Average Error: 47.1 → 17.2
Time: 1.4m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -6.971184209373122 \cdot 10^{-156}:\\ \;\;\;\;\frac{\frac{2}{\frac{k \cdot \left(t \cdot k\right)}{{\ell}^{2}}}}{\frac{{\left(\sin k\right)}^{2}}{\cos k}}\\ \mathbf{elif}\;\ell \le 1.2890514495577676 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\sqrt[3]{{\left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \frac{k \cdot \left(t \cdot k\right)}{\cos k}\right)}^{3}}}\\ \mathbf{elif}\;\ell \le 1.2583157289541618 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{2}{\frac{k \cdot \left(t \cdot k\right)}{{\ell}^{2}}}}{\frac{{\left(\sin k\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) - 1\right) \cdot \left(\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right) \cdot \tan k\right)}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if l < -6.971184209373122e-156 or 1.2890514495577676e-161 < l < 1.2583157289541618e+159

    1. Initial program 46.4

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

    2. Taylor expanded around -inf 20.7

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

    4. Applied associate-*r*19.0

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

    6. Applied unpow219.0

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

    7. Applied associate-*r*15.7

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

    9. Applied times-frac14.6

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

    10. Applied associate-/r*14.6

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

    if -6.971184209373122e-156 < l < 1.2890514495577676e-161

    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. Taylor expanded around -inf 19.3

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

    4. Applied associate-*r*18.5

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

    6. Applied unpow218.5

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

    7. Applied associate-*r*18.5

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

    9. Applied add-cbrt-cube18.6

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

    10. Applied add-cbrt-cube21.2

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

    11. Applied cbrt-undiv21.2

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

    12. Simplified14.3

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

    if 1.2583157289541618e+159 < l

    1. Initial program 62.3

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

    3. Applied add-cube-cbrt62.3

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

    4. Applied times-frac55.6

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

    5. Simplified55.6

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

    6. Simplified51.1

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -6.971184209373122 \cdot 10^{-156}:\\ \;\;\;\;\frac{\frac{2}{\frac{k \cdot \left(t \cdot k\right)}{{\ell}^{2}}}}{\frac{{\left(\sin k\right)}^{2}}{\cos k}}\\ \mathbf{elif}\;\ell \le 1.2890514495577676 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\sqrt[3]{{\left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \frac{k \cdot \left(t \cdot k\right)}{\cos k}\right)}^{3}}}\\ \mathbf{elif}\;\ell \le 1.2583157289541618 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{2}{\frac{k \cdot \left(t \cdot k\right)}{{\ell}^{2}}}}{\frac{{\left(\sin k\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) - 1\right) \cdot \left(\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right) \cdot \tan k\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 
(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))))