Average Error: 16.8 → 8.7
Time: 51.0s
Precision: 64
Internal Precision: 576
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;J \le -3.067009712891087 \cdot 10^{-186} \lor \neg \left(J \le -3.1959466840378386 \cdot 10^{-283}\right):\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1^2 + \left(\frac{\frac{U}{2}}{\cos \left(\frac{K}{2}\right) \cdot J}\right)^2}^*\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

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 < -3.067009712891087e-186 or -3.1959466840378386e-283 < J

    1. Initial program 15.1

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

    2. Initial simplification6.5

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

    4. Applied associate-/l/6.4

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

    if -3.067009712891087e-186 < J < -3.1959466840378386e-283

    1. Initial program 37.2

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

    2. Initial simplification20.5

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

    3. Taylor expanded around inf 35.6

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

    4. Simplified35.6

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;J \le -3.067009712891087 \cdot 10^{-186} \lor \neg \left(J \le -3.1959466840378386 \cdot 10^{-283}\right):\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1^2 + \left(\frac{\frac{U}{2}}{\cos \left(\frac{K}{2}\right) \cdot J}\right)^2}^*\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

Runtime

Time bar (total: 51.0s)Debug logProfile

herbie shell --seed 2018242 +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)))))