Average Error: 17.3 → 5.7
Time: 56.8s
Precision: 64
Internal Precision: 384
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \sqrt[3]{\sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*}\right) \cdot \left(\left(\sqrt[3]{\sqrt{1^2 + \left(\frac{\frac{1}{2}}{J} \cdot U\right)^2}^*} \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt[3]{\sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*}\right) = -\infty:\\ \;\;\;\;\frac{\frac{1}{2} \cdot -2}{\frac{1}{U}}\\ \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \sqrt[3]{\sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*}\right) \cdot \left(\left(\sqrt[3]{\sqrt{1^2 + \left(\frac{\frac{1}{2}}{J} \cdot U\right)^2}^*} \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt[3]{\sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*}\right) \le +\infty:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot -2}{\frac{1}{U}}\\ \end{array}\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

  1. Split input into 2 regimes

  2. if (* (* (cos (/ K 2)) (cbrt (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))))) (* (* (cbrt (hypot 1 (* (/ 1/2 J) U))) (* -2 J)) (cbrt (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2))))))) < -inf.0 or +inf.0 < (* (* (cos (/ K 2)) (cbrt (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))))) (* (* (cbrt (hypot 1 (* (/ 1/2 J) U))) (* -2 J)) (cbrt (hypot 1 (/ (/ (/ U 2) J) (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 simplify59.7

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

    4. Applied associate-*l*59.7

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

    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 simplify30.2

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

    if -inf.0 < (* (* (cos (/ K 2)) (cbrt (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))))) (* (* (cbrt (hypot 1 (* (/ 1/2 J) U))) (* -2 J)) (cbrt (hypot 1 (/ (/ (/ U 2) J) (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: 56.8s)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)))))