Average Error: 31.9 → 12.1
Time: 4.5m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{\ell}{t} \cdot \cos k\right) \cdot \frac{\frac{\frac{2}{t}}{\frac{\sin k}{\frac{\ell}{t}}}}{(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\sin k\right) + \left(\sin k\right))_*} \le -2.873184526993056 \cdot 10^{+293}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{if}\;\left(\frac{\ell}{t} \cdot \cos k\right) \cdot \frac{\frac{\frac{2}{t}}{\frac{\sin k}{\frac{\ell}{t}}}}{(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\sin k\right) + \left(\sin k\right))_*} \le 3.568245556703523 \cdot 10^{+301}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \cos k\right) \cdot \frac{\frac{\frac{2}{t}}{\frac{\sin k}{\frac{\ell}{t}}}}{(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\sin k\right) + \left(\sin k\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\sin \left(-k\right)}}{{\left(\frac{-1}{t}\right)}^{3} \cdot \left({-1}^{3} \cdot \tan k\right)}}{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + \left(1 + 1\right))_*}{\ell \cdot \ell}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if (* (* (/ l t) (cos k)) (/ (/ (/ 2 t) (/ (sin k) (/ l t))) (fma (fma (/ k t) (/ k t) 1) (sin k) (sin k)))) < -2.873184526993056e+293

    1. Initial program 49.9

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

    3. Applied add-cube-cbrt50.0

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

    4. Applied times-frac38.6

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

    5. Applied simplify38.4

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

    6. Applied simplify25.8

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

    if -2.873184526993056e+293 < (* (* (/ l t) (cos k)) (/ (/ (/ 2 t) (/ (sin k) (/ l t))) (fma (fma (/ k t) (/ k t) 1) (sin k) (sin k)))) < 3.568245556703523e+301

    1. Initial program 26.6

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

    3. Applied add-cube-cbrt26.7

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

    4. Applied times-frac23.5

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

    5. Applied simplify23.4

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

    6. Applied simplify12.8

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

    8. Applied tan-quot12.8

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

    9. Applied associate-*l/12.8

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

    10. Applied associate-*l/10.6

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

    11. Applied frac-times10.2

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

    12. Applied associate-*l/8.6

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{t} \cdot \cos k}}}\]
    13. Using strategy rm

    14. Applied associate-*l*5.5

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

    15. Taylor expanded around -inf 5.4

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

    16. Applied simplify4.9

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

    if 3.568245556703523e+301 < (* (* (/ l t) (cos k)) (/ (/ (/ 2 t) (/ (sin k) (/ l t))) (fma (fma (/ k t) (/ k t) 1) (sin k) (sin k))))

    1. Initial program 61.8

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

    2. Taylor expanded around -inf 62.3

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

    3. Applied simplify53.2

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

Runtime

Time bar (total: 4.5m)Debug logProfile

herbie shell --seed 2018195 +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))))