Average Error: 19.8 → 0.2
Time: 13.6s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{\frac{x}{x + y}}{x + y} \cdot y}{\left(x + y\right) + 1}\]
\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{\frac{x}{x + y}}{x + y} \cdot y}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r323564 = x;
        double r323565 = y;
        double r323566 = r323564 * r323565;
        double r323567 = r323564 + r323565;
        double r323568 = r323567 * r323567;
        double r323569 = 1.0;
        double r323570 = r323567 + r323569;
        double r323571 = r323568 * r323570;
        double r323572 = r323566 / r323571;
        return r323572;
}
↓
double f(double x, double y) {
        double r323573 = x;
        double r323574 = y;
        double r323575 = r323573 + r323574;
        double r323576 = r323573 / r323575;
        double r323577 = r323576 / r323575;
        double r323578 = r323577 * r323574;
        double r323579 = 1.0;
        double r323580 = r323575 + r323579;
        double r323581 = r323578 / r323580;
        return r323581;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
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 19.8

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

    \[\leadsto \frac{\frac{x}{x + y}}{x + y} \cdot \color{blue}{\frac{1}{\frac{\left(x + y\right) + 1}{y}}}\]
  8. Using strategy rm

  9. Applied div-inv0.2

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

  10. Applied associate-*l*0.2

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

  11. Simplified0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019303 
(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))))