Average Error: 17.4 → 7.8
Time: 40.9s
Precision: 64
Internal Precision: 576
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;J \le -8.757656270012445 \cdot 10^{-272}:\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\sqrt{1^2 + \left(\frac{\frac{U}{2 \cdot \cos \left(\frac{K}{2}\right)}}{J}\right)^2}^* \cdot \left(-2 \cdot J\right)\right)\\ \mathbf{elif}\;J \le 4.7060693406199846 \cdot 10^{-272}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\sqrt{1^2 + \left(\frac{\frac{U}{2 \cdot \cos \left(\frac{K}{2}\right)}}{J}\right)^2}^* \cdot \left(-2 \cdot J\right)\right)\\ \end{array}\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if J < -8.757656270012445e-272 or 4.7060693406199846e-272 < J

    1. Initial program 15.7

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]

    2. Initial simplification6.3

      \[\leadsto \sqrt{1^2 + \left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot 2\right) \cdot J}\right)^2}^* \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt6.5

      \[\leadsto \color{blue}{\left(\sqrt{\sqrt{1^2 + \left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot 2\right) \cdot J}\right)^2}^*} \cdot \sqrt{\sqrt{1^2 + \left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot 2\right) \cdot J}\right)^2}^*}\right)} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\]

    5. Applied associate-*l*6.5

      \[\leadsto \color{blue}{\sqrt{\sqrt{1^2 + \left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot 2\right) \cdot J}\right)^2}^*} \cdot \left(\sqrt{\sqrt{1^2 + \left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot 2\right) \cdot J}\right)^2}^*} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\right)}\]
    6. Using strategy rm

    7. Applied pow16.5

      \[\leadsto \sqrt{\sqrt{1^2 + \left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot 2\right) \cdot J}\right)^2}^*} \cdot \left(\sqrt{\sqrt{1^2 + \left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot 2\right) \cdot J}\right)^2}^*} \cdot \color{blue}{{\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)}^{1}}\right)\]

    8. Applied pow16.5

      \[\leadsto \sqrt{\sqrt{1^2 + \left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot 2\right) \cdot J}\right)^2}^*} \cdot \left(\color{blue}{{\left(\sqrt{\sqrt{1^2 + \left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot 2\right) \cdot J}\right)^2}^*}\right)}^{1}} \cdot {\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)}^{1}\right)\]

    9. Applied pow-prod-down6.5

      \[\leadsto \sqrt{\sqrt{1^2 + \left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot 2\right) \cdot J}\right)^2}^*} \cdot \color{blue}{{\left(\sqrt{\sqrt{1^2 + \left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot 2\right) \cdot J}\right)^2}^*} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\right)}^{1}}\]

    10. Applied pow16.5

      \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{1^2 + \left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot 2\right) \cdot J}\right)^2}^*}\right)}^{1}} \cdot {\left(\sqrt{\sqrt{1^2 + \left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot 2\right) \cdot J}\right)^2}^*} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\right)}^{1}\]

    11. Applied pow-prod-down6.5

      \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{1^2 + \left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot 2\right) \cdot J}\right)^2}^*} \cdot \left(\sqrt{\sqrt{1^2 + \left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot 2\right) \cdot J}\right)^2}^*} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\right)\right)}^{1}}\]

    12. Simplified6.4

      \[\leadsto {\color{blue}{\left(\left(\left(-2 \cdot J\right) \cdot \sqrt{1^2 + \left(\frac{\frac{U}{2 \cdot \cos \left(\frac{K}{2}\right)}}{J}\right)^2}^*\right) \cdot \cos \left(\frac{K}{2}\right)\right)}}^{1}\]

    if -8.757656270012445e-272 < J < 4.7060693406199846e-272

    1. Initial program 44.2

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]

    2. Initial simplification28.1

      \[\leadsto \sqrt{1^2 + \left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot 2\right) \cdot J}\right)^2}^* \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\]

    3. Taylor expanded around -inf 30.3

      \[\leadsto \color{blue}{-1 \cdot U}\]

    4. Simplified30.3

      \[\leadsto \color{blue}{-U}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;J \le -8.757656270012445 \cdot 10^{-272}:\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\sqrt{1^2 + \left(\frac{\frac{U}{2 \cdot \cos \left(\frac{K}{2}\right)}}{J}\right)^2}^* \cdot \left(-2 \cdot J\right)\right)\\ \mathbf{elif}\;J \le 4.7060693406199846 \cdot 10^{-272}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\sqrt{1^2 + \left(\frac{\frac{U}{2 \cdot \cos \left(\frac{K}{2}\right)}}{J}\right)^2}^* \cdot \left(-2 \cdot J\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 40.9s)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  (* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))