Average Error: 0.0 → 0.0
Time: 8.9s
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 r1316653 = 1.0;
        double r1316654 = 2.0;
        double r1316655 = t;
        double r1316656 = r1316654 / r1316655;
        double r1316657 = r1316653 / r1316655;
        double r1316658 = r1316653 + r1316657;
        double r1316659 = r1316656 / r1316658;
        double r1316660 = r1316654 - r1316659;
        double r1316661 = r1316660 * r1316660;
        double r1316662 = r1316654 + r1316661;
        double r1316663 = r1316653 / r1316662;
        double r1316664 = r1316653 - r1316663;
        return r1316664;
}
↓
double f(double t) {
        double r1316665 = 1.0;
        double r1316666 = 2.0;
        double r1316667 = t;
        double r1316668 = r1316665 + r1316667;
        double r1316669 = r1316666 / r1316668;
        double r1316670 = r1316666 - r1316669;
        double r1316671 = fma(r1316670, r1316670, r1316666);
        double r1316672 = r1316665 / r1316671;
        double r1316673 = r1316665 - r1316672;
        return r1316673;
}

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 2019158 +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)))))))))