Average Error: 0.0 → 0.0
Time: 10.4s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2\right)}\]
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}{\mathsf{fma}\left(1, t, 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2\right)}
double f(double t) {
        double r1475896 = 1.0;
        double r1475897 = 2.0;
        double r1475898 = t;
        double r1475899 = r1475897 / r1475898;
        double r1475900 = r1475896 / r1475898;
        double r1475901 = r1475896 + r1475900;
        double r1475902 = r1475899 / r1475901;
        double r1475903 = r1475897 - r1475902;
        double r1475904 = r1475903 * r1475903;
        double r1475905 = r1475897 + r1475904;
        double r1475906 = r1475896 / r1475905;
        double r1475907 = r1475896 - r1475906;
        return r1475907;
}
↓
double f(double t) {
        double r1475908 = 1.0;
        double r1475909 = 2.0;
        double r1475910 = t;
        double r1475911 = fma(r1475908, r1475910, r1475908);
        double r1475912 = r1475909 / r1475911;
        double r1475913 = r1475909 - r1475912;
        double r1475914 = fma(r1475913, r1475913, r1475909);
        double r1475915 = r1475908 / r1475914;
        double r1475916 = r1475908 - r1475915;
        return r1475916;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 3"
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))