Average Error: 20.0 → 0.2
Time: 3.9s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x}{x + y}}{x + y} \cdot \frac{1}{\frac{\left(x + y\right) + 1}{y}}\]
\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{x}{x + y}}{x + y} \cdot \frac{1}{\frac{\left(x + y\right) + 1}{y}}
double f(double x, double y) {
        double r451074 = x;
        double r451075 = y;
        double r451076 = r451074 * r451075;
        double r451077 = r451074 + r451075;
        double r451078 = r451077 * r451077;
        double r451079 = 1.0;
        double r451080 = r451077 + r451079;
        double r451081 = r451078 * r451080;
        double r451082 = r451076 / r451081;
        return r451082;
}
↓
double f(double x, double y) {
        double r451083 = x;
        double r451084 = y;
        double r451085 = r451083 + r451084;
        double r451086 = r451083 / r451085;
        double r451087 = r451086 / r451085;
        double r451088 = 1.0;
        double r451089 = 1.0;
        double r451090 = r451085 + r451089;
        double r451091 = r451090 / r451084;
        double r451092 = r451088 / r451091;
        double r451093 = r451087 * r451092;
        return r451093;
}

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-frac8.1

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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 clear-num0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2020060 
(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))))