Average Error: 0.0 → 0.0
Time: 11.2s
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 r1132325 = 1.0;
        double r1132326 = 2.0;
        double r1132327 = t;
        double r1132328 = r1132326 / r1132327;
        double r1132329 = r1132325 / r1132327;
        double r1132330 = r1132325 + r1132329;
        double r1132331 = r1132328 / r1132330;
        double r1132332 = r1132326 - r1132331;
        double r1132333 = r1132332 * r1132332;
        double r1132334 = r1132326 + r1132333;
        double r1132335 = r1132325 / r1132334;
        double r1132336 = r1132325 - r1132335;
        return r1132336;
}
↓
double f(double t) {
        double r1132337 = 1.0;
        double r1132338 = 2.0;
        double r1132339 = t;
        double r1132340 = r1132337 + r1132339;
        double r1132341 = r1132338 / r1132340;
        double r1132342 = r1132338 - r1132341;
        double r1132343 = fma(r1132342, r1132342, r1132338);
        double r1132344 = r1132337 / r1132343;
        double r1132345 = r1132337 - r1132344;
        return r1132345;
}

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 2019129 +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)))))))))