Average Error: 0.0 → 0.0
Time: 23.7s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(\frac{8 - \frac{2}{t + 1} \cdot \left(\frac{2}{t + 1} \cdot \frac{2}{t + 1}\right)}{\mathsf{fma}\left(\frac{2}{t + 1} + 2, \frac{2}{t + 1}, 4\right)}, 2 - \frac{2}{t + 1}, 2\right)}\]
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(\frac{8 - \frac{2}{t + 1} \cdot \left(\frac{2}{t + 1} \cdot \frac{2}{t + 1}\right)}{\mathsf{fma}\left(\frac{2}{t + 1} + 2, \frac{2}{t + 1}, 4\right)}, 2 - \frac{2}{t + 1}, 2\right)}
double f(double t) {
        double r1818919 = 1.0;
        double r1818920 = 2.0;
        double r1818921 = t;
        double r1818922 = r1818920 / r1818921;
        double r1818923 = r1818919 / r1818921;
        double r1818924 = r1818919 + r1818923;
        double r1818925 = r1818922 / r1818924;
        double r1818926 = r1818920 - r1818925;
        double r1818927 = r1818926 * r1818926;
        double r1818928 = r1818920 + r1818927;
        double r1818929 = r1818919 / r1818928;
        double r1818930 = r1818919 - r1818929;
        return r1818930;
}
↓
double f(double t) {
        double r1818931 = 1.0;
        double r1818932 = 8.0;
        double r1818933 = 2.0;
        double r1818934 = t;
        double r1818935 = r1818934 + r1818931;
        double r1818936 = r1818933 / r1818935;
        double r1818937 = r1818936 * r1818936;
        double r1818938 = r1818936 * r1818937;
        double r1818939 = r1818932 - r1818938;
        double r1818940 = r1818936 + r1818933;
        double r1818941 = 4.0;
        double r1818942 = fma(r1818940, r1818936, r1818941);
        double r1818943 = r1818939 / r1818942;
        double r1818944 = r1818933 - r1818936;
        double r1818945 = fma(r1818943, r1818944, r1818933);
        double r1818946 = r1818931 / r1818945;
        double r1818947 = r1818931 - r1818946;
        return r1818947;
}

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

    \[\leadsto \color{blue}{1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{1 + t}, 2 - \frac{2}{1 + t}, 2\right)}}\]
  3. Using strategy rm

  4. Applied flip3--0.0

    \[\leadsto 1 - \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{{2}^{3} - {\left(\frac{2}{1 + t}\right)}^{3}}{2 \cdot 2 + \left(\frac{2}{1 + t} \cdot \frac{2}{1 + t} + 2 \cdot \frac{2}{1 + t}\right)}}, 2 - \frac{2}{1 + t}, 2\right)}\]

  5. Simplified0.0

    \[\leadsto 1 - \frac{1}{\mathsf{fma}\left(\frac{\color{blue}{8 - \left(\frac{2}{1 + t} \cdot \frac{2}{1 + t}\right) \cdot \frac{2}{1 + t}}}{2 \cdot 2 + \left(\frac{2}{1 + t} \cdot \frac{2}{1 + t} + 2 \cdot \frac{2}{1 + t}\right)}, 2 - \frac{2}{1 + t}, 2\right)}\]

  6. Simplified0.0

    \[\leadsto 1 - \frac{1}{\mathsf{fma}\left(\frac{8 - \left(\frac{2}{1 + t} \cdot \frac{2}{1 + t}\right) \cdot \frac{2}{1 + t}}{\color{blue}{\mathsf{fma}\left(2 + \frac{2}{1 + t}, \frac{2}{1 + t}, 4\right)}}, 2 - \frac{2}{1 + t}, 2\right)}\]

  7. Final simplification0.0

    \[\leadsto 1 - \frac{1}{\mathsf{fma}\left(\frac{8 - \frac{2}{t + 1} \cdot \left(\frac{2}{t + 1} \cdot \frac{2}{t + 1}\right)}{\mathsf{fma}\left(\frac{2}{t + 1} + 2, \frac{2}{t + 1}, 4\right)}, 2 - \frac{2}{t + 1}, 2\right)}\]

Reproduce

herbie shell --seed 2019139 +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)))))))))