Average Error: 47.5 → 18.7
Time: 5.0m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\left|\frac{k}{t}\right| \cdot t\right) \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)\right)\right) \cdot \left|\frac{k}{t}\right| \le 2.8349053476030865 \cdot 10^{-297}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left|\frac{k}{t}\right|}{\frac{\ell}{t}} \cdot \left|\frac{k}{t}\right|}\\ \mathbf{if}\;\left(\left(\left|\frac{k}{t}\right| \cdot t\right) \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)\right)\right) \cdot \left|\frac{k}{t}\right| \le 1.7478937790522089 \cdot 10^{+137}:\\ \;\;\;\;\frac{2}{\left(\left(\left|\frac{k}{t}\right| \cdot t\right) \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)\right)\right) \cdot \left|\frac{k}{t}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left|\frac{k}{t}\right|}{\frac{\ell}{t}} \cdot \left|\frac{k}{t}\right|}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if (* (* (* (fabs (/ k t)) t) (* (* (/ t l) (/ t l)) (* (tan k) (sin k)))) (fabs (/ k t))) < 2.8349053476030865e-297 or 1.7478937790522089e+137 < (* (* (* (fabs (/ k t)) t) (* (* (/ t l) (/ t l)) (* (tan k) (sin k)))) (fabs (/ k t)))

    1. Initial program 47.0

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

    3. Applied add-sqr-sqrt47.0

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

    4. Applied simplify47.0

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

    5. Applied simplify39.4

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

    7. Applied add-cube-cbrt39.4

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

    8. Applied times-frac37.8

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

    9. Applied simplify37.8

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

    10. Applied simplify30.4

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

    12. Applied associate-*r*25.3

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

    14. Applied associate-*l/25.3

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

    15. Applied associate-*l/24.7

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

    16. Applied associate-*l/24.2

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

    17. Applied associate-*l/20.4

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

    if 2.8349053476030865e-297 < (* (* (* (fabs (/ k t)) t) (* (* (/ t l) (/ t l)) (* (tan k) (sin k)))) (fabs (/ k t))) < 1.7478937790522089e+137

    1. Initial program 51.9

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

    3. Applied add-sqr-sqrt51.9

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

    4. Applied simplify51.9

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

    5. Applied simplify44.7

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

    7. Applied add-cube-cbrt44.9

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

    8. Applied times-frac40.9

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

    9. Applied simplify40.8

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

    10. Applied simplify22.6

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

    12. Applied associate-*r*17.6

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

    14. Applied *-un-lft-identity17.6

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

    15. Applied associate-*r*17.6

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

    16. Applied simplify1.9

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

Runtime

Time bar (total: 5.0m)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' 
(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))))