Average Error: 0.0 → 0.0
Time: 6.0s
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 r53553 = 1.0;
        double r53554 = 2.0;
        double r53555 = t;
        double r53556 = r53554 / r53555;
        double r53557 = r53553 / r53555;
        double r53558 = r53553 + r53557;
        double r53559 = r53556 / r53558;
        double r53560 = r53554 - r53559;
        double r53561 = r53560 * r53560;
        double r53562 = r53554 + r53561;
        double r53563 = r53553 / r53562;
        double r53564 = r53553 - r53563;
        return r53564;
}
↓
double f(double t) {
        double r53565 = 1.0;
        double r53566 = 2.0;
        double r53567 = t;
        double r53568 = r53566 / r53567;
        double r53569 = r53565 / r53567;
        double r53570 = r53565 + r53569;
        double r53571 = r53568 / r53570;
        double r53572 = r53566 - r53571;
        double r53573 = r53572 * r53572;
        double r53574 = r53566 + r53573;
        double r53575 = r53565 / r53574;
        double r53576 = r53565 - r53575;
        return r53576;
}

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 2020049 +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)))))))))