Average Error: 17.3 → 5.7
Time: 56.1s
Precision: 64
Internal Precision: 384
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\sqrt[3]{\sqrt{1^2 + \left(\frac{\frac{\frac{U}{J}}{2}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*} \cdot \sqrt[3]{\sqrt{1^2 + \left(\frac{\frac{\frac{U}{J}}{2}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\right) \cdot \sqrt[3]{\sqrt{1^2 + \left(\frac{\sqrt[3]{\frac{\frac{U}{2}}{J}} \cdot \sqrt[3]{\frac{\frac{U}{2}}{J}}}{\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right)}} \cdot \frac{\sqrt[3]{\frac{\frac{U}{2}}{J}}}{\sqrt[3]{\cos \left(\frac{K}{2}\right)}}\right)^2}^*} = -\infty:\\ \;\;\;\;\frac{\frac{1}{2} \cdot -2}{\frac{1}{U}}\\ \mathbf{if}\;\left(\left(\sqrt[3]{\sqrt{1^2 + \left(\frac{\frac{\frac{U}{J}}{2}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*} \cdot \sqrt[3]{\sqrt{1^2 + \left(\frac{\frac{\frac{U}{J}}{2}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\right) \cdot \sqrt[3]{\sqrt{1^2 + \left(\frac{\sqrt[3]{\frac{\frac{U}{2}}{J}} \cdot \sqrt[3]{\frac{\frac{U}{2}}{J}}}{\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right)}} \cdot \frac{\sqrt[3]{\frac{\frac{U}{2}}{J}}}{\sqrt[3]{\cos \left(\frac{K}{2}\right)}}\right)^2}^*} \le +\infty:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot -2}{\frac{1}{U}}\\ \end{array}\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

  1. Split input into 2 regimes

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

    1. Initial program 59.8

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

    2. Applied simplify59.8

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

    4. Applied associate-*l*59.8

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

    5. Taylor expanded around inf 61.5

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

    6. Applied simplify30.1

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot -2}{\frac{1}{U}}}\]

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

    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 simplify4.1

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

    4. Applied associate-*l*4.1

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

Runtime

Time bar (total: 56.1s)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' +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)))))