Average Error: 19.8 → 0.2
Time: 15.0s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{x \cdot \frac{\frac{y}{x + y}}{x + y}}{1 + \left(x + y\right)}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{x \cdot \frac{\frac{y}{x + y}}{x + y}}{1 + \left(x + y\right)}
double f(double x, double y) {
        double r367586 = x;
        double r367587 = y;
        double r367588 = r367586 * r367587;
        double r367589 = r367586 + r367587;
        double r367590 = r367589 * r367589;
        double r367591 = 1.0;
        double r367592 = r367589 + r367591;
        double r367593 = r367590 * r367592;
        double r367594 = r367588 / r367593;
        return r367594;
}
↓
double f(double x, double y) {
        double r367595 = x;
        double r367596 = y;
        double r367597 = r367595 + r367596;
        double r367598 = r367596 / r367597;
        double r367599 = r367598 / r367597;
        double r367600 = r367595 * r367599;
        double r367601 = 1.0;
        double r367602 = r367601 + r367597;
        double r367603 = r367600 / r367602;
        return r367603;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
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 19.8

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

  2. Simplified19.8

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

  4. Applied times-frac7.8

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

  5. Simplified7.8

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

  6. Simplified0.2

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

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

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

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

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

  10. Applied times-frac0.2

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

  11. Applied associate-*l*0.2

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

  12. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019196 +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))))