Average Error: 0.0 → 0.0
Time: 4.1m
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 r8551166 = 1.0;
        double r8551167 = 2.0;
        double r8551168 = t;
        double r8551169 = r8551167 / r8551168;
        double r8551170 = r8551166 / r8551168;
        double r8551171 = r8551166 + r8551170;
        double r8551172 = r8551169 / r8551171;
        double r8551173 = r8551167 - r8551172;
        double r8551174 = r8551173 * r8551173;
        double r8551175 = r8551167 + r8551174;
        double r8551176 = r8551166 / r8551175;
        double r8551177 = r8551166 - r8551176;
        return r8551177;
}
↓
double f(double t) {
        double r8551178 = 1.0;
        double r8551179 = 2.0;
        double r8551180 = t;
        double r8551181 = r8551178 + r8551180;
        double r8551182 = r8551179 / r8551181;
        double r8551183 = r8551179 - r8551182;
        double r8551184 = fma(r8551183, r8551183, r8551179);
        double r8551185 = r8551178 / r8551184;
        double r8551186 = log1p(r8551185);
        double r8551187 = expm1(r8551186);
        double r8551188 = r8551178 - r8551187;
        return r8551188;
}

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 2019128 +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)))))))))