Average Error: 20.4 → 0.1
Time: 14.1s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{1}{\left(x + y\right) + 1} \cdot \frac{\left(\frac{y}{x + y} \cdot \frac{x}{x - y}\right) \cdot \left(x - y\right)}{x + y}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{1}{\left(x + y\right) + 1} \cdot \frac{\left(\frac{y}{x + y} \cdot \frac{x}{x - y}\right) \cdot \left(x - y\right)}{x + y}
double f(double x, double y) {
        double r341631 = x;
        double r341632 = y;
        double r341633 = r341631 * r341632;
        double r341634 = r341631 + r341632;
        double r341635 = r341634 * r341634;
        double r341636 = 1.0;
        double r341637 = r341634 + r341636;
        double r341638 = r341635 * r341637;
        double r341639 = r341633 / r341638;
        return r341639;
}
↓
double f(double x, double y) {
        double r341640 = 1.0;
        double r341641 = x;
        double r341642 = y;
        double r341643 = r341641 + r341642;
        double r341644 = 1.0;
        double r341645 = r341643 + r341644;
        double r341646 = r341640 / r341645;
        double r341647 = r341642 / r341643;
        double r341648 = r341641 - r341642;
        double r341649 = r341641 / r341648;
        double r341650 = r341647 * r341649;
        double r341651 = r341650 * r341648;
        double r341652 = r341651 / r341643;
        double r341653 = r341646 * r341652;
        return r341653;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.4
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 20.4

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

  3. Applied times-frac8.1

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

  4. Simplified8.1

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

  5. Simplified8.1

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

  7. Applied div-inv8.1

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

  8. Applied associate-*r*8.1

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

  9. Simplified0.1

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

  11. Applied flip-+10.2

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

  12. Applied associate-/r/10.2

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

  13. Applied associate-*r*10.2

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2019195 
(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))))