Average Error: 0.0 → 0.0
Time: 5.9s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}\]
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}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}
double f(double t) {
        double r924554 = 1.0;
        double r924555 = 2.0;
        double r924556 = t;
        double r924557 = r924555 / r924556;
        double r924558 = r924554 / r924556;
        double r924559 = r924554 + r924558;
        double r924560 = r924557 / r924559;
        double r924561 = r924555 - r924560;
        double r924562 = r924561 * r924561;
        double r924563 = r924555 + r924562;
        double r924564 = r924554 / r924563;
        double r924565 = r924554 - r924564;
        return r924565;
}
↓
double f(double t) {
        double r924566 = 1.0;
        double r924567 = 2.0;
        double r924568 = t;
        double r924569 = r924566 + r924568;
        double r924570 = r924567 / r924569;
        double r924571 = r924567 - r924570;
        double r924572 = fma(r924571, r924571, r924567);
        double r924573 = r924566 / r924572;
        double r924574 = r924566 - r924573;
        return r924574;
}

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

    \[\leadsto \color{blue}{1 - \frac{1}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}}\]

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019133 +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)))))))))