Average Error: 32.1 → 15.4
Time: 4.6m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)\right) \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \tan k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right) \le -1.4027502456406147 \cdot 10^{+170} \lor \neg \left(\left(\left(\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)\right) \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \tan k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right) \le 3.2871527882312187 \cdot 10^{-229}\right):\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt[3]{\left(\tan k \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)} \cdot \left(\sqrt[3]{\left(\tan k \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)} \cdot \sqrt[3]{\left(\tan k \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\right)}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (* (* (* (* t (/ t l)) (tan k)) (+ (+ 1 1) (* (/ k t) (/ k t)))) (* (/ t l) (sin k))) < -1.4027502456406147e+170 or 3.2871527882312187e-229 < (* (* (* (* t (/ t l)) (tan k)) (+ (+ 1 1) (* (/ k t) (/ k t)))) (* (/ t l) (sin k)))

    1. Initial program 30.2

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

    3. Applied add-cube-cbrt30.3

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

    4. Applied times-frac28.1

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

    5. Applied simplify28.1

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

    6. Applied simplify20.4

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

    8. Applied associate-*l*19.0

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

    10. Applied associate-*l*18.1

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

    12. Applied associate-*l*16.4

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

    if -1.4027502456406147e+170 < (* (* (* (* t (/ t l)) (tan k)) (+ (+ 1 1) (* (/ k t) (/ k t)))) (* (/ t l) (sin k))) < 3.2871527882312187e-229

    1. Initial program 44.8

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

    3. Applied add-cube-cbrt44.9

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

    4. Applied times-frac37.8

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

    5. Applied simplify37.6

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

    6. Applied simplify16.5

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

    8. Applied associate-*l*7.8

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

    10. Applied add-cube-cbrt8.4

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

  4. Applied simplify15.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(\left(\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)\right) \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \tan k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right) \le -1.4027502456406147 \cdot 10^{+170} \lor \neg \left(\left(\left(\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)\right) \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \tan k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right) \le 3.2871527882312187 \cdot 10^{-229}\right):\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt[3]{\left(\tan k \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)} \cdot \left(\sqrt[3]{\left(\tan k \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)} \cdot \sqrt[3]{\left(\tan k \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\right)}\\ \end{array}}\]

Runtime

Time bar (total: 4.6m)Debug logProfile

herbie shell --seed 2019053 
(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))))