Average Error: 19.7 → 0.1
Time: 4.0s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}
double f(double x, double y) {
        double r364989 = x;
        double r364990 = y;
        double r364991 = r364989 * r364990;
        double r364992 = r364989 + r364990;
        double r364993 = r364992 * r364992;
        double r364994 = 1.0;
        double r364995 = r364992 + r364994;
        double r364996 = r364993 * r364995;
        double r364997 = r364991 / r364996;
        return r364997;
}
↓
double f(double x, double y) {
        double r364998 = x;
        double r364999 = y;
        double r365000 = r364998 + r364999;
        double r365001 = r364998 / r365000;
        double r365002 = 1.0;
        double r365003 = r365000 + r365002;
        double r365004 = r364999 / r365003;
        double r365005 = r365004 / r365000;
        double r365006 = r365001 * r365005;
        return r365006;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.7
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 19.7

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
  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 associate-/r*0.2

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

  7. Applied associate-*l/0.1

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

  9. Applied *-un-lft-identity0.1

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

  10. Applied times-frac0.1

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020062 
(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))))