Average Error: 46.8 → 1.5
Time: 3.9m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -1.4270879212242336 \cdot 10^{-96}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\cos k}{\sin k}} \cdot \left(\left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{\ell}{k}}{\sin k}\right) \cdot \left(\sqrt[3]{\frac{\cos k}{\sin k}} \cdot \sqrt[3]{\frac{\cos k}{\sin k}}\right)\right)\right) \cdot 2\\ \mathbf{elif}\;k \le 1.6878013775025593 \cdot 10^{-90}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right) \cdot \cos k}{\sin k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\ell}{k}}{t} \cdot \left(\frac{\cos k}{\sin k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right)\right) \cdot 2\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if k < -1.4270879212242336e-96

    1. Initial program 45.4

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

    2. Simplified28.9

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

    3. Taylor expanded around -inf 19.5

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

    4. Simplified18.0

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

    6. Applied associate-*l/18.3

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

    7. Applied associate-/r/18.2

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

    8. Applied *-un-lft-identity18.2

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

    9. Applied times-frac18.2

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

    10. Simplified7.1

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

    12. Applied times-frac0.7

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

    14. Applied add-cube-cbrt1.0

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

    15. Applied associate-*r*1.0

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

    if -1.4270879212242336e-96 < k < 1.6878013775025593e-90

    1. Initial program 62.2

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

    2. Simplified54.7

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

    3. Taylor expanded around -inf 61.8

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

    4. Simplified32.4

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

    6. Applied associate-*l/39.5

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

    7. Applied associate-/r/39.6

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

    8. Applied *-un-lft-identity39.6

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

    9. Applied times-frac39.6

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

    10. Simplified10.5

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

    12. Applied times-frac7.8

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

    14. Applied associate-*l/8.2

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

    15. Applied frac-times8.0

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

    if 1.6878013775025593e-90 < k

    1. Initial program 45.1

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

    2. Simplified28.1

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

    3. Taylor expanded around -inf 20.1

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

    4. Simplified19.1

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

    6. Applied associate-*l/19.3

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

    7. Applied associate-/r/19.2

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

    8. Applied *-un-lft-identity19.2

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

    9. Applied times-frac19.2

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

    10. Simplified7.3

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

    12. Applied times-frac0.8

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

    14. Applied associate-*l*0.6

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1.4270879212242336 \cdot 10^{-96}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\cos k}{\sin k}} \cdot \left(\left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{\ell}{k}}{\sin k}\right) \cdot \left(\sqrt[3]{\frac{\cos k}{\sin k}} \cdot \sqrt[3]{\frac{\cos k}{\sin k}}\right)\right)\right) \cdot 2\\ \mathbf{elif}\;k \le 1.6878013775025593 \cdot 10^{-90}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right) \cdot \cos k}{\sin k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\ell}{k}}{t} \cdot \left(\frac{\cos k}{\sin k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right)\right) \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019051 
(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))))