Average Error: 0.0 → 0.0
Time: 3.3s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
double f(double t) {
        double r45573 = 1.0;
        double r45574 = 2.0;
        double r45575 = t;
        double r45576 = r45574 / r45575;
        double r45577 = r45573 / r45575;
        double r45578 = r45573 + r45577;
        double r45579 = r45576 / r45578;
        double r45580 = r45574 - r45579;
        double r45581 = r45580 * r45580;
        double r45582 = r45574 + r45581;
        double r45583 = r45573 / r45582;
        double r45584 = r45573 - r45583;
        return r45584;
}
↓
double f(double t) {
        double r45585 = 1.0;
        double r45586 = 2.0;
        double r45587 = t;
        double r45588 = r45586 / r45587;
        double r45589 = r45585 / r45587;
        double r45590 = r45585 + r45589;
        double r45591 = r45588 / r45590;
        double r45592 = r45586 - r45591;
        double r45593 = r45592 * r45592;
        double r45594 = r45586 + r45593;
        double r45595 = r45585 / r45594;
        double r45596 = r45585 - r45595;
        return r45596;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]

  2. Final simplification0.0

    \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))