Average Error: 19.8 → 0.8
Time: 23.9s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
↓
\[\left(\left(\frac{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{y}}{\sqrt[3]{x + y}} \cdot \frac{\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{x + y}}}{\sqrt[3]{x + y}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{1.0 + \left(x + y\right)}{\sqrt[3]{y}}}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}
↓
\left(\left(\frac{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{y}}{\sqrt[3]{x + y}} \cdot \frac{\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{x + y}}}{\sqrt[3]{x + y}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{1.0 + \left(x + y\right)}{\sqrt[3]{y}}}
double f(double x, double y) {
        double r23705796 = x;
        double r23705797 = y;
        double r23705798 = r23705796 * r23705797;
        double r23705799 = r23705796 + r23705797;
        double r23705800 = r23705799 * r23705799;
        double r23705801 = 1.0;
        double r23705802 = r23705799 + r23705801;
        double r23705803 = r23705800 * r23705802;
        double r23705804 = r23705798 / r23705803;
        return r23705804;
}
↓
double f(double x, double y) {
        double r23705805 = x;
        double r23705806 = cbrt(r23705805);
        double r23705807 = cbrt(r23705806);
        double r23705808 = y;
        double r23705809 = cbrt(r23705808);
        double r23705810 = r23705807 * r23705809;
        double r23705811 = r23705805 + r23705808;
        double r23705812 = cbrt(r23705811);
        double r23705813 = r23705810 / r23705812;
        double r23705814 = r23705806 * r23705806;
        double r23705815 = cbrt(r23705814);
        double r23705816 = r23705815 / r23705812;
        double r23705817 = r23705816 / r23705812;
        double r23705818 = r23705813 * r23705817;
        double r23705819 = r23705811 / r23705809;
        double r23705820 = r23705806 / r23705819;
        double r23705821 = r23705818 * r23705820;
        double r23705822 = 1.0;
        double r23705823 = r23705822 + r23705811;
        double r23705824 = r23705823 / r23705809;
        double r23705825 = r23705806 / r23705824;
        double r23705826 = r23705821 * r23705825;
        return r23705826;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.1
Herbie0.8
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 19.8

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
  2. Using strategy rm

  3. Applied associate-/l*11.6

    \[\leadsto \color{blue}{\frac{x}{\frac{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}{y}}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt12.0

    \[\leadsto \frac{x}{\frac{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}}\]

  6. Applied times-frac10.0

    \[\leadsto \frac{x}{\color{blue}{\frac{\left(x + y\right) \cdot \left(x + y\right)}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}}\]

  7. Applied add-cube-cbrt10.1

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{\left(x + y\right) \cdot \left(x + y\right)}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}\]

  8. Applied times-frac8.3

    \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{\left(x + y\right) \cdot \left(x + y\right)}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}}\]

  9. Simplified0.9

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{y}}}\right)} \cdot \frac{\sqrt[3]{x}}{\frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}\]
  10. Using strategy rm

  11. Applied *-un-lft-identity0.9

    \[\leadsto \left(\frac{\sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{\color{blue}{1 \cdot y}}}} \cdot \frac{\sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}\]

  12. Applied cbrt-prod0.9

    \[\leadsto \left(\frac{\sqrt[3]{x}}{\frac{x + y}{\color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{y}}}} \cdot \frac{\sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}\]

  13. Applied add-cube-cbrt0.8

    \[\leadsto \left(\frac{\sqrt[3]{x}}{\frac{\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}{\sqrt[3]{1} \cdot \sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}\]

  14. Applied times-frac0.8

    \[\leadsto \left(\frac{\sqrt[3]{x}}{\color{blue}{\frac{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}{\sqrt[3]{1}} \cdot \frac{\sqrt[3]{x + y}}{\sqrt[3]{y}}}} \cdot \frac{\sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}\]

  15. Applied add-cube-cbrt0.8

    \[\leadsto \left(\frac{\sqrt[3]{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}{\frac{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}{\sqrt[3]{1}} \cdot \frac{\sqrt[3]{x + y}}{\sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}\]

  16. Applied cbrt-prod0.8

    \[\leadsto \left(\frac{\color{blue}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}}{\frac{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}{\sqrt[3]{1}} \cdot \frac{\sqrt[3]{x + y}}{\sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}\]

  17. Applied times-frac0.8

    \[\leadsto \left(\color{blue}{\left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\frac{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}{\sqrt[3]{1}}} \cdot \frac{\sqrt[3]{\sqrt[3]{x}}}{\frac{\sqrt[3]{x + y}}{\sqrt[3]{y}}}\right)} \cdot \frac{\sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}\]

  18. Simplified0.8

    \[\leadsto \left(\left(\color{blue}{\frac{\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{y + x}}}{\sqrt[3]{y + x}}} \cdot \frac{\sqrt[3]{\sqrt[3]{x}}}{\frac{\sqrt[3]{x + y}}{\sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}\]

  19. Simplified0.8

    \[\leadsto \left(\left(\frac{\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{y + x}}}{\sqrt[3]{y + x}} \cdot \color{blue}{\frac{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{y}}{\sqrt[3]{y + x}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}\]

  20. Final simplification0.8

    \[\leadsto \left(\left(\frac{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{y}}{\sqrt[3]{x + y}} \cdot \frac{\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{x + y}}}{\sqrt[3]{x + y}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{1.0 + \left(x + y\right)}{\sqrt[3]{y}}}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"

  :herbie-target
  (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))