Average Error: 20.2 → 0.2
Time: 16.1s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{y \cdot \frac{\frac{x}{y + x}}{y + x}}{\left(y + x\right) + 1}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{y \cdot \frac{\frac{x}{y + x}}{y + x}}{\left(y + x\right) + 1}
double f(double x, double y) {
        double r16278517 = x;
        double r16278518 = y;
        double r16278519 = r16278517 * r16278518;
        double r16278520 = r16278517 + r16278518;
        double r16278521 = r16278520 * r16278520;
        double r16278522 = 1.0;
        double r16278523 = r16278520 + r16278522;
        double r16278524 = r16278521 * r16278523;
        double r16278525 = r16278519 / r16278524;
        return r16278525;
}
↓
double f(double x, double y) {
        double r16278526 = y;
        double r16278527 = x;
        double r16278528 = r16278526 + r16278527;
        double r16278529 = r16278527 / r16278528;
        double r16278530 = r16278529 / r16278528;
        double r16278531 = r16278526 * r16278530;
        double r16278532 = 1.0;
        double r16278533 = r16278528 + r16278532;
        double r16278534 = r16278531 / r16278533;
        return r16278534;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.2
Target0.2
Herbie0.2
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 20.2

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
  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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019172 +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))))