Average Error: 47.4 → 25.3
Time: 2.6m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.6115286184264376 \cdot 10^{-169} \lor \neg \left(t \le 5.00501987548527 \cdot 10^{-180}\right):\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{\frac{t}{\frac{\ell}{t}}}}{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}}{\left(\sin k \cdot \left(\tan k \cdot \frac{t}{\ell}\right)\right) \cdot \left(\sqrt[3]{\frac{k}{t}} \cdot \sqrt[3]{\frac{k}{t}}\right)}}{\sqrt[3]{\left|\frac{k}{t}\right|} \cdot \sqrt[3]{\left|\frac{k}{t}\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot t\right) \cdot \left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)\right)\right)\right) \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if t < -3.6115286184264376e-169 or 5.00501987548527e-180 < t

    1. Initial program 44.2

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

    3. Applied add-cube-cbrt44.2

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

    4. Applied associate-*r*44.2

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

    5. Applied simplify36.4

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

    7. Applied add-sqr-sqrt36.4

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

    8. Applied cbrt-prod36.4

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

    9. Applied simplify36.4

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

    10. Applied simplify22.4

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

    11. Taylor expanded around 0 22.9

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

    12. Applied simplify22.1

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

    14. Applied cbrt-prod22.2

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

    if -3.6115286184264376e-169 < t < 5.00501987548527e-180

    1. Initial program 62.7

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

    3. Applied add-cube-cbrt62.7

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

    4. Applied associate-*r*62.7

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

    5. Applied simplify48.3

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

    7. Applied associate-*r*40.1

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

  4. Applied simplify25.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;t \le -3.6115286184264376 \cdot 10^{-169} \lor \neg \left(t \le 5.00501987548527 \cdot 10^{-180}\right):\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{\frac{t}{\frac{\ell}{t}}}}{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}}{\left(\sin k \cdot \left(\tan k \cdot \frac{t}{\ell}\right)\right) \cdot \left(\sqrt[3]{\frac{k}{t}} \cdot \sqrt[3]{\frac{k}{t}}\right)}}{\sqrt[3]{\left|\frac{k}{t}\right|} \cdot \sqrt[3]{\left|\frac{k}{t}\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot t\right) \cdot \left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)\right)\right)\right) \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}\\ \end{array}}\]

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))