Average Error: 47.0 → 26.7
Time: 5.4m
Precision: 64
Internal Precision: 4224
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \sin k\right) \cdot \left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{\tan k}{\frac{\ell}{t}}\right)\right)\right) \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \le -2.954411983378822 \cdot 10^{-216}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left|\frac{k}{t}\right|\right)\right) \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}\\ \mathbf{if}\;\left(\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \sin k\right) \cdot \left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{\tan k}{\frac{\ell}{t}}\right)\right)\right) \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \le 7.162293700860324 \cdot 10^{-281}:\\ \;\;\;\;\frac{2}{\sqrt[3]{{\left(\left(\tan k \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}\\ \mathbf{if}\;\left(\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \sin k\right) \cdot \left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{\tan k}{\frac{\ell}{t}}\right)\right)\right) \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \le +\infty:\\ \;\;\;\;\frac{2}{\left(\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \sin k\right) \cdot \left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{\tan k}{\frac{\ell}{t}}\right)\right)\right) \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sqrt[3]{\frac{\ell}{k} \cdot 12} \cdot \sqrt[3]{\frac{\ell}{k} \cdot 12}\right) \cdot \sqrt[3]{\frac{\ell}{k} \cdot 12}\right) \cdot \frac{\frac{\ell}{k}}{t}}{{k}^{4}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 4 regimes

  2. if (* (* (* (cbrt (* (/ k t) (/ k t))) (sin k)) (* (cbrt (* (/ k t) (/ k t))) (* (/ t (/ l t)) (/ (tan k) (/ l t))))) (cbrt (- (+ 1 (pow (/ k t) 2)) 1))) < -2.954411983378822e-216

    1. Initial program 24.5

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

    3. Applied add-sqr-sqrt24.5

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

    4. Applied associate-*r*24.5

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

    5. Applied simplify6.8

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

    if -2.954411983378822e-216 < (* (* (* (cbrt (* (/ k t) (/ k t))) (sin k)) (* (cbrt (* (/ k t) (/ k t))) (* (/ t (/ l t)) (/ (tan k) (/ l t))))) (cbrt (- (+ 1 (pow (/ k t) 2)) 1))) < 7.162293700860324e-281

    1. Initial program 61.1

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

    3. Applied add-cbrt-cube61.4

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

    4. Applied simplify45.5

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

    if 7.162293700860324e-281 < (* (* (* (cbrt (* (/ k t) (/ k t))) (sin k)) (* (cbrt (* (/ k t) (/ k t))) (* (/ t (/ l t)) (/ (tan k) (/ l t))))) (cbrt (- (+ 1 (pow (/ k t) 2)) 1))) < +inf.0

    1. Initial program 24.1

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

    3. Applied add-cube-cbrt24.2

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

    4. Applied associate-*r*24.2

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

    5. Applied simplify5.6

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

    if +inf.0 < (* (* (* (cbrt (* (/ k t) (/ k t))) (sin k)) (* (cbrt (* (/ k t) (/ k t))) (* (/ t (/ l t)) (/ (tan k) (/ l t))))) (cbrt (- (+ 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 59.5

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

    3. Applied simplify39.9

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

    5. Applied frac-times39.8

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

    6. Applied associate-/l/40.0

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

    7. Applied simplify39.8

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

    9. Applied associate-/r/39.8

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

    10. Applied associate-/r*39.5

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

    11. Applied simplify39.4

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

    13. Applied add-cube-cbrt39.4

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

Runtime

Time bar (total: 5.4m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' +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))))