Average Error: 49.5 → 24.2
Time: 7.3m
Precision: 64
Internal Precision: 3968
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -9.727779105883051 \cdot 10^{-155}:\\ \;\;\;\;\frac{2}{\left(\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right) \cdot \left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \tan k\right)\right) \cdot \left(\sqrt[3]{\left|\frac{k}{t}\right|} \cdot \sqrt[3]{\left|\frac{k}{t}\right|}\right)}\\ \mathbf{if}\;t \le 1.2381270714685696 \cdot 10^{-204}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{1}{{\ell}^{2} \cdot {t}^{3}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right) \cdot \left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \tan k\right)\right) \cdot \sqrt[3]{\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if t < -9.727779105883051e-155

    1. Initial program 43.3

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

    3. Applied add-cube-cbrt43.4

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

    4. Applied associate-*r*43.4

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

    5. Applied simplify37.3

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

    7. Applied add-sqr-sqrt37.3

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

    8. Applied cbrt-prod37.3

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

    9. Applied simplify37.3

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

    10. Applied simplify25.2

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

    if -9.727779105883051e-155 < t < 1.2381270714685696e-204

    1. Initial program 62.9

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

    2. Taylor expanded around inf 21.1

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

    if 1.2381270714685696e-204 < t

    1. Initial program 44.8

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

    3. Applied add-cube-cbrt44.8

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

    4. Applied associate-*r*44.8

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

    5. Applied simplify38.1

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

    7. Applied add-cube-cbrt38.1

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

    8. Applied simplify38.1

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

    9. Applied simplify25.6

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

Runtime

Time bar (total: 7.3m)Debug logProfile

herbie shell --seed '#(1062930989 876886121 3990119081 3032829768 3060892583 1929069376)' 
(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))))