Average Error: 0.0 → 0.0
Time: 51.4s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{(\left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right) + 2)_*}\]
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}{(\left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right) + 2)_*}
double f(double t) {
        double r1412428 = 1.0;
        double r1412429 = 2.0;
        double r1412430 = t;
        double r1412431 = r1412429 / r1412430;
        double r1412432 = r1412428 / r1412430;
        double r1412433 = r1412428 + r1412432;
        double r1412434 = r1412431 / r1412433;
        double r1412435 = r1412429 - r1412434;
        double r1412436 = r1412435 * r1412435;
        double r1412437 = r1412429 + r1412436;
        double r1412438 = r1412428 / r1412437;
        double r1412439 = r1412428 - r1412438;
        return r1412439;
}
↓
double f(double t) {
        double r1412440 = 1.0;
        double r1412441 = 2.0;
        double r1412442 = t;
        double r1412443 = r1412440 + r1412442;
        double r1412444 = r1412441 / r1412443;
        double r1412445 = r1412441 - r1412444;
        double r1412446 = fma(r1412445, r1412445, r1412441);
        double r1412447 = r1412440 / r1412446;
        double r1412448 = r1412440 - r1412447;
        return r1412448;
}

Error

Bits error versus t

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. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 3"
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))