Average Error: 0.0 → 0.0
Time: 11.5s
Precision: 64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, \left(\left(-\frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right) + \frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right) + \left(2 - \frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right), 1\right)}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2\right)}\]
\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
\frac{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, \left(\left(-\frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right) + \frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right) + \left(2 - \frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right), 1\right)}{\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 r29601 = 1.0;
        double r29602 = 2.0;
        double r29603 = t;
        double r29604 = r29602 / r29603;
        double r29605 = r29601 / r29603;
        double r29606 = r29601 + r29605;
        double r29607 = r29604 / r29606;
        double r29608 = r29602 - r29607;
        double r29609 = r29608 * r29608;
        double r29610 = r29601 + r29609;
        double r29611 = r29602 + r29609;
        double r29612 = r29610 / r29611;
        return r29612;
}
↓
double f(double t) {
        double r29613 = 2.0;
        double r29614 = 1.0;
        double r29615 = t;
        double r29616 = fma(r29614, r29615, r29614);
        double r29617 = r29613 / r29616;
        double r29618 = r29613 - r29617;
        double r29619 = cbrt(r29616);
        double r29620 = 3.0;
        double r29621 = pow(r29619, r29620);
        double r29622 = r29613 / r29621;
        double r29623 = -r29622;
        double r29624 = r29623 + r29622;
        double r29625 = r29613 - r29622;
        double r29626 = r29624 + r29625;
        double r29627 = fma(r29618, r29626, r29614);
        double r29628 = fma(r29618, r29618, r29613);
        double r29629 = r29627 / r29628;
        return r29629;
}

Error

Bits error versus t

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  4. Applied add-cube-cbrt0.0

    \[\leadsto \frac{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 2 - \frac{2}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(1, t, \frac{1}{1}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}}}, 1\right)}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 2\right)}\]

  5. Applied add-cube-cbrt0.0

    \[\leadsto \frac{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 2 - \frac{\color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}}{\left(\sqrt[3]{\mathsf{fma}\left(1, t, \frac{1}{1}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}}, 1\right)}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 2\right)}\]

  6. Applied times-frac0.0

    \[\leadsto \frac{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 2 - \color{blue}{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{\mathsf{fma}\left(1, t, \frac{1}{1}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}}}, 1\right)}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 2\right)}\]

  7. Applied add-cube-cbrt0.8

    \[\leadsto \frac{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, \color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}} - \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{\mathsf{fma}\left(1, t, \frac{1}{1}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}}, 1\right)}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 2\right)}\]

  8. Applied prod-diff0.8

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

  9. Simplified0.0

    \[\leadsto \frac{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, \color{blue}{\left(2 - \frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right)} + \mathsf{fma}\left(-\frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}}, \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{\mathsf{fma}\left(1, t, \frac{1}{1}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}}, \frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{\mathsf{fma}\left(1, t, \frac{1}{1}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}}\right), 1\right)}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 2\right)}\]

  10. Simplified0.0

    \[\leadsto \frac{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, \left(2 - \frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right) + \color{blue}{\left(\left(-\frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right) + \frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right)}, 1\right)}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, \frac{1}{1}\right)}, 2\right)}\]

  11. Final simplification0.0

    \[\leadsto \frac{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, \left(\left(-\frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right) + \frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right) + \left(2 - \frac{2}{{\left(\sqrt[3]{\mathsf{fma}\left(1, t, 1\right)}\right)}^{3}}\right), 1\right)}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, t, 1\right)}, 2\right)}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 2"
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))