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 r1014242 = 1.0;
        double r1014243 = 2.0;
        double r1014244 = t;
        double r1014245 = r1014243 / r1014244;
        double r1014246 = r1014242 / r1014244;
        double r1014247 = r1014242 + r1014246;
        double r1014248 = r1014245 / r1014247;
        double r1014249 = r1014243 - r1014248;
        double r1014250 = r1014249 * r1014249;
        double r1014251 = r1014243 + r1014250;
        double r1014252 = r1014242 / r1014251;
        double r1014253 = r1014242 - r1014252;
        return r1014253;
}
↓
double f(double t) {
        double r1014254 = 1.0;
        double r1014255 = 2.0;
        double r1014256 = t;
        double r1014257 = r1014254 + r1014256;
        double r1014258 = r1014255 / r1014257;
        double r1014259 = r1014255 - r1014258;
        double r1014260 = fma(r1014259, r1014259, r1014255);
        double r1014261 = r1014254 / r1014260;
        double r1014262 = r1014254 - r1014261;
        return r1014262;
}

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 2019130 +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)))))))))