Average Error: 19.6 → 0.1
Time: 13.6s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{y}{\left(y + 1\right) + x}}{y + x} \cdot \frac{x}{y + x}\]
\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(y + 1\right) + x}}{y + x} \cdot \frac{x}{y + x}
double f(double x, double y) {
        double r448602 = x;
        double r448603 = y;
        double r448604 = r448602 * r448603;
        double r448605 = r448602 + r448603;
        double r448606 = r448605 * r448605;
        double r448607 = 1.0;
        double r448608 = r448605 + r448607;
        double r448609 = r448606 * r448608;
        double r448610 = r448604 / r448609;
        return r448610;
}
↓
double f(double x, double y) {
        double r448611 = y;
        double r448612 = 1.0;
        double r448613 = r448611 + r448612;
        double r448614 = x;
        double r448615 = r448613 + r448614;
        double r448616 = r448611 / r448615;
        double r448617 = r448611 + r448614;
        double r448618 = r448616 / r448617;
        double r448619 = r448614 / r448617;
        double r448620 = r448618 * r448619;
        return r448620;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
Target0.1
Herbie0.1
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 19.6

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

  3. Applied times-frac7.8

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  7. Applied div-inv0.2

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

  8. Applied associate-*l*0.2

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))