Average Error: 19.4 → 0.2
Time: 17.2s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
↓
\[\frac{y}{1.0 + \left(y + x\right)} \cdot \frac{\frac{x}{y + x}}{y + x}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}
↓
\frac{y}{1.0 + \left(y + x\right)} \cdot \frac{\frac{x}{y + x}}{y + x}
double f(double x, double y) {
        double r20961460 = x;
        double r20961461 = y;
        double r20961462 = r20961460 * r20961461;
        double r20961463 = r20961460 + r20961461;
        double r20961464 = r20961463 * r20961463;
        double r20961465 = 1.0;
        double r20961466 = r20961463 + r20961465;
        double r20961467 = r20961464 * r20961466;
        double r20961468 = r20961462 / r20961467;
        return r20961468;
}
↓
double f(double x, double y) {
        double r20961469 = y;
        double r20961470 = 1.0;
        double r20961471 = x;
        double r20961472 = r20961469 + r20961471;
        double r20961473 = r20961470 + r20961472;
        double r20961474 = r20961469 / r20961473;
        double r20961475 = r20961471 / r20961472;
        double r20961476 = r20961475 / r20961472;
        double r20961477 = r20961474 * r20961476;
        return r20961477;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.4
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 19.4

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
  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 *-un-lft-identity7.6

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

  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. Using strategy rm

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

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

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

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

  10. Applied times-frac0.2

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

  11. Applied associate-*r*0.2

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

  12. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019164 
(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))))