Average Error: 46.9 → 27.7
Time: 5.3m
Precision: 64
Internal Precision: 4224
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.9206503186740354 \cdot 10^{-237}:\\ \;\;\;\;\frac{2}{\left(\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right) \cdot \left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \tan k\right)\right) \cdot \left(\sqrt[3]{\left|\frac{k}{t}\right|} \cdot \left(\sqrt{\sqrt[3]{\left|\frac{k}{t}\right|}} \cdot \sqrt{\sqrt[3]{\left|\frac{k}{t}\right|}}\right)\right)}\\ \mathbf{if}\;t \le 7.608552185613352 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{1}{{\ell}^{2} \cdot {t}^{3}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right) \cdot \left(\frac{\sqrt[3]{\frac{k}{t} \cdot k}}{\sqrt[3]{t}} \cdot \tan k\right)\right) \cdot \left(\sqrt[3]{\left|\frac{k}{t}\right|} \cdot \sqrt[3]{\left|\frac{k}{t}\right|}\right)}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if t < -1.9206503186740354e-237

    1. Initial program 46.1

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

    3. Applied add-cube-cbrt46.1

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

    4. Applied associate-*r*46.1

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

    5. Applied simplify38.4

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

    7. Applied add-sqr-sqrt38.4

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

    8. Applied cbrt-prod38.4

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

    9. Applied simplify38.4

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

    10. Applied simplify26.6

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

    12. Applied add-sqr-sqrt26.6

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

    if -1.9206503186740354e-237 < t < 7.608552185613352e-138

    1. Initial program 62.6

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

    2. Taylor expanded around inf 41.3

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

    if 7.608552185613352e-138 < t

    1. Initial program 41.5

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

    3. Applied add-cube-cbrt41.6

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

    4. Applied associate-*r*41.6

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

    5. Applied simplify36.5

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

    7. Applied add-sqr-sqrt36.5

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

    8. Applied cbrt-prod36.5

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

    9. Applied simplify36.5

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

    10. Applied simplify22.9

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

    12. Applied associate-*r/23.6

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

    13. Applied cbrt-div23.6

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

Runtime

Time bar (total: 5.3m)Debug logProfile

herbie shell --seed '#(1063313015 2771194459 1594909340 1344785158 2223560818 546365448)' 
(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))))