Average Error: 47.5 → 1.7
Time: 4.2m
Precision: 64
Internal Precision: 4224
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{2}{\frac{k}{\ell}}}{\left(\frac{t}{\frac{\cos k}{\frac{k}{\ell}}} \cdot \sin k\right) \cdot \sin k}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Initial program 47.5

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

  3. Applied add-cbrt-cube48.9

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

  4. Applied simplify35.9

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

  5. Taylor expanded around inf 30.1

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

  6. Applied simplify9.6

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

  8. Applied div-inv9.6

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

  9. Applied associate-/l*3.8

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

  10. Applied simplify3.2

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

  12. Applied associate-*r*1.7

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

Runtime

Time bar (total: 4.2m)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' +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))))