Average Error: 17.5 → 5.0
Time: 1.0m
Precision: 64
Internal Precision: 384
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{\frac{\frac{\frac{1}{2}}{\cos \left(K \cdot \frac{1}{2}\right)}}{\frac{\frac{1}{U}}{\frac{1}{J}}}} \cdot \sqrt[3]{\sqrt{1^2 + \left(\frac{\frac{U}{J + J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*}\right) \cdot \left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt[3]{\sqrt{1^2 + \left(\frac{\frac{U}{J + J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*}\right) = -\infty:\\ \;\;\;\;\frac{\frac{1}{2} \cdot U}{\cos \left(\frac{1}{2} \cdot K\right)} \cdot \frac{\cos \left(\frac{K}{2}\right)}{\frac{1}{-2}}\\ \mathbf{if}\;\left(\sqrt[3]{\frac{\frac{\frac{1}{2}}{\cos \left(K \cdot \frac{1}{2}\right)}}{\frac{\frac{1}{U}}{\frac{1}{J}}}} \cdot \sqrt[3]{\sqrt{1^2 + \left(\frac{\frac{U}{J + J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*}\right) \cdot \left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt[3]{\sqrt{1^2 + \left(\frac{\frac{U}{J + J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*}\right) \le 2.1678690799149524 \cdot 10^{+272}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{U}{J + J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2 \cdot \left(U \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\frac{K}{2}\right)}{\frac{\cos \left(K \cdot \frac{1}{2}\right)}{1}}\\ \end{array}\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

  1. Split input into 3 regimes

  2. if (* (* (cbrt (/ (/ 1/2 (cos (* K 1/2))) (/ (/ 1 U) (/ 1 J)))) (cbrt (hypot 1 (/ (/ U (+ J J)) (cos (/ K 2)))))) (* (* (* -2 J) (cos (/ K 2))) (cbrt (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 62.1

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

    4. Applied simplify35.2

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

    if (* (* (cbrt (/ (/ 1/2 (cos (* K 1/2))) (/ (/ 1 U) (/ 1 J)))) (cbrt (hypot 1 (/ (/ U (+ J J)) (cos (/ K 2)))))) (* (* (* -2 J) (cos (/ K 2))) (cbrt (hypot 1 (/ (/ U (+ J J)) (cos (/ K 2))))))) < 2.1678690799149524e+272

    1. Initial program 11.1

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

    2. Applied simplify0.2

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

    4. Applied associate-*l*0.1

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

    if 2.1678690799149524e+272 < (* (* (cbrt (/ (/ 1/2 (cos (* K 1/2))) (/ (/ 1 U) (/ 1 J)))) (cbrt (hypot 1 (/ (/ U (+ J J)) (cos (/ K 2)))))) (* (* (* -2 J) (cos (/ K 2))) (cbrt (hypot 1 (/ (/ U (+ J J)) (cos (/ K 2)))))))

    1. Initial program 51.1

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

    2. Applied simplify47.0

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

    4. Applied associate-*l*47.0

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

    5. Taylor expanded around inf 56.4

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

    6. Applied simplify31.4

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +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)))))