Average Error: 17.6 → 4.5
Time: 1.5m
Precision: 64
Internal Precision: 576
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(J \cdot -2\right) \cdot \left(\left(\sqrt[3]{\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*} \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*}\right) \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*}\right) \le -1.771671894238396 \cdot 10^{+308}:\\ \;\;\;\;-2 \cdot \left(\frac{1}{2} \cdot U\right)\\ \mathbf{if}\;\left(J \cdot -2\right) \cdot \left(\left(\sqrt[3]{\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*} \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*}\right) \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*}\right) \le 1.779761014579861 \cdot 10^{+308}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{U}{2}}{J} \cdot \frac{1}{\cos \left(\frac{K}{2}\right)}\right)^2}^*\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\frac{1}{2} \cdot U\right)\\ \end{array}\]

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)) (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))))) (cbrt (* (cos (/ K 2)) (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2))))))) (cbrt (* (cos (/ K 2)) (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))))))) < -1.771671894238396e+308 or 1.779761014579861e+308 < (* (* J -2) (* (* (cbrt (* (cos (/ K 2)) (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))))) (cbrt (* (cos (/ K 2)) (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2))))))) (cbrt (* (cos (/ K 2)) (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2))))))))

    1. Initial program 59.9

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

    2. Applied simplify59.9

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

    4. Applied associate-*l*59.9

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

    5. Taylor expanded around inf 62.0

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

    6. Applied simplify33.9

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

    if -1.771671894238396e+308 < (* (* J -2) (* (* (cbrt (* (cos (/ K 2)) (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))))) (cbrt (* (cos (/ K 2)) (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2))))))) (cbrt (* (cos (/ K 2)) (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))))))) < 1.779761014579861e+308

    1. Initial program 11.3

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

    2. Applied simplify0.2

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

    4. Applied associate-*l*0.1

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

    6. Applied div-inv0.2

      \[\leadsto \left(J \cdot -2\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \color{blue}{\left(\frac{\frac{U}{2}}{J} \cdot \frac{1}{\cos \left(\frac{K}{2}\right)}\right)}^2}^*\right)\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018207 +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)))))