Average Error: 47.0 → 25.2
Time: 3.0m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{\left(\sqrt[3]{1} \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \left(\left(\frac{\sqrt[3]{\frac{k}{t} \cdot k}}{\sqrt[3]{t}} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \frac{t}{\ell}\right)\right) \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right)\right)} \le 1.7774314990542808 \cdot 10^{+308}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right)\right) \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{\left(\frac{-1}{t}\right)}^{3}}{\frac{\ell \cdot \ell}{{-1}^{3}}} \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \end{array}\]

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 2 regimes

  2. if (/ 2 (* (* (* (cbrt (* (/ k t) (/ k t))) (* (/ t l) t)) (* (* (/ t l) (* (tan k) (sin k))) (/ (cbrt (* (/ k t) k)) (cbrt t)))) (* (cbrt 1) (cbrt (* (/ k t) (/ k t)))))) < 1.7774314990542808e+308

    1. Initial program 40.0

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

    3. Applied add-cube-cbrt40.0

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

    4. Applied associate-*r*40.0

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

    5. Applied simplify26.1

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

    7. Applied *-un-lft-identity26.1

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

    8. Applied cbrt-prod26.1

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

    9. Applied simplify10.3

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

    if 1.7774314990542808e+308 < (/ 2 (* (* (* (cbrt (* (/ k t) (/ k t))) (* (/ t l) t)) (* (* (/ t l) (* (tan k) (sin k))) (/ (cbrt (* (/ k t) k)) (cbrt t)))) (* (cbrt 1) (cbrt (* (/ k t) (/ k t))))))

    1. Initial program 62.5

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

    2. Taylor expanded around -inf 62.5

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

    3. Applied simplify58.2

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{\left(\frac{-1}{t}\right)}^{3}}{\frac{\ell \cdot \ell}{{-1}^{3}}} \cdot \left(\frac{k}{t} \cdot \frac{k}{t} - 0\right)\right) \cdot \left(\sin k \cdot \tan k\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify25.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{2}{\left(\sqrt[3]{1} \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \left(\left(\frac{\sqrt[3]{\frac{k}{t} \cdot k}}{\sqrt[3]{t}} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \frac{t}{\ell}\right)\right) \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right)\right)} \le 1.7774314990542808 \cdot 10^{+308}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right)\right) \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{\left(\frac{-1}{t}\right)}^{3}}{\frac{\ell \cdot \ell}{{-1}^{3}}} \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \end{array}}\]

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed 2018195 
(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))))