Average Error: 19.7 → 0.1
Time: 5.0s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{x + y} \cdot \frac{y}{x + y}}{\left(x + y\right) + 1}\]
\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}{x + y} \cdot \frac{y}{x + y}}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r401860 = x;
        double r401861 = y;
        double r401862 = r401860 * r401861;
        double r401863 = r401860 + r401861;
        double r401864 = r401863 * r401863;
        double r401865 = 1.0;
        double r401866 = r401863 + r401865;
        double r401867 = r401864 * r401866;
        double r401868 = r401862 / r401867;
        return r401868;
}
↓
double f(double x, double y) {
        double r401869 = x;
        double r401870 = y;
        double r401871 = r401869 + r401870;
        double r401872 = r401869 / r401871;
        double r401873 = r401870 / r401871;
        double r401874 = r401872 * r401873;
        double r401875 = 1.0;
        double r401876 = r401871 + r401875;
        double r401877 = r401874 / r401876;
        return r401877;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.7
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 19.7

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

  5. Applied associate-/r*0.2

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

  7. Applied associate-*r/0.2

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

  9. Applied div-inv0.2

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

  10. Applied associate-*l*0.2

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019356 
(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))))