Average Error: 19.2 → 0.1
Time: 15.4s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) + 1}}{x + y}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) + 1}}{x + y}
double f(double x, double y) {
        double r268332 = x;
        double r268333 = y;
        double r268334 = r268332 * r268333;
        double r268335 = r268332 + r268333;
        double r268336 = r268335 * r268335;
        double r268337 = 1.0;
        double r268338 = r268335 + r268337;
        double r268339 = r268336 * r268338;
        double r268340 = r268334 / r268339;
        return r268340;
}
↓
double f(double x, double y) {
        double r268341 = x;
        double r268342 = y;
        double r268343 = r268341 + r268342;
        double r268344 = r268341 / r268343;
        double r268345 = r268344 * r268342;
        double r268346 = 1.0;
        double r268347 = r268343 + r268346;
        double r268348 = r268345 / r268347;
        double r268349 = r268348 / r268343;
        return r268349;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.2
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 19.2

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
  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 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 associate-*r/0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019305 +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))))