Average Error: 19.3 → 0.2
Time: 16.2s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
↓
\[\frac{y}{1.0 + \left(y + x\right)} \cdot \left(\frac{x}{y + x} \cdot \frac{1}{y + x}\right)\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}
↓
\frac{y}{1.0 + \left(y + x\right)} \cdot \left(\frac{x}{y + x} \cdot \frac{1}{y + x}\right)
double f(double x, double y) {
        double r17112923 = x;
        double r17112924 = y;
        double r17112925 = r17112923 * r17112924;
        double r17112926 = r17112923 + r17112924;
        double r17112927 = r17112926 * r17112926;
        double r17112928 = 1.0;
        double r17112929 = r17112926 + r17112928;
        double r17112930 = r17112927 * r17112929;
        double r17112931 = r17112925 / r17112930;
        return r17112931;
}
↓
double f(double x, double y) {
        double r17112932 = y;
        double r17112933 = 1.0;
        double r17112934 = x;
        double r17112935 = r17112932 + r17112934;
        double r17112936 = r17112933 + r17112935;
        double r17112937 = r17112932 / r17112936;
        double r17112938 = r17112934 / r17112935;
        double r17112939 = 1.0;
        double r17112940 = r17112939 / r17112935;
        double r17112941 = r17112938 * r17112940;
        double r17112942 = r17112937 * r17112941;
        return r17112942;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.3
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 19.3

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

  3. Applied times-frac7.6

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

  5. Applied associate-/r*0.2

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

  7. Applied div-inv0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(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))))