Average Error: 19.4 → 0.2
Time: 13.1s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{1}{\frac{\left(x + y\right) + 1}{y}} \cdot \frac{x}{x + y}}{x + y}\]
\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{1}{\frac{\left(x + y\right) + 1}{y}} \cdot \frac{x}{x + y}}{x + y}
double f(double x, double y) {
        double r371498 = x;
        double r371499 = y;
        double r371500 = r371498 * r371499;
        double r371501 = r371498 + r371499;
        double r371502 = r371501 * r371501;
        double r371503 = 1.0;
        double r371504 = r371501 + r371503;
        double r371505 = r371502 * r371504;
        double r371506 = r371500 / r371505;
        return r371506;
}
↓
double f(double x, double y) {
        double r371507 = 1.0;
        double r371508 = x;
        double r371509 = y;
        double r371510 = r371508 + r371509;
        double r371511 = 1.0;
        double r371512 = r371510 + r371511;
        double r371513 = r371512 / r371509;
        double r371514 = r371507 / r371513;
        double r371515 = r371508 / r371510;
        double r371516 = r371514 * r371515;
        double r371517 = r371516 / r371510;
        return r371517;
}

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
Target0.2
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

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
  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 *-un-lft-identity7.8

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

  6. Applied times-frac0.2

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

  7. Applied associate-*l*0.2

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

  9. Applied associate-*l/7.2

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

  10. Applied associate-*r/7.2

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

  11. Simplified0.1

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

  13. Applied clear-num0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2019235 
(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))))