Average Error: 0.0 → 0.0
Time: 5.8s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{2 + \mathsf{fma}\left(\sqrt{2}, \sqrt{2}, -\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
1 - \frac{1}{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}{2 + \mathsf{fma}\left(\sqrt{2}, \sqrt{2}, -\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
double f(double t) {
        double r63885 = 1.0;
        double r63886 = 2.0;
        double r63887 = t;
        double r63888 = r63886 / r63887;
        double r63889 = r63885 / r63887;
        double r63890 = r63885 + r63889;
        double r63891 = r63888 / r63890;
        double r63892 = r63886 - r63891;
        double r63893 = r63892 * r63892;
        double r63894 = r63886 + r63893;
        double r63895 = r63885 / r63894;
        double r63896 = r63885 - r63895;
        return r63896;
}
↓
double f(double t) {
        double r63897 = 1.0;
        double r63898 = 2.0;
        double r63899 = sqrt(r63898);
        double r63900 = t;
        double r63901 = r63898 / r63900;
        double r63902 = r63897 / r63900;
        double r63903 = r63897 + r63902;
        double r63904 = r63901 / r63903;
        double r63905 = -r63904;
        double r63906 = fma(r63899, r63899, r63905);
        double r63907 = r63898 - r63904;
        double r63908 = r63906 * r63907;
        double r63909 = r63898 + r63908;
        double r63910 = r63897 / r63909;
        double r63911 = r63897 - r63910;
        return r63911;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.0

    \[\leadsto 1 - \frac{1}{2 + \left(\color{blue}{\sqrt{2} \cdot \sqrt{2}} - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]

  4. Applied fma-neg0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020060 +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)))))))))