Average Error: 20.2 → 0.1
Time: 16.5s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{y}{\left(y + x\right) + 1}}{y + x} \cdot \frac{x}{y + x}\]
\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 + x\right) + 1}}{y + x} \cdot \frac{x}{y + x}
double f(double x, double y) {
        double r18349316 = x;
        double r18349317 = y;
        double r18349318 = r18349316 * r18349317;
        double r18349319 = r18349316 + r18349317;
        double r18349320 = r18349319 * r18349319;
        double r18349321 = 1.0;
        double r18349322 = r18349319 + r18349321;
        double r18349323 = r18349320 * r18349322;
        double r18349324 = r18349318 / r18349323;
        return r18349324;
}
↓
double f(double x, double y) {
        double r18349325 = y;
        double r18349326 = x;
        double r18349327 = r18349325 + r18349326;
        double r18349328 = 1.0;
        double r18349329 = r18349327 + r18349328;
        double r18349330 = r18349325 / r18349329;
        double r18349331 = r18349330 / r18349327;
        double r18349332 = r18349326 / r18349327;
        double r18349333 = r18349331 * r18349332;
        return r18349333;
}

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

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-frac8.2

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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 div-inv0.2

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

  8. Applied associate-*l*0.2

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019192 +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))))