Average Error: 19.3 → 0.2
Time: 16.6s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
↓
\[\frac{\left(\frac{x}{y + x} \cdot \frac{1}{y + x}\right) \cdot y}{1.0 + \left(y + x\right)}\]
\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{\left(\frac{x}{y + x} \cdot \frac{1}{y + x}\right) \cdot y}{1.0 + \left(y + x\right)}
double f(double x, double y) {
        double r19049392 = x;
        double r19049393 = y;
        double r19049394 = r19049392 * r19049393;
        double r19049395 = r19049392 + r19049393;
        double r19049396 = r19049395 * r19049395;
        double r19049397 = 1.0;
        double r19049398 = r19049395 + r19049397;
        double r19049399 = r19049396 * r19049398;
        double r19049400 = r19049394 / r19049399;
        return r19049400;
}
↓
double f(double x, double y) {
        double r19049401 = x;
        double r19049402 = y;
        double r19049403 = r19049402 + r19049401;
        double r19049404 = r19049401 / r19049403;
        double r19049405 = 1.0;
        double r19049406 = r19049405 / r19049403;
        double r19049407 = r19049404 * r19049406;
        double r19049408 = r19049407 * r19049402;
        double r19049409 = 1.0;
        double r19049410 = r19049409 + r19049403;
        double r19049411 = r19049408 / r19049410;
        return r19049411;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.3
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 19.3

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

  3. Applied times-frac7.6

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1.0}}\]
  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 associate-*r/0.1

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

  9. Applied div-inv0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019163 
(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))))