Average Error: 0.0 → 0.0
Time: 12.7s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{1 + t}, 2 - \frac{2}{1 + t}, 2\right)}\]
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}{1 + t}, 2 - \frac{2}{1 + t}, 2\right)}
double f(double t) {
        double r1385279 = 1.0;
        double r1385280 = 2.0;
        double r1385281 = t;
        double r1385282 = r1385280 / r1385281;
        double r1385283 = r1385279 / r1385281;
        double r1385284 = r1385279 + r1385283;
        double r1385285 = r1385282 / r1385284;
        double r1385286 = r1385280 - r1385285;
        double r1385287 = r1385286 * r1385286;
        double r1385288 = r1385280 + r1385287;
        double r1385289 = r1385279 / r1385288;
        double r1385290 = r1385279 - r1385289;
        return r1385290;
}
↓
double f(double t) {
        double r1385291 = 1.0;
        double r1385292 = 2.0;
        double r1385293 = t;
        double r1385294 = r1385291 + r1385293;
        double r1385295 = r1385292 / r1385294;
        double r1385296 = r1385292 - r1385295;
        double r1385297 = fma(r1385296, r1385296, r1385292);
        double r1385298 = r1385291 / r1385297;
        double r1385299 = r1385291 - r1385298;
        return r1385299;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019162 +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)))))))))