Average Error: 17.0 → 3.9
Time: 38.1s
Precision: 64
Internal Precision: 576
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{1^2 + \left(\frac{\sqrt[3]{\frac{\frac{U}{2}}{J}}}{\cos \left(\frac{K}{2}\right)} \cdot \left(\sqrt[3]{\frac{\frac{U}{2}}{J}} \cdot \sqrt[3]{\frac{\frac{U}{2}}{J}}\right)\right)^2}^* \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \le -1.7801884027861585 \cdot 10^{+308}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{1^2 + \left(\frac{\sqrt[3]{\frac{\frac{U}{2}}{J}}}{\cos \left(\frac{K}{2}\right)} \cdot \left(\sqrt[3]{\frac{\frac{U}{2}}{J}} \cdot \sqrt[3]{\frac{\frac{U}{2}}{J}}\right)\right)^2}^* \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \le 1.7824102095180647 \cdot 10^{+308}:\\ \;\;\;\;\sqrt{\sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*} \cdot \left(\sqrt{\sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

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 (* (hypot 1 (* (* (cbrt (/ (/ U 2) J)) (cbrt (/ (/ U 2) J))) (/ (cbrt (/ (/ U 2) J)) (cos (/ K 2))))) (* (cos (/ K 2)) (* -2 J))) < -1.7801884027861585e+308 or 1.7824102095180647e+308 < (* (hypot 1 (* (* (cbrt (/ (/ U 2) J)) (cbrt (/ (/ U 2) J))) (/ (cbrt (/ (/ U 2) J)) (cos (/ K 2))))) (* (cos (/ K 2)) (* -2 J)))

    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. Initial simplification59.9

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

    3. Taylor expanded around inf 29.6

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

    4. Simplified29.6

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

    if -1.7801884027861585e+308 < (* (hypot 1 (* (* (cbrt (/ (/ U 2) J)) (cbrt (/ (/ U 2) J))) (/ (cbrt (/ (/ U 2) J)) (cos (/ K 2))))) (* (cos (/ K 2)) (* -2 J))) < 1.7824102095180647e+308

    1. Initial program 11.0

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

    2. Initial simplification0.2

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

    4. Applied add-sqr-sqrt0.3

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

    5. Applied associate-*l*0.3

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

  4. Final simplification3.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1^2 + \left(\frac{\sqrt[3]{\frac{\frac{U}{2}}{J}}}{\cos \left(\frac{K}{2}\right)} \cdot \left(\sqrt[3]{\frac{\frac{U}{2}}{J}} \cdot \sqrt[3]{\frac{\frac{U}{2}}{J}}\right)\right)^2}^* \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \le -1.7801884027861585 \cdot 10^{+308}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{1^2 + \left(\frac{\sqrt[3]{\frac{\frac{U}{2}}{J}}}{\cos \left(\frac{K}{2}\right)} \cdot \left(\sqrt[3]{\frac{\frac{U}{2}}{J}} \cdot \sqrt[3]{\frac{\frac{U}{2}}{J}}\right)\right)^2}^* \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \le 1.7824102095180647 \cdot 10^{+308}:\\ \;\;\;\;\sqrt{\sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*} \cdot \left(\sqrt{\sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

Runtime

Time bar (total: 38.1s)Debug logProfile

herbie shell --seed 2018216 +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)))))