Average Error: 0.0 → 0.0
Time: 13.7s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, 1, t \cdot 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, 1, t \cdot 1\right)}, 2\right)}\]
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(2 - \frac{2}{\mathsf{fma}\left(1, 1, t \cdot 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, 1, t \cdot 1\right)}, 2\right)}
double f(double t) {
        double r1511663 = 1.0;
        double r1511664 = 2.0;
        double r1511665 = t;
        double r1511666 = r1511664 / r1511665;
        double r1511667 = r1511663 / r1511665;
        double r1511668 = r1511663 + r1511667;
        double r1511669 = r1511666 / r1511668;
        double r1511670 = r1511664 - r1511669;
        double r1511671 = r1511670 * r1511670;
        double r1511672 = r1511664 + r1511671;
        double r1511673 = r1511663 / r1511672;
        double r1511674 = r1511663 - r1511673;
        return r1511674;
}
↓
double f(double t) {
        double r1511675 = 1.0;
        double r1511676 = 2.0;
        double r1511677 = 1.0;
        double r1511678 = t;
        double r1511679 = r1511678 * r1511675;
        double r1511680 = fma(r1511677, r1511675, r1511679);
        double r1511681 = r1511676 / r1511680;
        double r1511682 = r1511676 - r1511681;
        double r1511683 = fma(r1511682, r1511682, r1511676);
        double r1511684 = r1511675 / r1511683;
        double r1511685 = r1511675 - r1511684;
        return r1511685;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 3"
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))