Average Error: 19.7 → 0.1
Time: 17.0s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
↓
\[\frac{\frac{y}{\left(y + x\right) + 1.0}}{y + x} \cdot \frac{x}{y + x}\]
\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}{\left(y + x\right) + 1.0}}{y + x} \cdot \frac{x}{y + x}
double f(double x, double y) {
        double r21843522 = x;
        double r21843523 = y;
        double r21843524 = r21843522 * r21843523;
        double r21843525 = r21843522 + r21843523;
        double r21843526 = r21843525 * r21843525;
        double r21843527 = 1.0;
        double r21843528 = r21843525 + r21843527;
        double r21843529 = r21843526 * r21843528;
        double r21843530 = r21843524 / r21843529;
        return r21843530;
}
↓
double f(double x, double y) {
        double r21843531 = y;
        double r21843532 = x;
        double r21843533 = r21843531 + r21843532;
        double r21843534 = 1.0;
        double r21843535 = r21843533 + r21843534;
        double r21843536 = r21843531 / r21843535;
        double r21843537 = r21843536 / r21843533;
        double r21843538 = r21843532 / r21843533;
        double r21843539 = r21843537 * r21843538;
        return r21843539;
}

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

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019162 
(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))))