Average Error: 0.0 → 0.0
Time: 4.7s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{1 \cdot \left(t + 1\right)}, 2 - \frac{2}{1 \cdot \left(t + 1\right)}, 2\right)}\]
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(2 - \frac{2}{1 \cdot \left(t + 1\right)}, 2 - \frac{2}{1 \cdot \left(t + 1\right)}, 2\right)}
double f(double t) {
        double r29354 = 1.0;
        double r29355 = 2.0;
        double r29356 = t;
        double r29357 = r29355 / r29356;
        double r29358 = r29354 / r29356;
        double r29359 = r29354 + r29358;
        double r29360 = r29357 / r29359;
        double r29361 = r29355 - r29360;
        double r29362 = r29361 * r29361;
        double r29363 = r29355 + r29362;
        double r29364 = r29354 / r29363;
        double r29365 = r29354 - r29364;
        return r29365;
}
↓
double f(double t) {
        double r29366 = 1.0;
        double r29367 = 2.0;
        double r29368 = t;
        double r29369 = 1.0;
        double r29370 = r29368 + r29369;
        double r29371 = r29366 * r29370;
        double r29372 = r29367 / r29371;
        double r29373 = r29367 - r29372;
        double r29374 = fma(r29373, r29373, r29367);
        double r29375 = r29366 / r29374;
        double r29376 = r29366 - r29375;
        return r29376;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +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)))))))))