Average Error: 19.6 → 15.3
Time: 7.0s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{\frac{y \cdot x}{x + y}}{\left(x + y\right) + 1}}{x + y}\]
\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{\frac{y \cdot x}{x + y}}{\left(x + y\right) + 1}}{x + y}
double f(double x, double y) {
        double r338331 = x;
        double r338332 = y;
        double r338333 = r338331 * r338332;
        double r338334 = r338331 + r338332;
        double r338335 = r338334 * r338334;
        double r338336 = 1.0;
        double r338337 = r338334 + r338336;
        double r338338 = r338335 * r338337;
        double r338339 = r338333 / r338338;
        return r338339;
}
↓
double f(double x, double y) {
        double r338340 = y;
        double r338341 = x;
        double r338342 = r338340 * r338341;
        double r338343 = r338341 + r338340;
        double r338344 = r338342 / r338343;
        double r338345 = 1.0;
        double r338346 = r338343 + r338345;
        double r338347 = r338344 / r338346;
        double r338348 = r338347 / r338343;
        return r338348;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
Target0.1
Herbie15.3
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 19.6

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

  3. Applied times-frac7.9

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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 clear-num0.2

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

  10. Final simplification15.3

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

Reproduce

herbie shell --seed 2019308 
(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))))