Average Error: 19.7 → 0.5
Time: 9.4s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{1}{\frac{x + y}{\frac{x}{x + y}}} \cdot \frac{y}{\left(x + y\right) + 1}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{1}{\frac{x + y}{\frac{x}{x + y}}} \cdot \frac{y}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r400318 = x;
        double r400319 = y;
        double r400320 = r400318 * r400319;
        double r400321 = r400318 + r400319;
        double r400322 = r400321 * r400321;
        double r400323 = 1.0;
        double r400324 = r400321 + r400323;
        double r400325 = r400322 * r400324;
        double r400326 = r400320 / r400325;
        return r400326;
}
↓
double f(double x, double y) {
        double r400327 = 1.0;
        double r400328 = x;
        double r400329 = y;
        double r400330 = r400328 + r400329;
        double r400331 = r400328 / r400330;
        double r400332 = r400330 / r400331;
        double r400333 = r400327 / r400332;
        double r400334 = 1.0;
        double r400335 = r400330 + r400334;
        double r400336 = r400329 / r400335;
        double r400337 = r400333 * r400336;
        return r400337;
}

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
Herbie0.5
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 19.7

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
  2. Using strategy rm

  3. Applied times-frac7.5

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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 *-un-lft-identity0.2

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

  8. Applied *-un-lft-identity0.2

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

  9. Applied times-frac0.2

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

  10. Applied associate-/l*0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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))))