Average Error: 47.0 → 5.5
Time: 4.6m
Precision: 64
Internal Precision: 4224
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -8.322406385231813 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{\frac{\sin k \cdot t}{\frac{\cos k}{\sin k}}}\\ \mathbf{if}\;k \le -1.5241799393084268 \cdot 10^{-86}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2}}{\frac{\ell}{t \cdot {\left(\sin k\right)}^{2}}}}{\ell \cdot \cos k}}\\ \mathbf{if}\;k \le 3.1432359405476076 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{\frac{\sin k \cdot t}{\frac{\cos k}{\sin k}}}\\ \mathbf{if}\;k \le 4.883041309493023 \cdot 10^{+153}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2}}{\frac{\ell}{t \cdot {\left(\sin k\right)}^{2}}}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{\frac{\sin k \cdot t}{\frac{\cos k}{\sin k}}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if k < -8.322406385231813e+101 or -1.5241799393084268e-86 < k < 3.1432359405476076e-108 or 4.883041309493023e+153 < k

    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-cbrt-cube43.9

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

    4. Applied simplify35.2

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \left(\left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}}\]
    5. Using strategy rm

    6. Applied associate-*l/37.6

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \left(\color{blue}{\frac{t \cdot \left(t \cdot t\right)}{\ell}} \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}\]

    7. Applied associate-*l/37.6

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \color{blue}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}}\right)}^{3}}}\]

    8. Applied tan-quot37.6

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}\right)}^{3}}}\]

    9. Applied associate-*r/37.6

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\color{blue}{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k}{\cos k}} \cdot \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}\right)}^{3}}}\]

    10. Applied frac-times37.4

      \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)}{\cos k \cdot \ell}\right)}}^{3}}}\]

    11. Applied cube-div40.8

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{\frac{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}{{\left(\cos k \cdot \ell\right)}^{3}}}}}\]

    12. Applied cbrt-div40.8

      \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}{\sqrt[3]{{\left(\cos k \cdot \ell\right)}^{3}}}}}\]

    13. Applied simplify34.2

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}}{\sqrt[3]{{\left(\cos k \cdot \ell\right)}^{3}}}}\]

    14. Applied simplify29.2

      \[\leadsto \frac{2}{\frac{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}{\color{blue}{\ell \cdot \cos k}}}\]

    15. Taylor expanded around inf 25.4

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

    16. Taylor expanded around -inf 62.8

      \[\leadsto \frac{2}{\color{blue}{\frac{e^{2 \cdot \left(\log -1 - \log \left(\frac{-1}{k}\right)\right)} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{{\ell}^{2} \cdot \cos k}}}\]

    17. Applied simplify6.5

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{\frac{\sin k \cdot t}{\frac{\cos k}{\sin k}}}}\]

    if -8.322406385231813e+101 < k < -1.5241799393084268e-86 or 3.1432359405476076e-108 < k < 4.883041309493023e+153

    1. Initial program 51.6

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

    3. Applied add-cbrt-cube54.1

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

    4. Applied simplify43.0

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \left(\left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}}\]
    5. Using strategy rm

    6. Applied associate-*l/45.1

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \left(\color{blue}{\frac{t \cdot \left(t \cdot t\right)}{\ell}} \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}\]

    7. Applied associate-*l/45.1

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \color{blue}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}}\right)}^{3}}}\]

    8. Applied tan-quot45.1

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}\right)}^{3}}}\]

    9. Applied associate-*r/45.1

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\color{blue}{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k}{\cos k}} \cdot \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}\right)}^{3}}}\]

    10. Applied frac-times44.9

      \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)}{\cos k \cdot \ell}\right)}}^{3}}}\]

    11. Applied cube-div47.0

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{\frac{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}{{\left(\cos k \cdot \ell\right)}^{3}}}}}\]

    12. Applied cbrt-div46.3

      \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}{\sqrt[3]{{\left(\cos k \cdot \ell\right)}^{3}}}}}\]

    13. Applied simplify37.7

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}}{\sqrt[3]{{\left(\cos k \cdot \ell\right)}^{3}}}}\]

    14. Applied simplify28.0

      \[\leadsto \frac{2}{\frac{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}{\color{blue}{\ell \cdot \cos k}}}\]

    15. Taylor expanded around inf 9.5

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

    17. Applied associate-/l*4.2

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

Runtime

Time bar (total: 4.6m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +o rules:numerics
(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))))