Average Error: 0.0 → 0.0
Time: 14.6s
Precision: 64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 1\right)}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(\left(2 - \frac{\frac{2}{\sqrt[3]{1 + t}}}{\sqrt[3]{1 + t} \cdot \sqrt[3]{1 + t}}\right) + \mathsf{fma}\left(\left(\frac{-1}{\sqrt[3]{1 + t} \cdot \sqrt[3]{1 + t}}\right), \left(\frac{2}{\sqrt[3]{1 + t}}\right), \left(\frac{\frac{2}{\sqrt[3]{1 + t}}}{\sqrt[3]{1 + t} \cdot \sqrt[3]{1 + t}}\right)\right)\right), 2\right)}\]
\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(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 1\right)}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(\left(2 - \frac{\frac{2}{\sqrt[3]{1 + t}}}{\sqrt[3]{1 + t} \cdot \sqrt[3]{1 + t}}\right) + \mathsf{fma}\left(\left(\frac{-1}{\sqrt[3]{1 + t} \cdot \sqrt[3]{1 + t}}\right), \left(\frac{2}{\sqrt[3]{1 + t}}\right), \left(\frac{\frac{2}{\sqrt[3]{1 + t}}}{\sqrt[3]{1 + t} \cdot \sqrt[3]{1 + t}}\right)\right)\right), 2\right)}
double f(double t) {
        double r1313735 = 1.0;
        double r1313736 = 2.0;
        double r1313737 = t;
        double r1313738 = r1313736 / r1313737;
        double r1313739 = r1313735 / r1313737;
        double r1313740 = r1313735 + r1313739;
        double r1313741 = r1313738 / r1313740;
        double r1313742 = r1313736 - r1313741;
        double r1313743 = r1313742 * r1313742;
        double r1313744 = r1313735 + r1313743;
        double r1313745 = r1313736 + r1313743;
        double r1313746 = r1313744 / r1313745;
        return r1313746;
}
↓
double f(double t) {
        double r1313747 = 2.0;
        double r1313748 = 1.0;
        double r1313749 = t;
        double r1313750 = r1313748 + r1313749;
        double r1313751 = r1313747 / r1313750;
        double r1313752 = r1313747 - r1313751;
        double r1313753 = fma(r1313752, r1313752, r1313748);
        double r1313754 = cbrt(r1313750);
        double r1313755 = r1313747 / r1313754;
        double r1313756 = r1313754 * r1313754;
        double r1313757 = r1313755 / r1313756;
        double r1313758 = r1313747 - r1313757;
        double r1313759 = -1.0;
        double r1313760 = r1313759 / r1313756;
        double r1313761 = fma(r1313760, r1313755, r1313757);
        double r1313762 = r1313758 + r1313761;
        double r1313763 = fma(r1313752, r1313762, r1313747);
        double r1313764 = r1313753 / r1313763;
        return r1313764;
}

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

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied *-un-lft-identity0.0

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

  6. Applied times-frac0.0

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

  7. Applied *-un-lft-identity0.0

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

  8. Applied prod-diff0.0

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

  9. Simplified0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 2"
  (/ (+ 1 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))) (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))))))