Average Error: 20.0 → 0.1
Time: 13.1s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{\frac{y}{x + y}}{\frac{x + y}{x}}}{1 + \left(x + y\right)}\]
\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}{x + y}}{\frac{x + y}{x}}}{1 + \left(x + y\right)}
double f(double x, double y) {
        double r370765 = x;
        double r370766 = y;
        double r370767 = r370765 * r370766;
        double r370768 = r370765 + r370766;
        double r370769 = r370768 * r370768;
        double r370770 = 1.0;
        double r370771 = r370768 + r370770;
        double r370772 = r370769 * r370771;
        double r370773 = r370767 / r370772;
        return r370773;
}
↓
double f(double x, double y) {
        double r370774 = y;
        double r370775 = x;
        double r370776 = r370775 + r370774;
        double r370777 = r370774 / r370776;
        double r370778 = r370776 / r370775;
        double r370779 = r370777 / r370778;
        double r370780 = 1.0;
        double r370781 = r370780 + r370776;
        double r370782 = r370779 / r370781;
        return r370782;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
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 20.0

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

  2. Simplified20.0

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

  4. Applied times-frac8.0

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

  5. Simplified8.0

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

  6. Simplified0.2

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

  8. Applied associate-*l/0.2

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

  9. Simplified0.1

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

  11. Applied div-inv0.2

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

  12. Simplified0.2

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

  14. Applied *-un-lft-identity0.2

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

  15. Applied associate-/r*0.2

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

  16. Simplified0.1

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

  17. Final simplification0.1

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

Reproduce

herbie shell --seed 2019179 
(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))))