Average Error: 47.5 → 17.2
Time: 5.2m
Precision: 64
Internal Precision: 4224
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}\right)\right) \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \le +\infty:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k\right) \cdot \left|\frac{k}{t}\right|}{\frac{\ell}{t} \cdot \cos k} \cdot \left|\frac{k}{t}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left|\frac{k}{t}\right| \cdot t\right) \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)\right)\right) \cdot \left|\frac{k}{t}\right|}\\ \end{array}\]

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

    1. Initial program 36.9

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

    3. Applied add-sqr-sqrt36.9

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

    4. Applied simplify36.9

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

    5. Applied simplify34.2

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

    7. Applied add-cube-cbrt34.3

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

    8. Applied times-frac31.3

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

    9. Applied simplify31.2

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

    10. Applied simplify17.4

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

    12. Applied associate-*r*13.1

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

    14. Applied tan-quot13.1

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

    15. Applied associate-*l/13.1

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

    16. Applied associate-*l/11.9

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

    17. Applied frac-times11.1

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

    18. Applied associate-*l/4.0

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

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

    3. Applied add-sqr-sqrt62.7

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

    4. Applied simplify62.7

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

    5. Applied simplify48.3

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

    7. Applied add-cube-cbrt48.3

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

    8. Applied times-frac47.9

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

    9. Applied simplify47.9

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

    10. Applied simplify47.9

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

    12. Applied associate-*r*41.6

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

    14. Applied *-un-lft-identity41.6

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

    15. Applied associate-*r*41.6

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

    16. Applied simplify36.2

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

Runtime

Time bar (total: 5.2m)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' 
(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))))