Average Error: 16.9 → 8.1
Time: 21.6s
Precision: 64
Internal Precision: 576
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;J \le -3.138216443546463 \cdot 10^{-214} \lor \neg \left(J \le 2.5854915047549 \cdot 10^{-275}\right):\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \cdot \sqrt{1^2 + \left(\frac{1}{2} \cdot \frac{U}{J \cdot \cos \left(K \cdot \frac{1}{2}\right)}\right)^2}^*\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

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 J < -3.138216443546463e-214 or 2.5854915047549e-275 < J

    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. Initial simplification5.4

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

    3. Taylor expanded around -inf 5.4

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

    if -3.138216443546463e-214 < J < 2.5854915047549e-275

    1. Initial program 42.0

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

    2. Initial simplification25.0

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

    4. Applied add-cube-cbrt25.8

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

    5. Taylor expanded around inf 33.0

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

    6. Simplified33.0

      \[\leadsto \color{blue}{-U}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;J \le -3.138216443546463 \cdot 10^{-214} \lor \neg \left(J \le 2.5854915047549 \cdot 10^{-275}\right):\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \cdot \sqrt{1^2 + \left(\frac{1}{2} \cdot \frac{U}{J \cdot \cos \left(K \cdot \frac{1}{2}\right)}\right)^2}^*\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

Runtime

Time bar (total: 21.6s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes7.38.13.44.0-19.9%
herbie shell --seed 2018286 +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)))))