Average Error: 19.9 → 0.2
Time: 10.8s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{\frac{x}{x + y}}{x + y} \cdot y}{\left(x + y\right) + 1}\]
\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{\frac{x}{x + y}}{x + y} \cdot y}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r371794 = x;
        double r371795 = y;
        double r371796 = r371794 * r371795;
        double r371797 = r371794 + r371795;
        double r371798 = r371797 * r371797;
        double r371799 = 1.0;
        double r371800 = r371797 + r371799;
        double r371801 = r371798 * r371800;
        double r371802 = r371796 / r371801;
        return r371802;
}
↓
double f(double x, double y) {
        double r371803 = x;
        double r371804 = y;
        double r371805 = r371803 + r371804;
        double r371806 = r371803 / r371805;
        double r371807 = r371806 / r371805;
        double r371808 = r371807 * r371804;
        double r371809 = 1.0;
        double r371810 = r371805 + r371809;
        double r371811 = r371808 / r371810;
        return r371811;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
Target0.1
Herbie0.2
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 19.9

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

  3. Applied times-frac8.0

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

  5. Applied *-un-lft-identity8.0

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

  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

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2020047 +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))))