Average Error: 17.1 → 6.0
Time: 28.0s
Precision: 64
Internal Precision: 384
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1^2 + \left(\frac{\frac{U}{J + J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^* = -\infty:\\ \;\;\;\;\frac{\frac{1}{2} \cdot U}{\cos \left(\frac{1}{2} \cdot K\right)} \cdot \frac{\cos \left(\frac{K}{2}\right)}{\frac{1}{-2}}\\ \mathbf{if}\;\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1^2 + \left(\frac{\frac{U}{J + J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^* \le +\infty:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \sqrt{1^2 + \left(\frac{\frac{U}{J + J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot U}{\cos \left(\frac{1}{2} \cdot K\right)} \cdot \frac{\cos \left(\frac{K}{2}\right)}{\frac{1}{-2}}\\ \end{array}\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

  1. Split input into 2 regimes

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

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

    3. Taylor expanded around inf 61.9

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

    4. Applied simplify32.3

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

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

    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 simplify4.2

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

    4. Applied associate-*l*4.2

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

Runtime

Time bar (total: 28.0s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +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)))))