Average Error: 0.0 → 0.0
Time: 10.9s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{1 \cdot \left(t + 1\right)}, \mathsf{fma}\left(\sqrt{2}, \sqrt{2}, -\frac{2}{\mathsf{fma}\left(t, 1, 1\right)}\right), 2\right)}\]
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}{1 \cdot \left(t + 1\right)}, \mathsf{fma}\left(\sqrt{2}, \sqrt{2}, -\frac{2}{\mathsf{fma}\left(t, 1, 1\right)}\right), 2\right)}
double f(double t) {
        double r65804 = 1.0;
        double r65805 = 2.0;
        double r65806 = t;
        double r65807 = r65805 / r65806;
        double r65808 = r65804 / r65806;
        double r65809 = r65804 + r65808;
        double r65810 = r65807 / r65809;
        double r65811 = r65805 - r65810;
        double r65812 = r65811 * r65811;
        double r65813 = r65805 + r65812;
        double r65814 = r65804 / r65813;
        double r65815 = r65804 - r65814;
        return r65815;
}
↓
double f(double t) {
        double r65816 = 1.0;
        double r65817 = 2.0;
        double r65818 = t;
        double r65819 = 1.0;
        double r65820 = r65818 + r65819;
        double r65821 = r65816 * r65820;
        double r65822 = r65817 / r65821;
        double r65823 = r65817 - r65822;
        double r65824 = sqrt(r65817);
        double r65825 = fma(r65818, r65816, r65816);
        double r65826 = r65817 / r65825;
        double r65827 = -r65826;
        double r65828 = fma(r65824, r65824, r65827);
        double r65829 = fma(r65823, r65828, r65817);
        double r65830 = r65816 / r65829;
        double r65831 = r65816 - r65830;
        return r65831;
}

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

    \[\leadsto \color{blue}{1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{1 \cdot \left(t + 1\right)}, 2 - \frac{2}{1 \cdot \left(t + 1\right)}, 2\right)}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.0

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

  5. Applied fma-neg0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020042 +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)))))))))