Average Error: 20.2 → 0.2
Time: 11.0s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{x \cdot \frac{1}{x + y}}{x + y} \cdot y}{\left(x + y\right) + 1}\]
\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{x \cdot \frac{1}{x + y}}{x + y} \cdot y}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r310354 = x;
        double r310355 = y;
        double r310356 = r310354 * r310355;
        double r310357 = r310354 + r310355;
        double r310358 = r310357 * r310357;
        double r310359 = 1.0;
        double r310360 = r310357 + r310359;
        double r310361 = r310358 * r310360;
        double r310362 = r310356 / r310361;
        return r310362;
}
↓
double f(double x, double y) {
        double r310363 = x;
        double r310364 = 1.0;
        double r310365 = y;
        double r310366 = r310363 + r310365;
        double r310367 = r310364 / r310366;
        double r310368 = r310363 * r310367;
        double r310369 = r310368 / r310366;
        double r310370 = r310369 * r310365;
        double r310371 = 1.0;
        double r310372 = r310366 + r310371;
        double r310373 = r310370 / r310372;
        return r310373;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original20.2
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 20.2

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
  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. 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 associate-*r/0.2

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

  9. Applied div-inv0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019209 
(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))))