Average Error: 19.6 → 0.1
Time: 3.8s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}
double f(double x, double y) {
        double r465840 = x;
        double r465841 = y;
        double r465842 = r465840 * r465841;
        double r465843 = r465840 + r465841;
        double r465844 = r465843 * r465843;
        double r465845 = 1.0;
        double r465846 = r465843 + r465845;
        double r465847 = r465844 * r465846;
        double r465848 = r465842 / r465847;
        return r465848;
}
↓
double f(double x, double y) {
        double r465849 = y;
        double r465850 = x;
        double r465851 = r465850 + r465849;
        double r465852 = 1.0;
        double r465853 = r465851 + r465852;
        double r465854 = r465849 / r465853;
        double r465855 = r465850 / r465851;
        double r465856 = r465854 * r465855;
        double r465857 = r465856 / r465851;
        return r465857;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
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 19.6

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

  3. Applied times-frac7.6

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

  5. Applied *-un-lft-identity7.6

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

  6. Applied times-frac0.2

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

  8. Applied associate-*r/0.2

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

  9. Applied associate-*l/0.2

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(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))))