Average Error: 0.0 → 0.0
Time: 6.5s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}\]
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(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}
double f(double t) {
        double r834141 = 1.0;
        double r834142 = 2.0;
        double r834143 = t;
        double r834144 = r834142 / r834143;
        double r834145 = r834141 / r834143;
        double r834146 = r834141 + r834145;
        double r834147 = r834144 / r834146;
        double r834148 = r834142 - r834147;
        double r834149 = r834148 * r834148;
        double r834150 = r834142 + r834149;
        double r834151 = r834141 / r834150;
        double r834152 = r834141 - r834151;
        return r834152;
}
↓
double f(double t) {
        double r834153 = 1.0;
        double r834154 = 2.0;
        double r834155 = t;
        double r834156 = r834153 + r834155;
        double r834157 = r834154 / r834156;
        double r834158 = r834154 - r834157;
        double r834159 = fma(r834158, r834158, r834154);
        double r834160 = r834153 / r834159;
        double r834161 = r834153 - r834160;
        return r834161;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019132 +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)))))))))