Average Error: 17.4 → 13.3
Time: 45.5s
Precision: 64
Internal Precision: 320
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) = -\infty:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{U}{\cos \left(K \cdot \frac{1}{2}\right) \cdot J}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \le 8.341369795932956 \cdot 10^{+299}:\\ \;\;\;\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{U}{\cos \left(K \cdot \frac{1}{2}\right) \cdot J}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \end{array}\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))) < -inf.0 or 8.341369795932956e+299 < (* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))

    1. Initial program 58.9

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

    2. Taylor expanded around -inf 44.9

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

    if -inf.0 < (* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))) < 8.341369795932956e+299

    1. Initial program 0.1

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

  4. Final simplification13.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) = -\infty:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{U}{\cos \left(K \cdot \frac{1}{2}\right) \cdot J}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \le 8.341369795932956 \cdot 10^{+299}:\\ \;\;\;\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{U}{\cos \left(K \cdot \frac{1}{2}\right) \cdot J}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 45.5s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes17.413.312.45.081.4%
herbie shell --seed 2018274 
(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)))))