Average Error: 46.7 → 26.8
Time: 3.4m
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}\;\frac{\frac{\frac{\frac{2}{\frac{t}{\ell}}}{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\left(\sin k \cdot t\right) \cdot \tan k\right)}}{\frac{t}{\ell} \cdot e^{\left(\log \left(\frac{-1}{t}\right) - \log \left(\frac{-1}{k}\right)\right) \cdot \left(\frac{1}{3} \cdot 2\right)}}}{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \sqrt[3]{1}} \le -3.27275983052784 \cdot 10^{-128}:\\ \;\;\;\;\frac{2}{\frac{e^{\left(\log \left(\frac{-1}{t}\right) - \log \left(\frac{-1}{k}\right)\right) \cdot \left(\frac{1}{3} \cdot 2\right)}}{\frac{\frac{\ell}{t}}{t}} \cdot \left(\frac{\sin k \cdot \sin k}{\cos k} \cdot \left(\frac{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t} - 0}\right)\right)}\\ \mathbf{if}\;\frac{\frac{\frac{\frac{2}{\frac{t}{\ell}}}{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\left(\sin k \cdot t\right) \cdot \tan k\right)}}{\frac{t}{\ell} \cdot e^{\left(\log \left(\frac{-1}{t}\right) - \log \left(\frac{-1}{k}\right)\right) \cdot \left(\frac{1}{3} \cdot 2\right)}}}{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \sqrt[3]{1}} \le 0.0:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{\frac{t}{\ell}}}{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\left(\sin k \cdot t\right) \cdot \tan k\right)}}{\frac{t}{\ell} \cdot e^{\left(\log \left(\frac{-1}{t}\right) - \log \left(\frac{-1}{k}\right)\right) \cdot \left(\frac{1}{3} \cdot 2\right)}}}{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \sqrt[3]{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sqrt[3]{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}} \cdot \sqrt[3]{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}\right) \cdot \sqrt[3]{\left(\left(\frac{t}{\ell} \cdot t\right) \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 \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right)}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if (/ (/ (/ (/ 2 (/ t l)) (* (cbrt (* (/ k t) (/ k t))) (* (* (sin k) t) (tan k)))) (* (/ t l) (exp (* (- (log (/ -1 t)) (log (/ -1 k))) (* 1/3 2))))) (* (cbrt (* (/ k t) (/ k t))) (cbrt 1))) < -3.27275983052784e-128

    1. Initial program 59.1

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

    3. Applied add-cube-cbrt59.1

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

    4. Applied associate-*r*59.1

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

    5. Applied simplify50.3

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

    6. Taylor expanded around -inf 55.0

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

    7. Applied simplify30.8

      \[\leadsto \color{blue}{\frac{2}{\frac{e^{\left(\log \left(\frac{-1}{t}\right) - \log \left(\frac{-1}{k}\right)\right) \cdot \left(\frac{1}{3} \cdot 2\right)}}{\frac{\frac{\ell}{t}}{t}} \cdot \left(\frac{\sin k \cdot \sin k}{\cos k} \cdot \left(\frac{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t} - 0}\right)\right)}}\]

    if -3.27275983052784e-128 < (/ (/ (/ (/ 2 (/ t l)) (* (cbrt (* (/ k t) (/ k t))) (* (* (sin k) t) (tan k)))) (* (/ t l) (exp (* (- (log (/ -1 t)) (log (/ -1 k))) (* 1/3 2))))) (* (cbrt (* (/ k t) (/ k t))) (cbrt 1))) < 0.0

    1. Initial program 39.1

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

    3. Applied add-cube-cbrt39.1

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

    4. Applied associate-*r*39.1

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

    5. Applied simplify30.7

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

    7. Applied *-un-lft-identity30.7

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

    8. Applied cbrt-prod30.7

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

    9. Applied simplify18.6

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

    10. Taylor expanded around -inf 18.8

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

    11. Applied simplify15.9

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

    if 0.0 < (/ (/ (/ (/ 2 (/ t l)) (* (cbrt (* (/ k t) (/ k t))) (* (* (sin k) t) (tan k)))) (* (/ t l) (exp (* (- (log (/ -1 t)) (log (/ -1 k))) (* 1/3 2))))) (* (cbrt (* (/ k t) (/ k t))) (cbrt 1)))

    1. Initial program 47.4

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

    3. Applied add-cube-cbrt47.4

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

    4. Applied associate-*r*47.4

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

    5. Applied simplify38.9

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

    7. Applied *-un-lft-identity38.9

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

    8. Applied cbrt-prod38.9

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

    9. Applied simplify28.4

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

    11. Applied add-cube-cbrt28.5

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

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' 
(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))))