Average Error: 0.0 → 0.0
Time: 20.0s
Precision: 64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 1\right)}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2\right)}\]
\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
\frac{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 1\right)}{\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 r1754896 = 1.0;
        double r1754897 = 2.0;
        double r1754898 = t;
        double r1754899 = r1754897 / r1754898;
        double r1754900 = r1754896 / r1754898;
        double r1754901 = r1754896 + r1754900;
        double r1754902 = r1754899 / r1754901;
        double r1754903 = r1754897 - r1754902;
        double r1754904 = r1754903 * r1754903;
        double r1754905 = r1754896 + r1754904;
        double r1754906 = r1754897 + r1754904;
        double r1754907 = r1754905 / r1754906;
        return r1754907;
}
↓
double f(double t) {
        double r1754908 = 2.0;
        double r1754909 = 1.0;
        double r1754910 = t;
        double r1754911 = fma(r1754909, r1754910, r1754909);
        double r1754912 = r1754908 / r1754911;
        double r1754913 = r1754908 - r1754912;
        double r1754914 = fma(r1754913, r1754913, r1754909);
        double r1754915 = fma(r1754913, r1754913, r1754908);
        double r1754916 = r1754914 / r1754915;
        return r1754916;
}

Error

Bits error versus t

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 2"
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))