Average Error: 47.1 → 1.4
Time: 6.6m
Precision: 64
Internal Precision: 4480
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{2}{k}}{\frac{t}{\cos k}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right) \le -3.562950036072969 \cdot 10^{+274}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{k}{\ell}}}{k}}{\frac{t}{\cos k}} \cdot \frac{\ell}{\sin k \cdot \sin k}\\ \mathbf{if}\;\frac{\frac{2}{k}}{\frac{t}{\cos k}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right) \le -1.1162674882265014 \cdot 10^{-308}:\\ \;\;\;\;\frac{\frac{2}{k}}{\frac{t}{\cos k}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right)\\ \mathbf{if}\;\frac{\frac{2}{k}}{\frac{t}{\cos k}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right) \le -0.0:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell}}}{\frac{t}{\cos k}} \cdot \frac{\frac{\ell}{\sin k}}{\sin k \cdot k}\\ \mathbf{if}\;\frac{\frac{2}{k}}{\frac{t}{\cos k}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right) \le 1.286753789222229 \cdot 10^{+222}:\\ \;\;\;\;\frac{\frac{2}{k}}{\frac{t}{\cos k}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell}}}{\frac{t}{\cos k}} \cdot \frac{\frac{\ell}{\sin k}}{\sin k \cdot k}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if (* (/ (/ 2 k) (/ t (cos k))) (* (/ l (sin k)) (/ (/ l k) (sin k)))) < -3.562950036072969e+274

    1. Initial program 62.4

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

    3. Applied add-cbrt-cube62.8

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

    4. Applied simplify59.0

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left(\tan k \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}}\]

    5. Taylor expanded around inf 60.6

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

    6. Applied simplify30.1

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

    8. Applied associate-/r/30.1

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

    9. Applied times-frac5.7

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{k}{\ell}}}{k}}{\frac{t}{\cos k}} \cdot \frac{\ell}{\sin k \cdot \sin k}}\]

    if -3.562950036072969e+274 < (* (/ (/ 2 k) (/ t (cos k))) (* (/ l (sin k)) (/ (/ l k) (sin k)))) < -1.1162674882265014e-308 or -0.0 < (* (/ (/ 2 k) (/ t (cos k))) (* (/ l (sin k)) (/ (/ l k) (sin k)))) < 1.286753789222229e+222

    1. Initial program 55.9

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

    3. Applied add-cbrt-cube59.8

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

    4. Applied simplify54.5

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left(\tan k \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}}\]

    5. Taylor expanded around inf 48.8

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

    6. Applied simplify12.9

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

    8. Applied div-inv12.9

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

    9. Applied div-inv12.9

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

    10. Applied times-frac15.3

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

    11. Applied times-frac9.7

      \[\leadsto \color{blue}{\frac{\frac{2}{k}}{\frac{t}{\cos k}} \cdot \frac{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{1}{\ell}}}{\sin k \cdot \sin k}}\]

    12. Applied simplify1.2

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

    if -1.1162674882265014e-308 < (* (/ (/ 2 k) (/ t (cos k))) (* (/ l (sin k)) (/ (/ l k) (sin k)))) < -0.0 or 1.286753789222229e+222 < (* (/ (/ 2 k) (/ t (cos k))) (* (/ l (sin k)) (/ (/ l k) (sin k))))

    1. Initial program 40.2

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

    3. Applied add-cbrt-cube40.2

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

    4. Applied simplify21.3

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left(\tan k \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}}\]

    5. Taylor expanded around inf 15.1

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

    6. Applied simplify6.0

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

    8. Applied div-inv6.0

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

    9. Applied times-frac1.2

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{t}{\cos k}} \cdot \frac{\frac{1}{\frac{k}{\ell}}}{\sin k \cdot \sin k}}\]

    10. Applied simplify1.2

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t}{\cos k}} \cdot \color{blue}{\frac{\frac{\ell}{\sin k}}{\sin k \cdot k}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 6.6m)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' +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))))