Average Error: 0.0 → 0.0
Time: 2.1m
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 r3677151 = 1.0;
        double r3677152 = 2.0;
        double r3677153 = t;
        double r3677154 = r3677152 / r3677153;
        double r3677155 = r3677151 / r3677153;
        double r3677156 = r3677151 + r3677155;
        double r3677157 = r3677154 / r3677156;
        double r3677158 = r3677152 - r3677157;
        double r3677159 = r3677158 * r3677158;
        double r3677160 = r3677152 + r3677159;
        double r3677161 = r3677151 / r3677160;
        double r3677162 = r3677151 - r3677161;
        return r3677162;
}
↓
double f(double t) {
        double r3677163 = 1.0;
        double r3677164 = 2.0;
        double r3677165 = t;
        double r3677166 = r3677163 + r3677165;
        double r3677167 = r3677164 / r3677166;
        double r3677168 = r3677164 - r3677167;
        double r3677169 = fma(r3677168, r3677168, r3677164);
        double r3677170 = r3677163 / r3677169;
        double r3677171 = r3677163 - r3677170;
        return r3677171;
}

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 2019121 +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)))))))))