Average Error: 20.0 → 0.2
Time: 4.3s
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 r532699 = x;
        double r532700 = y;
        double r532701 = r532699 * r532700;
        double r532702 = r532699 + r532700;
        double r532703 = r532702 * r532702;
        double r532704 = 1.0;
        double r532705 = r532702 + r532704;
        double r532706 = r532703 * r532705;
        double r532707 = r532701 / r532706;
        return r532707;
}
↓
double f(double x, double y) {
        double r532708 = x;
        double r532709 = y;
        double r532710 = r532708 + r532709;
        double r532711 = r532708 / r532710;
        double r532712 = r532711 / r532710;
        double r532713 = 1.0;
        double r532714 = 1.0;
        double r532715 = r532710 + r532714;
        double r532716 = r532715 / r532709;
        double r532717 = r532713 / r532716;
        double r532718 = r532712 * r532717;
        return r532718;
}

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 +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))))