Average Error: 47.1 → 17.4
Time: 4.7m
Precision: 64
Internal precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{{\ell}^2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^2\right) - 1\right)}\]
⬇
\[\begin{array}{l} \mathbf{if}\;k \le -8.399687047689758 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(\sqrt[3]{\tan k} \cdot \left(\sqrt[3]{\frac{k}{t}} \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{t \cdot \sin k}}{\sqrt[3]{t} \cdot \sqrt[3]{{\left(\frac{\ell}{t}\right)}^2}}\right)}^3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\frac{k}{t} \cdot k}\right) \cdot \sqrt[3]{t \cdot \sin k}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{\frac{\ell}{t}}\right) \cdot \sqrt[3]{\frac{\ell}{t}}}\right)}^3}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes.

  2. if k < -8.399687047689758e+102

    1. Initial program 41.4

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{{\ell}^2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^2\right) - 1\right)}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt 41.4

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

    4. Applied add-cube-cbrt 41.4

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

    5. Applied add-cube-cbrt 41.4

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

    6. Applied cube-unprod 41.4

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

    7. Applied cube-unprod 41.4

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

    8. Applied simplify 23.9

      \[\leadsto \frac{2}{{\color{blue}{\left(\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \sqrt[3]{\frac{t \cdot \sin k}{{\left(\frac{\ell}{t}\right)}^2}}\right)}}^3}\]
    9. Using strategy rm

    10. Applied cbrt-div 20.5

      \[\leadsto \frac{2}{{\left(\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \color{blue}{\frac{\sqrt[3]{t \cdot \sin k}}{\sqrt[3]{{\left(\frac{\ell}{t}\right)}^2}}}\right)}^3}\]

    11. Applied associate-*r/ 21.6

      \[\leadsto \frac{2}{{\left(\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\color{blue}{\frac{\frac{k}{t} \cdot k}{t}}}\right) \cdot \frac{\sqrt[3]{t \cdot \sin k}}{\sqrt[3]{{\left(\frac{\ell}{t}\right)}^2}}\right)}^3}\]

    12. Applied cbrt-div 20.9

      \[\leadsto \frac{2}{{\left(\left(\sqrt[3]{\tan k} \cdot \color{blue}{\frac{\sqrt[3]{\frac{k}{t} \cdot k}}{\sqrt[3]{t}}}\right) \cdot \frac{\sqrt[3]{t \cdot \sin k}}{\sqrt[3]{{\left(\frac{\ell}{t}\right)}^2}}\right)}^3}\]

    13. Applied associate-*r/ 20.9

      \[\leadsto \frac{2}{{\left(\color{blue}{\frac{\sqrt[3]{\tan k} \cdot \sqrt[3]{\frac{k}{t} \cdot k}}{\sqrt[3]{t}}} \cdot \frac{\sqrt[3]{t \cdot \sin k}}{\sqrt[3]{{\left(\frac{\ell}{t}\right)}^2}}\right)}^3}\]

    14. Applied frac-times 20.9

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\frac{k}{t} \cdot k}\right) \cdot \sqrt[3]{t \cdot \sin k}}{\sqrt[3]{t} \cdot \sqrt[3]{{\left(\frac{\ell}{t}\right)}^2}}\right)}}^3}\]
    15. Using strategy rm

    16. Applied cbrt-prod 17.9

      \[\leadsto \frac{2}{{\left(\frac{\left(\sqrt[3]{\tan k} \cdot \color{blue}{\left(\sqrt[3]{\frac{k}{t}} \cdot \sqrt[3]{k}\right)}\right) \cdot \sqrt[3]{t \cdot \sin k}}{\sqrt[3]{t} \cdot \sqrt[3]{{\left(\frac{\ell}{t}\right)}^2}}\right)}^3}\]

    if -8.399687047689758e+102 < k

    1. Initial program 49.1

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{{\ell}^2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^2\right) - 1\right)}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt 49.1

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

    4. Applied add-cube-cbrt 49.1

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

    5. Applied add-cube-cbrt 49.2

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

    6. Applied cube-unprod 49.2

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

    7. Applied cube-unprod 49.1

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

    8. Applied simplify 32.2

      \[\leadsto \frac{2}{{\color{blue}{\left(\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \sqrt[3]{\frac{t \cdot \sin k}{{\left(\frac{\ell}{t}\right)}^2}}\right)}}^3}\]
    9. Using strategy rm

    10. Applied cbrt-div 29.6

      \[\leadsto \frac{2}{{\left(\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \color{blue}{\frac{\sqrt[3]{t \cdot \sin k}}{\sqrt[3]{{\left(\frac{\ell}{t}\right)}^2}}}\right)}^3}\]

    11. Applied associate-*r/ 30.0

      \[\leadsto \frac{2}{{\left(\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\color{blue}{\frac{\frac{k}{t} \cdot k}{t}}}\right) \cdot \frac{\sqrt[3]{t \cdot \sin k}}{\sqrt[3]{{\left(\frac{\ell}{t}\right)}^2}}\right)}^3}\]

    12. Applied cbrt-div 23.1

      \[\leadsto \frac{2}{{\left(\left(\sqrt[3]{\tan k} \cdot \color{blue}{\frac{\sqrt[3]{\frac{k}{t} \cdot k}}{\sqrt[3]{t}}}\right) \cdot \frac{\sqrt[3]{t \cdot \sin k}}{\sqrt[3]{{\left(\frac{\ell}{t}\right)}^2}}\right)}^3}\]

    13. Applied associate-*r/ 23.1

      \[\leadsto \frac{2}{{\left(\color{blue}{\frac{\sqrt[3]{\tan k} \cdot \sqrt[3]{\frac{k}{t} \cdot k}}{\sqrt[3]{t}}} \cdot \frac{\sqrt[3]{t \cdot \sin k}}{\sqrt[3]{{\left(\frac{\ell}{t}\right)}^2}}\right)}^3}\]

    14. Applied frac-times 23.1

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\frac{k}{t} \cdot k}\right) \cdot \sqrt[3]{t \cdot \sin k}}{\sqrt[3]{t} \cdot \sqrt[3]{{\left(\frac{\ell}{t}\right)}^2}}\right)}}^3}\]
    15. Using strategy rm

    16. Applied square-mult 23.1

      \[\leadsto \frac{2}{{\left(\frac{\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\frac{k}{t} \cdot k}\right) \cdot \sqrt[3]{t \cdot \sin k}}{\sqrt[3]{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)}^3}\]

    17. Applied cbrt-prod 17.3

      \[\leadsto \frac{2}{{\left(\frac{\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\frac{k}{t} \cdot k}\right) \cdot \sqrt[3]{t \cdot \sin k}}{\sqrt[3]{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}\right)}}\right)}^3}\]

    18. Applied associate-*r* 17.3

      \[\leadsto \frac{2}{{\left(\frac{\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\frac{k}{t} \cdot k}\right) \cdot \sqrt[3]{t \cdot \sin k}}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{\frac{\ell}{t}}\right) \cdot \sqrt[3]{\frac{\ell}{t}}}}\right)}^3}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 4.7m) Debug log

Please include this information when filing a bug report:

herbie --seed '#(2839807893 3966094546 3130744521 2290871878 3103855764 887452522)'
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2 (* (* (* (/ (pow t 3) (sqr l)) (sin k)) (tan k)) (- (+ 1 (sqr (/ k t))) 1))))