Average Error: 17.2 → 6.0
Time: 1.1m
Precision: 64
Internal Precision: 320
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(J \cdot -2\right) \cdot \left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right)}\right)\right) \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right)}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^* \le 1.7714660295444198 \cdot 10^{+308}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\frac{1}{2} \cdot U\right)\\ \end{array}\]

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 -2) (* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2))))) (cbrt (cos (/ K 2)))) (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2))))) < 1.7714660295444198e+308

    1. Initial program 14.5

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

    2. Applied simplify4.2

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

    4. Applied associate-*l*4.2

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

    if 1.7714660295444198e+308 < (* (* (* (* J -2) (* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2))))) (cbrt (cos (/ K 2)))) (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))))

    1. Initial program 59.7

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

    2. Applied simplify59.7

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

    4. Applied associate-*l*59.7

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

    5. Taylor expanded around inf 61.9

      \[\leadsto \left(J \cdot -2\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{U}{J}\right)}\]

    6. Applied simplify34.1

      \[\leadsto \color{blue}{-2 \cdot \left(\frac{1}{2} \cdot U\right)}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018166 +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)))))