Average Error: 0.0 → 0.0
Time: 2.4m
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{1}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}\right)\right)\right)\right)\]
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 - \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{1}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}\right)\right)\right)\right)
double f(double t) {
        double r6198378 = 1.0;
        double r6198379 = 2.0;
        double r6198380 = t;
        double r6198381 = r6198379 / r6198380;
        double r6198382 = r6198378 / r6198380;
        double r6198383 = r6198378 + r6198382;
        double r6198384 = r6198381 / r6198383;
        double r6198385 = r6198379 - r6198384;
        double r6198386 = r6198385 * r6198385;
        double r6198387 = r6198379 + r6198386;
        double r6198388 = r6198378 / r6198387;
        double r6198389 = r6198378 - r6198388;
        return r6198389;
}
↓
double f(double t) {
        double r6198390 = 1.0;
        double r6198391 = 2.0;
        double r6198392 = t;
        double r6198393 = r6198390 + r6198392;
        double r6198394 = r6198391 / r6198393;
        double r6198395 = r6198391 - r6198394;
        double r6198396 = fma(r6198395, r6198395, r6198391);
        double r6198397 = r6198390 / r6198396;
        double r6198398 = log1p(r6198397);
        double r6198399 = expm1(r6198398);
        double r6198400 = r6198390 - r6198399;
        return r6198400;
}

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. Using strategy rm

  4. Applied expm1-log1p-u0.0

    \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{1}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}\right)\right)\right)\right)}\]

  5. Final simplification0.0

    \[\leadsto 1 - \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{1}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019124 +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)))))))))