Average Error: 0.0 → 0.0
Time: 18.1s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2\right)}\]
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, t, 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2\right)}
double f(double t) {
        double r31642 = 1.0;
        double r31643 = 2.0;
        double r31644 = t;
        double r31645 = r31643 / r31644;
        double r31646 = r31642 / r31644;
        double r31647 = r31642 + r31646;
        double r31648 = r31645 / r31647;
        double r31649 = r31643 - r31648;
        double r31650 = r31649 * r31649;
        double r31651 = r31643 + r31650;
        double r31652 = r31642 / r31651;
        double r31653 = r31642 - r31652;
        return r31653;
}
↓
double f(double t) {
        double r31654 = 1.0;
        double r31655 = 2.0;
        double r31656 = t;
        double r31657 = fma(r31654, r31656, r31654);
        double r31658 = r31655 / r31657;
        double r31659 = r31655 - r31658;
        double r31660 = fma(r31659, r31659, r31655);
        double r31661 = r31654 / r31660;
        double r31662 = r31654 - r31661;
        return r31662;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))