Average Error: 47.0 → 25.1
Time: 3.0m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{\sqrt[3]{\frac{k}{t} \cdot k}}{\sqrt[3]{t}} \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left(\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right)\right) \cdot \left(\sqrt{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}} \cdot \sqrt{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}\right)} \le +\infty:\\ \;\;\;\;\frac{2}{\left(\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left(\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(\sqrt{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}} \cdot \sqrt{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}\right)\right)\right) \cdot \left(\sqrt{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}} \cdot \sqrt{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{t}{\ell}\right) \cdot t\right) \cdot \left(\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right)\right) \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if (/ 2 (* (* (* (/ (cbrt (* (/ k t) k)) (cbrt t)) (* (/ t l) t)) (* (* (/ t l) (* (tan k) (sin k))) (cbrt (* (/ k t) (/ k t))))) (* (sqrt (cbrt (* (/ k t) (/ k t)))) (sqrt (cbrt (* (/ k t) (/ k t))))))) < +inf.0

    1. Initial program 40.6

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

    3. Applied add-cube-cbrt40.6

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

    4. Applied associate-*r*40.6

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

    5. Applied simplify27.5

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

    7. Applied add-sqr-sqrt27.5

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

    8. Applied simplify27.5

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

    9. Applied simplify11.4

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

    11. Applied add-sqr-sqrt11.4

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

    if +inf.0 < (/ 2 (* (* (* (/ (cbrt (* (/ k t) k)) (cbrt t)) (* (/ t l) t)) (* (* (/ t l) (* (tan k) (sin k))) (cbrt (* (/ k t) (/ k t))))) (* (sqrt (cbrt (* (/ k t) (/ k t)))) (sqrt (cbrt (* (/ k t) (/ k t)))))))

    1. Initial program 62.6

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

    3. Applied add-cube-cbrt62.6

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

    4. Applied associate-*r*62.6

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

    5. Applied simplify62.6

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

    7. Applied associate-*r*58.0

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

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' 
(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))))