Average Error: 17.1 → 5.7
Time: 56.9s
Precision: 64
Internal Precision: 384
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{1^2 + \left(\frac{\frac{\frac{U}{J}}{2}}{\cos \left(\frac{K}{2}\right)}\right)^2}^* \cdot \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) = -\infty:\\ \;\;\;\;\left(-2 \cdot U\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*\right)\\ \end{array}\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

  1. Split input into 2 regimes

  2. if (* (hypot 1 (/ (/ (/ U J) 2) (cos (/ K 2)))) (* (* -2 J) (cos (/ K 2)))) < -inf.0

    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 add-cube-cbrt59.9

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

    5. Taylor expanded around 0 63.9

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{e^{\log J + \log -2} \cdot U}{J}}\]

    6. Applied simplify30.6

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

    if -inf.0 < (* (hypot 1 (/ (/ (/ U J) 2) (cos (/ K 2)))) (* (* -2 J) (cos (/ K 2))))

    1. Initial program 14.2

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

    2. Applied simplify4.1

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

    4. Applied associate-*l*4.0

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

Runtime

Time bar (total: 56.9s)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' +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)))))