Average Error: 19.7 → 0.1
Time: 17.7s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
↓
\[\frac{\frac{y}{y + \left(x + 1.0\right)}}{y + x} \cdot \frac{x}{y + x}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}
↓
\frac{\frac{y}{y + \left(x + 1.0\right)}}{y + x} \cdot \frac{x}{y + x}
double f(double x, double y) {
        double r19169493 = x;
        double r19169494 = y;
        double r19169495 = r19169493 * r19169494;
        double r19169496 = r19169493 + r19169494;
        double r19169497 = r19169496 * r19169496;
        double r19169498 = 1.0;
        double r19169499 = r19169496 + r19169498;
        double r19169500 = r19169497 * r19169499;
        double r19169501 = r19169495 / r19169500;
        return r19169501;
}
↓
double f(double x, double y) {
        double r19169502 = y;
        double r19169503 = x;
        double r19169504 = 1.0;
        double r19169505 = r19169503 + r19169504;
        double r19169506 = r19169502 + r19169505;
        double r19169507 = r19169502 / r19169506;
        double r19169508 = r19169502 + r19169503;
        double r19169509 = r19169507 / r19169508;
        double r19169510 = r19169503 / r19169508;
        double r19169511 = r19169509 * r19169510;
        return r19169511;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.7
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 19.7

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
  2. Using strategy rm

  3. Applied times-frac8.1

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1.0}}\]
  4. Using strategy rm

  5. Applied associate-/r*0.2

    \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \cdot \frac{y}{\left(x + y\right) + 1.0}\]
  6. Using strategy rm

  7. Applied div-inv0.2

    \[\leadsto \color{blue}{\left(\frac{x}{x + y} \cdot \frac{1}{x + y}\right)} \cdot \frac{y}{\left(x + y\right) + 1.0}\]

  8. Applied associate-*l*0.2

    \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \left(\frac{1}{x + y} \cdot \frac{y}{\left(x + y\right) + 1.0}\right)}\]

  9. Simplified0.1

    \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{\frac{y}{\left(1.0 + x\right) + y}}{x + y}}\]

  10. Final simplification0.1

    \[\leadsto \frac{\frac{y}{y + \left(x + 1.0\right)}}{y + x} \cdot \frac{x}{y + x}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"

  :herbie-target
  (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x))))

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