Average Error: 19.5 → 0.1
Time: 14.7s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{y \cdot \frac{x}{x + y}}{x + y}}{\left(x + y\right) + 1}\]
\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{y \cdot \frac{x}{x + y}}{x + y}}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r317722 = x;
        double r317723 = y;
        double r317724 = r317722 * r317723;
        double r317725 = r317722 + r317723;
        double r317726 = r317725 * r317725;
        double r317727 = 1.0;
        double r317728 = r317725 + r317727;
        double r317729 = r317726 * r317728;
        double r317730 = r317724 / r317729;
        return r317730;
}
↓
double f(double x, double y) {
        double r317731 = y;
        double r317732 = x;
        double r317733 = r317732 + r317731;
        double r317734 = r317732 / r317733;
        double r317735 = r317731 * r317734;
        double r317736 = r317735 / r317733;
        double r317737 = 1.0;
        double r317738 = r317733 + r317737;
        double r317739 = r317736 / r317738;
        return r317739;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.5
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 19.5

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

  3. Applied times-frac7.7

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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 div-inv0.2

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

  8. Applied associate-*r*0.2

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

  10. Applied div-inv0.2

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019291 
(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))))