734 = x;
        double r20274735 = y;
        double r20274736 = r20274734 * r20274735;
        double r20274737 = r20274734 + r20274735;
        double r20274738 = r20274737 * r20274737;
        double r20274739 = 1.0;
        double r20274740 = r20274737 + r20274739;
        double r20274741 = r20274738 * r20274740;
        double r20274742 = r20274736 / r20274741;
        return r20274742;
}
↓
double f(double x, double y) {
        double r20274743 = x;
        double r20274744 = y;
        double r20274745 = r20274744 + r20274743;
        double r20274746 = r20274743 / r20274745;
        double r20274747 = r20274746 / r20274745;
        double r20274748 = 1.0;
        double r20274749 = 1.0;
        double r20274750 = r20274745 + r20274749;
        double r20274751 = r20274750 / r20274744;
        double r20274752 = r20274748 / r20274751;
        double r20274753 = r20274747 * r20274752;
        return r20274753;
}

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

Derivation

  1. Initial program 19.2

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
  2. Using strategy rm

  3. Applied times-frac7.7

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1.0}}\]
  4. Using strategy rm

  5. Applied associate-/r*0.2

    \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \cdot \frac{y}{\left(x + y\right) + 1.0}\]
  6. Using strategy rm

  7. Applied clear-num0.2

    \[\leadsto \frac{\frac{x}{x + y}}{x + y} \cdot \color{blue}{\frac{1}{\frac{\left(x + y\right) + 1.0}{y}}}\]

  8. Final simplification0.2

    \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot \frac{1}{\frac{\left(y + x\right) + 1.0}{y}}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"

  :herbie-target
  (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))