Error

Derivation

1. Split input into 3 regimes

2. if (* (* (* (* (tan k) (sin k)) (* (/ t l) t)) (* (/ t l) (fabs (/ k t)))) (sqrt (- (+ 1 (pow (/ k t) 2)) 1))) < -1.4945494869687503e-217 or 0.0 < (* (* (* (* (tan k) (sin k)) (* (/ t l) t)) (* (/ t l) (fabs (/ k t)))) (sqrt (- (+ 1 (pow (/ k t) 2)) 1))) < +inf.0

   1. Initial program 28.8

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

   3. Applied add-sqr-sqrt28.8

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

   4. Applied associate-*r*28.8

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

   5. Applied simplify10.2

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

   if -1.4945494869687503e-217 < (* (* (* (* (tan k) (sin k)) (* (/ t l) t)) (* (/ t l) (fabs (/ k t)))) (sqrt (- (+ 1 (pow (/ k t) 2)) 1))) < 0.0

   1. Initial program 61.5

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

   3. Applied associate-*l*61.3

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

   4. Applied simplify50.4

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

   if +inf.0 < (* (* (* (* (tan k) (sin k)) (* (/ t l) t)) (* (/ t l) (fabs (/ k t)))) (sqrt (- (+ 1 (pow (/ k t) 2)) 1)))

   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. Taylor expanded around 0 59.2

      \[\leadsto \frac{2}{\color{blue}{\frac{1}{6} \cdot \frac{{k}^{4} \cdot \left({t}^{3} \cdot e^{2 \cdot \left(\log k - \log t\right)}\right)}{{\ell}^{2}}}}\]

   3. Applied simplify39.2

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\frac{t}{\ell}} \cdot \frac{12}{{k}^{4}}}{\left(t \cdot \frac{k}{t}\right) \cdot \left(t \cdot \frac{k}{t}\right)}}\]
   4. Using strategy rm

   5. Applied add-cbrt-cube39.3

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\frac{\ell}{\frac{t}{\ell}} \cdot \frac{12}{{k}^{4}}}{\left(t \cdot \frac{k}{t}\right) \cdot \left(t \cdot \frac{k}{t}\right)} \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot \frac{12}{{k}^{4}}}{\left(t \cdot \frac{k}{t}\right) \cdot \left(t \cdot \frac{k}{t}\right)}\right) \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot \frac{12}{{k}^{4}}}{\left(t \cdot \frac{k}{t}\right) \cdot \left(t \cdot \frac{k}{t}\right)}}}\]

   6. Applied simplify39.2

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{12}{t \cdot {k}^{4}} \cdot \left(\frac{\frac{\ell}{t}}{\frac{k}{t}} \cdot \frac{\frac{\ell}{t}}{\frac{k}{t}}\right)\right)}^{3}}}\]
   7. Using strategy rm

   8. Applied unpow-prod-down39.7

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{12}{t \cdot {k}^{4}}\right)}^{3} \cdot {\left(\frac{\frac{\ell}{t}}{\frac{k}{t}} \cdot \frac{\frac{\ell}{t}}{\frac{k}{t}}\right)}^{3}}}\]

   9. Applied cbrt-prod39.7

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{12}{t \cdot {k}^{4}}\right)}^{3}} \cdot \sqrt[3]{{\left(\frac{\frac{\ell}{t}}{\frac{k}{t}} \cdot \frac{\frac{\ell}{t}}{\frac{k}{t}}\right)}^{3}}}\]

   10. Applied simplify39.3

       \[\leadsto \color{blue}{\frac{\frac{12}{t}}{{k}^{4}}} \cdot \sqrt[3]{{\left(\frac{\frac{\ell}{t}}{\frac{k}{t}} \cdot \frac{\frac{\ell}{t}}{\frac{k}{t}}\right)}^{3}}\]

   11. Applied simplify39.0

       \[\leadsto \frac{\frac{12}{t}}{{k}^{4}} \cdot \color{blue}{\left(\left(1 \cdot \frac{\ell}{k}\right) \cdot \left(1 \cdot \frac{\ell}{k}\right)\right)}\]
   12. Using strategy rm

   13. Applied add-cube-cbrt39.0

       \[\leadsto \frac{\frac{12}{t}}{\color{blue}{\left(\sqrt[3]{{k}^{4}} \cdot \sqrt[3]{{k}^{4}}\right) \cdot \sqrt[3]{{k}^{4}}}} \cdot \left(\left(1 \cdot \frac{\ell}{k}\right) \cdot \left(1 \cdot \frac{\ell}{k}\right)\right)\]

   14. Applied add-cube-cbrt39.0

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{12}{t}} \cdot \sqrt[3]{\frac{12}{t}}\right) \cdot \sqrt[3]{\frac{12}{t}}}}{\left(\sqrt[3]{{k}^{4}} \cdot \sqrt[3]{{k}^{4}}\right) \cdot \sqrt[3]{{k}^{4}}} \cdot \left(\left(1 \cdot \frac{\ell}{k}\right) \cdot \left(1 \cdot \frac{\ell}{k}\right)\right)\]

   15. Applied times-frac39.0

       \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{\frac{12}{t}} \cdot \sqrt[3]{\frac{12}{t}}}{\sqrt[3]{{k}^{4}} \cdot \sqrt[3]{{k}^{4}}} \cdot \frac{\sqrt[3]{\frac{12}{t}}}{\sqrt[3]{{k}^{4}}}\right)} \cdot \left(\left(1 \cdot \frac{\ell}{k}\right) \cdot \left(1 \cdot \frac{\ell}{k}\right)\right)\]

   16. Applied associate-*l*39.0

       \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{12}{t}} \cdot \sqrt[3]{\frac{12}{t}}}{\sqrt[3]{{k}^{4}} \cdot \sqrt[3]{{k}^{4}}} \cdot \left(\frac{\sqrt[3]{\frac{12}{t}}}{\sqrt[3]{{k}^{4}}} \cdot \left(\left(1 \cdot \frac{\ell}{k}\right) \cdot \left(1 \cdot \frac{\ell}{k}\right)\right)\right)}\]

   17. Applied simplify39.0

       \[\leadsto \frac{\sqrt[3]{\frac{12}{t}} \cdot \sqrt[3]{\frac{12}{t}}}{\sqrt[3]{{k}^{4}} \cdot \sqrt[3]{{k}^{4}}} \cdot \color{blue}{\frac{\frac{\sqrt[3]{\frac{12}{t}}}{\sqrt[3]{{k}^{4}}}}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}\]
3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 7.7m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +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))))