Average Error: 0.0 → 0.0
Time: 8.7s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, 1, t \cdot 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, 1, t \cdot 1\right)}, 2\right)}\]
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, 1, t \cdot 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, 1, t \cdot 1\right)}, 2\right)}
double f(double t) {
        double r33143 = 1.0;
        double r33144 = 2.0;
        double r33145 = t;
        double r33146 = r33144 / r33145;
        double r33147 = r33143 / r33145;
        double r33148 = r33143 + r33147;
        double r33149 = r33146 / r33148;
        double r33150 = r33144 - r33149;
        double r33151 = r33150 * r33150;
        double r33152 = r33144 + r33151;
        double r33153 = r33143 / r33152;
        double r33154 = r33143 - r33153;
        return r33154;
}
↓
double f(double t) {
        double r33155 = 1.0;
        double r33156 = 2.0;
        double r33157 = 1.0;
        double r33158 = t;
        double r33159 = r33158 * r33155;
        double r33160 = fma(r33157, r33155, r33159);
        double r33161 = r33156 / r33160;
        double r33162 = r33156 - r33161;
        double r33163 = fma(r33162, r33162, r33156);
        double r33164 = r33155 / r33163;
        double r33165 = r33155 - r33164;
        return r33165;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019194 +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)))))))))