Average Error: 0.0 → 0.1
Time: 6.3s
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 r57760 = 1.0;
        double r57761 = 2.0;
        double r57762 = t;
        double r57763 = r57761 / r57762;
        double r57764 = r57760 / r57762;
        double r57765 = r57760 + r57764;
        double r57766 = r57763 / r57765;
        double r57767 = r57761 - r57766;
        double r57768 = r57767 * r57767;
        double r57769 = r57761 + r57768;
        double r57770 = r57760 / r57769;
        double r57771 = r57760 - r57770;
        return r57771;
}
↓
double f(double t) {
        double r57772 = 1.0;
        double r57773 = 2.0;
        double r57774 = sqrt(r57773);
        double r57775 = t;
        double r57776 = r57773 / r57775;
        double r57777 = r57772 / r57775;
        double r57778 = r57772 + r57777;
        double r57779 = r57776 / r57778;
        double r57780 = -r57779;
        double r57781 = fma(r57774, r57774, r57780);
        double r57782 = r57773 - r57779;
        double r57783 = r57781 * r57782;
        double r57784 = r57773 + r57783;
        double r57785 = r57772 / r57784;
        double r57786 = r57772 - r57785;
        return r57786;
}

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-sqrt0.1

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

  4. Applied fma-neg0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2020083 +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)))))))))