Average Error: 16.7 → 5.5
Time: 45.4s
Precision: 64
Internal Precision: 384
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\sqrt{1^2 + \left(\frac{\frac{U}{J + J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*}\right) \cdot \sqrt{\sqrt{1^2 + \left(\frac{\frac{U}{J + J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*} = -\infty:\\ \;\;\;\;\frac{\frac{1}{2} \cdot U}{\cos \left(\frac{1}{2} \cdot K\right)} \cdot \frac{\cos \left(\frac{K}{2}\right)}{\frac{1}{-2}}\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{U}{J + J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*\right)\\ \end{array}\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

  1. Split input into 2 regimes

  2. if (* (* (* (* J -2) (cos (/ K 2))) (sqrt (hypot 1 (/ (/ U (+ J J)) (cos (/ K 2)))))) (sqrt (hypot 1 (/ (/ U (+ J 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 simplify59.8

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

    3. Taylor expanded around inf 61.8

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

    4. Applied simplify31.8

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

    if (* (* (* (* J -2) (cos (/ K 2))) (sqrt (hypot 1 (/ (/ U (+ J J)) (cos (/ K 2)))))) (sqrt (hypot 1 (/ (/ U (+ J J)) (cos (/ K 2))))))

    1. Initial program 14.1

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

    2. Applied simplify4.0

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

    4. Applied associate-*l*4.0

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

Runtime

Time bar (total: 45.4s)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' +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)))))