Average Error: 20.0 → 0.2
Time: 17.1s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}\]
\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{x}{x + y}}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r322934 = x;
        double r322935 = y;
        double r322936 = r322934 * r322935;
        double r322937 = r322934 + r322935;
        double r322938 = r322937 * r322937;
        double r322939 = 1.0;
        double r322940 = r322937 + r322939;
        double r322941 = r322938 * r322940;
        double r322942 = r322936 / r322941;
        return r322942;
}
↓
double f(double x, double y) {
        double r322943 = x;
        double r322944 = y;
        double r322945 = r322943 + r322944;
        double r322946 = r322943 / r322945;
        double r322947 = r322946 / r322945;
        double r322948 = 1.0;
        double r322949 = r322945 + r322948;
        double r322950 = r322944 / r322949;
        double r322951 = r322947 * r322950;
        return r322951;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
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 20.0

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

  3. Applied times-frac7.9

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 +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))))