Average Error: 19.3 → 0.1
Time: 15.1s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
↓
\[\frac{\frac{\frac{x}{y + x} \cdot y}{\left(y + x\right) + 1.0}}{y + x}\]
\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{\frac{x}{y + x} \cdot y}{\left(y + x\right) + 1.0}}{y + x}
double f(double x, double y) {
        double r18497324 = x;
        double r18497325 = y;
        double r18497326 = r18497324 * r18497325;
        double r18497327 = r18497324 + r18497325;
        double r18497328 = r18497327 * r18497327;
        double r18497329 = 1.0;
        double r18497330 = r18497327 + r18497329;
        double r18497331 = r18497328 * r18497330;
        double r18497332 = r18497326 / r18497331;
        return r18497332;
}
↓
double f(double x, double y) {
        double r18497333 = x;
        double r18497334 = y;
        double r18497335 = r18497334 + r18497333;
        double r18497336 = r18497333 / r18497335;
        double r18497337 = r18497336 * r18497334;
        double r18497338 = 1.0;
        double r18497339 = r18497335 + r18497338;
        double r18497340 = r18497337 / r18497339;
        double r18497341 = r18497340 / r18497335;
        return r18497341;
}

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

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-frac8.0

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

  5. Applied *-un-lft-identity8.0

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

  6. Applied times-frac0.2

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

  7. Applied associate-*l*0.2

    \[\leadsto \color{blue}{\frac{1}{x + y} \cdot \left(\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) + 1.0}\right)}\]
  8. Using strategy rm

  9. Applied associate-*l/7.2

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

  10. Applied associate-*r/7.2

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019168 
(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))))