Average Error: 20.0 → 11.5
Time: 12.4s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{y}{\left(x + y\right) + 1} \cdot x}{\left(x + y\right) \cdot \left(x + y\right)}\]
\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{y}{\left(x + y\right) + 1} \cdot x}{\left(x + y\right) \cdot \left(x + y\right)}
double f(double x, double y) {
        double r313060 = x;
        double r313061 = y;
        double r313062 = r313060 * r313061;
        double r313063 = r313060 + r313061;
        double r313064 = r313063 * r313063;
        double r313065 = 1.0;
        double r313066 = r313063 + r313065;
        double r313067 = r313064 * r313066;
        double r313068 = r313062 / r313067;
        return r313068;
}
↓
double f(double x, double y) {
        double r313069 = y;
        double r313070 = x;
        double r313071 = r313070 + r313069;
        double r313072 = 1.0;
        double r313073 = r313071 + r313072;
        double r313074 = r313069 / r313073;
        double r313075 = r313074 * r313070;
        double r313076 = r313071 * r313071;
        double r313077 = r313075 / r313076;
        return r313077;
}

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
Herbie11.5
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

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-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 div-inv0.2

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

  8. Final simplification11.5

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

Reproduce

herbie shell --seed 2019297 
(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))))