Average Error: 0.0 → 0.0
Time: 6.1s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{2 + \left(\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot 2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \frac{-{\left(\frac{\sqrt[3]{2}}{\sqrt[3]{t}}\right)}^{3}}{1 + \frac{1}{t}}\right) + \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \frac{-{\left(\frac{\sqrt[3]{2}}{\sqrt[3]{t}}\right)}^{3}}{1 + \frac{1}{t}} + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{1 + \frac{1}{t}}\right)\right)}\]
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 + \left(\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot 2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \frac{-{\left(\frac{\sqrt[3]{2}}{\sqrt[3]{t}}\right)}^{3}}{1 + \frac{1}{t}}\right) + \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \frac{-{\left(\frac{\sqrt[3]{2}}{\sqrt[3]{t}}\right)}^{3}}{1 + \frac{1}{t}} + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{1 + \frac{1}{t}}\right)\right)}
double f(double t) {
        double r49662 = 1.0;
        double r49663 = 2.0;
        double r49664 = t;
        double r49665 = r49663 / r49664;
        double r49666 = r49662 / r49664;
        double r49667 = r49662 + r49666;
        double r49668 = r49665 / r49667;
        double r49669 = r49663 - r49668;
        double r49670 = r49669 * r49669;
        double r49671 = r49663 + r49670;
        double r49672 = r49662 / r49671;
        double r49673 = r49662 - r49672;
        return r49673;
}
↓
double f(double t) {
        double r49674 = 1.0;
        double r49675 = 2.0;
        double r49676 = t;
        double r49677 = r49675 / r49676;
        double r49678 = r49674 / r49676;
        double r49679 = r49674 + r49678;
        double r49680 = r49677 / r49679;
        double r49681 = r49675 - r49680;
        double r49682 = r49681 * r49675;
        double r49683 = cbrt(r49675);
        double r49684 = cbrt(r49676);
        double r49685 = r49683 / r49684;
        double r49686 = 3.0;
        double r49687 = pow(r49685, r49686);
        double r49688 = -r49687;
        double r49689 = r49688 / r49679;
        double r49690 = r49681 * r49689;
        double r49691 = r49682 + r49690;
        double r49692 = r49683 * r49683;
        double r49693 = r49684 * r49684;
        double r49694 = r49692 / r49693;
        double r49695 = r49685 * r49694;
        double r49696 = r49695 / r49679;
        double r49697 = r49681 * r49696;
        double r49698 = r49690 + r49697;
        double r49699 = r49691 + r49698;
        double r49700 = r49675 + r49699;
        double r49701 = r49674 / r49700;
        double r49702 = r49674 - r49701;
        return r49702;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 *-un-lft-identity0.0

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied add-cube-cbrt0.0

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

  6. Applied times-frac0.0

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

  7. Applied times-frac0.0

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

  8. Applied add-sqr-sqrt0.0

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

  9. Applied prod-diff0.0

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

  10. Applied distribute-lft-in0.0

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

  11. Simplified0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +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)))))))))