如何处理中间计算中的大数字？

计算科学 数值分析 C 精确 枫 数值限制
2021-11-28 17:59:04

我有一个相当长的表达式（https://pastebin.com/jUsxdCCs），它是 Maple 以符号方式生成的一组微分方程的解析解。我需要在 32 位系统中用 C 语言求解一组这样的方程。

长表达式有一个变量omega2，当被替换时会得到一个我感兴趣的数字。当这个操作在 Maple 中执行时，它会产生一个相当不错的浮点数，但是当我尝试在 C 中执行它时，（尝试使用omega2 = 100)，我得到nan了，原因是长表达式中的某些子表达式超出了double2.3E-308 到 1.7E+308 的范围。我已经能够使用某些系统上可用的“long double”数据类型来解决这个问题，但这不是独立于平台的，并且绝对不能在我计划针对的具有 32 位 ARM 内核的微控制器上使用。

我可以想象的一种解决方案是使用 GMP 库，它可以处理无限精度的数字。只是想知道是否有任何其他 hack，因为 GMP'ing 整个代码可能相当麻烦，而且我没有足够的内存用于目标机器上的 GMP。

表达方式：

0.137197706359762e-3 * (-0.708981540362208e2 * cosh(0.354490770181104e2 * sqrt(omega2)) * sin(0.354490770181104e2 * sqrt(omega2)) * (-0.727331795502673e6 * cosh(0.354490770181104e2 * sqrt(omega2)) * sin(0.354490770181104e2 * sqrt(omega2)) - 0.209935630488671e8 * cosh(0.354490770181104e2 * sqrt(omega2)) * cos(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) - 0.727331795502673e6 * cos(0.354490770181104e2 * sqrt(omega2)) * sinh(0.354490770181104e2 * sqrt(omega2))) * sqrt(omega2) / (0.348294222590750e3 * cosh(0.354490770181104e2 * sqrt(omega2)) * sinh(0.354490770181104e2 * sqrt(omega2)) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sqrt(omega2) - 0.348294222590750e3 * cos(0.354490770181104e2 * sqrt(omega2)) * pow(sinh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.348294222590750e3 * pow(cosh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * cos(0.354490770181104e2 * sqrt(omega2)) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.348294222590750e3 * cosh(0.354490770181104e2 * sqrt(omega2)) * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sinh(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.120668159897616e2 * pow(cosh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) + 0.120668159897616e2 * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(sinh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) + 0.502654824574368e4 * pow(sinh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * omega2 + 0.502654824574368e4 * pow(cosh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * omega2) - 0.708981540362208e2 * cosh(0.354490770181104e2 * sqrt(omega2)) * sin(0.354490770181104e2 * sqrt(omega2)) / (0.348294222590750e3 * cosh(0.354490770181104e2 * sqrt(omega2)) * sinh(0.354490770181104e2 * sqrt(omega2)) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sqrt(omega2) - 0.348294222590750e3 * cos(0.354490770181104e2 * sqrt(omega2)) * pow(sinh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.348294222590750e3 * pow(cosh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * cos(0.354490770181104e2 * sqrt(omega2)) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.348294222590750e3 * cosh(0.354490770181104e2 * sqrt(omega2)) * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sinh(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.120668159897616e2 * pow(cosh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) + 0.120668159897616e2 * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(sinh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) + 0.502654824574368e4 * pow(sinh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * omega2 + 0.502654824574368e4 * pow(cosh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * omega2) * sqrt(omega2) * (-0.209935630488671e8 * sinh(0.354490770181104e2 * sqrt(omega2)) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) - 0.727331795502673e6 * cosh(0.354490770181104e2 * sqrt(omega2)) * sin(0.354490770181104e2 * sqrt(omega2)) + 0.727331795502673e6 * cos(0.354490770181104e2 * sqrt(omega2)) * sinh(0.354490770181104e2 * sqrt(omega2))) - 0.708981540362208e2 * cos(0.354490770181104e2 * sqrt(omega2)) * sinh(0.354490770181104e2 * sqrt(omega2)) * (-0.727331795502673e6 * cosh(0.354490770181104e2 * sqrt(omega2)) * sin(0.354490770181104e2 * sqrt(omega2)) - 0.209935630488671e8 * cosh(0.354490770181104e2 * sqrt(omega2)) * cos(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) - 0.727331795502673e6 * cos(0.354490770181104e2 * sqrt(omega2)) * sinh(0.354490770181104e2 * sqrt(omega2))) * sqrt(omega2) / (0.348294222590750e3 * cosh(0.354490770181104e2 * sqrt(omega2)) * sinh(0.354490770181104e2 * sqrt(omega2)) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sqrt(omega2) - 0.348294222590750e3 * cos(0.354490770181104e2 * sqrt(omega2)) * pow(sinh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.348294222590750e3 * pow(cosh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * cos(0.354490770181104e2 * sqrt(omega2)) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.348294222590750e3 * cosh(0.354490770181104e2 * sqrt(omega2)) * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sinh(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.120668159897616e2 * pow(cosh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) + 0.120668159897616e2 * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(sinh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) + 0.502654824574368e4 * pow(sinh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * omega2 + 0.502654824574368e4 * pow(cosh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * omega2) + 0.708981540362208e2 * cos(0.354490770181104e2 * sqrt(omega2)) * sinh(0.354490770181104e2 * sqrt(omega2)) / (0.348294222590750e3 * cosh(0.354490770181104e2 * sqrt(omega2)) * sinh(0.354490770181104e2 * sqrt(omega2)) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sqrt(omega2) - 0.348294222590750e3 * cos(0.354490770181104e2 * sqrt(omega2)) * pow(sinh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.348294222590750e3 * pow(cosh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * cos(0.354490770181104e2 * sqrt(omega2)) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.348294222590750e3 * cosh(0.354490770181104e2 * sqrt(omega2)) * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sinh(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.120668159897616e2 * pow(cosh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) + 0.120668159897616e2 * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(sinh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) + 0.502654824574368e4 * pow(sinh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * omega2 + 0.502654824574368e4 * pow(cosh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * omega2) * sqrt(omega2) * (-0.209935630488671e8 * sinh(0.354490770181104e2 * sqrt(omega2)) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) - 0.727331795502673e6 * cosh(0.354490770181104e2 * sqrt(omega2)) * sin(0.354490770181104e2 * sqrt(omega2)) + 0.727331795502673e6 * cos(0.354490770181104e2 * sqrt(omega2)) * sinh(0.354490770181104e2 * sqrt(omega2)))) * pow(omega2, -0.1e1 / 0.2e1) - 0.833338826540431e3;

简化 Maple 中的表达式，得到：

.414741582884880e315/(.702517310158453e311*pow(.487478610966388,2.)+.702517310158453e311*pow(-.873134928776922,2.)+502666.891390358*pow(.449112744897583e154,2.)*pow(.487478610966388,2.)+502666.891390358*pow(-.873134928776922,2.)*pow(.449112744897583e154,2.))*pow(.1e3,-.500000000000000)-833.338826540431

可以看出，sinh 和 cosh 项在内部爆炸，但最终抵消了。

2个回答

我认为有一个简单的方法可以做到这一点。你有一个相同 cosh/sinh 项的有理函数，其中每个表达式都是 cosh/sinh 中的齐次多项式，唯一的问题是这些指数项溢出。当这些项接近无穷大时，该函数不会发散，因此如果您将每个分子和分母除以相同的 cosh 幂，您将摆脱溢出并只留下无害的下溢。

由于 cosh/sinh 的幂必须匹配（表达式是有界的），我们不需要手动确定它们：我们可以（字面上）替换每个cosh⁡(⋯)经过1，并且每个sinh⁡(⋯)经过tanh⁡(⋯)，得到以下表达式：

0.137197706359762e-3 * (-0.708981540362208e2 * 1 * sin(0.354490770181104e2 * sqrt(omega2)) * (-0.727331795502673e6 * 1 * sin(0.354490770181104e2 * sqrt(omega2)) - 0.209935630488671e8 * 1 * cos(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) - 0.727331795502673e6 * cos(0.354490770181104e2 * sqrt(omega2)) * tanh(0.354490770181104e2 * sqrt(omega2))) * sqrt(omega2) / (0.348294222590750e3 * 1 * tanh(0.354490770181104e2 * sqrt(omega2)) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sqrt(omega2) - 0.348294222590750e3 * cos(0.354490770181104e2 * sqrt(omega2)) * pow(tanh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.348294222590750e3 * pow(1, 0.2e1) * cos(0.354490770181104e2 * sqrt(omega2)) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.348294222590750e3 * 1 * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * tanh(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.120668159897616e2 * pow(1, 0.2e1) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) + 0.120668159897616e2 * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(tanh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) + 0.502654824574368e4 * pow(tanh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * omega2 + 0.502654824574368e4 * pow(1, 0.2e1) * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * omega2) - 0.708981540362208e2 * 1 * sin(0.354490770181104e2 * sqrt(omega2)) / (0.348294222590750e3 * 1 * tanh(0.354490770181104e2 * sqrt(omega2)) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sqrt(omega2) - 0.348294222590750e3 * cos(0.354490770181104e2 * sqrt(omega2)) * pow(tanh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.348294222590750e3 * pow(1, 0.2e1) * cos(0.354490770181104e2 * sqrt(omega2)) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.348294222590750e3 * 1 * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * tanh(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.120668159897616e2 * pow(1, 0.2e1) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) + 0.120668159897616e2 * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(tanh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) + 0.502654824574368e4 * pow(tanh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * omega2 + 0.502654824574368e4 * pow(1, 0.2e1) * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * omega2) * sqrt(omega2) * (-0.209935630488671e8 * tanh(0.354490770181104e2 * sqrt(omega2)) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) - 0.727331795502673e6 * 1 * sin(0.354490770181104e2 * sqrt(omega2)) + 0.727331795502673e6 * cos(0.354490770181104e2 * sqrt(omega2)) * tanh(0.354490770181104e2 * sqrt(omega2))) - 0.708981540362208e2 * cos(0.354490770181104e2 * sqrt(omega2)) * tanh(0.354490770181104e2 * sqrt(omega2)) * (-0.727331795502673e6 * 1 * sin(0.354490770181104e2 * sqrt(omega2)) - 0.209935630488671e8 * 1 * cos(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) - 0.727331795502673e6 * cos(0.354490770181104e2 * sqrt(omega2)) * tanh(0.354490770181104e2 * sqrt(omega2))) * sqrt(omega2) / (0.348294222590750e3 * 1 * tanh(0.354490770181104e2 * sqrt(omega2)) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sqrt(omega2) - 0.348294222590750e3 * cos(0.354490770181104e2 * sqrt(omega2)) * pow(tanh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.348294222590750e3 * pow(1, 0.2e1) * cos(0.354490770181104e2 * sqrt(omega2)) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.348294222590750e3 * 1 * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * tanh(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.120668159897616e2 * pow(1, 0.2e1) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) + 0.120668159897616e2 * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(tanh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) + 0.502654824574368e4 * pow(tanh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * omega2 + 0.502654824574368e4 * pow(1, 0.2e1) * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * omega2) + 0.708981540362208e2 * cos(0.354490770181104e2 * sqrt(omega2)) * tanh(0.354490770181104e2 * sqrt(omega2)) / (0.348294222590750e3 * 1 * tanh(0.354490770181104e2 * sqrt(omega2)) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sqrt(omega2) - 0.348294222590750e3 * cos(0.354490770181104e2 * sqrt(omega2)) * pow(tanh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.348294222590750e3 * pow(1, 0.2e1) * cos(0.354490770181104e2 * sqrt(omega2)) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.348294222590750e3 * 1 * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * tanh(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) + 0.120668159897616e2 * pow(1, 0.2e1) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) + 0.120668159897616e2 * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(tanh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) + 0.502654824574368e4 * pow(tanh(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * pow(sin(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * omega2 + 0.502654824574368e4 * pow(1, 0.2e1) * pow(cos(0.354490770181104e2 * sqrt(omega2)), 0.2e1) * omega2) * sqrt(omega2) * (-0.209935630488671e8 * tanh(0.354490770181104e2 * sqrt(omega2)) * sin(0.354490770181104e2 * sqrt(omega2)) * sqrt(omega2) - 0.727331795502673e6 * 1 * sin(0.354490770181104e2 * sqrt(omega2)) + 0.727331795502673e6 * cos(0.354490770181104e2 * sqrt(omega2)) * tanh(0.354490770181104e2 * sqrt(omega2)))) * pow(omega2, -0.1e1 / 0.2e1) - 0.833338826540431e3;

这仅适用于 cosh/sinh 中的每个表达式都是齐次的并且分子和分母的度数相同时，但在我看来它们是相同的，并且在数字上也可以检查。我认为如果权力不匹配，同样的方法会起作用，但你可能必须手动取消权力或使用适当的 CAS。

如果您不想（或不能）处理表达式，只需使用您选择的多精度库。下面是 C++ 中 Boost.Multiprecision 的示例。由于运算符和函数重载，这具有您根本不必触及原始表达式的宝贵优势。

#include <iostream>
#include <cmath>

#include <boost/multiprecision/cpp_dec_float.hpp>

using boost::multiprecision::cpp_dec_float_50;

// Forward declaration for brevity
cpp_dec_float_50 expr(cpp_dec_float_50 const &omega2);


int main ()
{
  std::cout << expr (100) << '\n';
}

// Indentation by GNU indent
cpp_dec_float_50 expr(cpp_dec_float_50 const &omega2)
{
  return
    0.137197706359762e-3 * (-0.708981540362208e2 *
                cosh (0.354490770181104e2 * sqrt (omega2)) *
                sin (0.354490770181104e2 * sqrt (omega2)) *
                (-0.727331795502673e6 *
                 cosh (0.354490770181104e2 * sqrt (omega2)) *
                 sin (0.354490770181104e2 * sqrt (omega2)) -
                 0.209935630488671e8 * cosh (0.354490770181104e2 *
                             sqrt (omega2)) *
                 cos (0.354490770181104e2 * sqrt (omega2)) *
                 sqrt (omega2) -
                 0.727331795502673e6 * cos (0.354490770181104e2 *
                            sqrt (omega2)) *
                 sinh (0.354490770181104e2 * sqrt (omega2))) *
                sqrt (omega2) / (0.348294222590750e3 *
                         cosh (0.354490770181104e2 *
                           sqrt (omega2)) *
                         sinh (0.354490770181104e2 *
                           sqrt (omega2)) *
                         pow (sin
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) * sqrt (omega2) -
                         0.348294222590750e3 *
                         cos (0.354490770181104e2 *
                          sqrt (omega2)) *
                         pow (sinh
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) *
                         sin (0.354490770181104e2 *
                          sqrt (omega2)) *
                         sqrt (omega2) +
                         0.348294222590750e3 *
                         pow (cosh
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) *
                         cos (0.354490770181104e2 *
                          sqrt (omega2)) *
                         sin (0.354490770181104e2 *
                          sqrt (omega2)) *
                         sqrt (omega2) +
                         0.348294222590750e3 *
                         cosh (0.354490770181104e2 *
                           sqrt (omega2)) *
                         pow (cos
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) *
                         sinh (0.354490770181104e2 *
                           sqrt (omega2)) *
                         sqrt (omega2) +
                         0.120668159897616e2 *
                         pow (cosh
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) *
                         pow (sin
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) +
                         0.120668159897616e2 *
                         pow (cos
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) *
                         pow (sinh
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) +
                         0.502654824574368e4 *
                         pow (sinh
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) *
                         pow (sin
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) * omega2 +
                         0.502654824574368e4 *
                         pow (cosh
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) *
                         pow (cos
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) * omega2) -
                0.708981540362208e2 * cosh (0.354490770181104e2 *
                            sqrt (omega2)) *
                sin (0.354490770181104e2 * sqrt (omega2)) /
                (0.348294222590750e3 *
                 cosh (0.354490770181104e2 * sqrt (omega2)) *
                 sinh (0.354490770181104e2 * sqrt (omega2)) *
                 pow (sin (0.354490770181104e2 * sqrt (omega2)),
                  0.2e1) * sqrt (omega2) -
                 0.348294222590750e3 * cos (0.354490770181104e2 *
                            sqrt (omega2)) *
                 pow (sinh (0.354490770181104e2 * sqrt (omega2)),
                  0.2e1) * sin (0.354490770181104e2 *
                        sqrt (omega2)) *
                 sqrt (omega2) +
                 0.348294222590750e3 *
                 pow (cosh (0.354490770181104e2 * sqrt (omega2)),
                  0.2e1) * cos (0.354490770181104e2 *
                        sqrt (omega2)) *
                 sin (0.354490770181104e2 * sqrt (omega2)) *
                 sqrt (omega2) +
                 0.348294222590750e3 * cosh (0.354490770181104e2 *
                             sqrt (omega2)) *
                 pow (cos (0.354490770181104e2 * sqrt (omega2)),
                  0.2e1) * sinh (0.354490770181104e2 *
                         sqrt (omega2)) *
                 sqrt (omega2) +
                 0.120668159897616e2 *
                 pow (cosh (0.354490770181104e2 * sqrt (omega2)),
                  0.2e1) * pow (sin (0.354490770181104e2 *
                             sqrt (omega2)),
                        0.2e1) +
                 0.120668159897616e2 *
                 pow (cos (0.354490770181104e2 * sqrt (omega2)),
                  0.2e1) * pow (sinh (0.354490770181104e2 *
                              sqrt (omega2)),
                        0.2e1) +
                 0.502654824574368e4 *
                 pow (sinh (0.354490770181104e2 * sqrt (omega2)),
                  0.2e1) * pow (sin (0.354490770181104e2 *
                             sqrt (omega2)),
                        0.2e1) * omega2 +
                 0.502654824574368e4 *
                 pow (cosh (0.354490770181104e2 * sqrt (omega2)),
                  0.2e1) * pow (cos (0.354490770181104e2 *
                             sqrt (omega2)),
                        0.2e1) * omega2) *
                sqrt (omega2) * (-0.209935630488671e8 *
                         sinh (0.354490770181104e2 *
                           sqrt (omega2)) *
                         sin (0.354490770181104e2 *
                          sqrt (omega2)) *
                         sqrt (omega2) -
                         0.727331795502673e6 *
                         cosh (0.354490770181104e2 *
                           sqrt (omega2)) *
                         sin (0.354490770181104e2 *
                          sqrt (omega2)) +
                         0.727331795502673e6 *
                         cos (0.354490770181104e2 *
                          sqrt (omega2)) *
                         sinh (0.354490770181104e2 *
                           sqrt (omega2))) -
                0.708981540362208e2 * cos (0.354490770181104e2 *
                               sqrt (omega2)) *
                sinh (0.354490770181104e2 * sqrt (omega2)) *
                (-0.727331795502673e6 *
                 cosh (0.354490770181104e2 * sqrt (omega2)) *
                 sin (0.354490770181104e2 * sqrt (omega2)) -
                 0.209935630488671e8 * cosh (0.354490770181104e2 *
                             sqrt (omega2)) *
                 cos (0.354490770181104e2 * sqrt (omega2)) *
                 sqrt (omega2) -
                 0.727331795502673e6 * cos (0.354490770181104e2 *
                            sqrt (omega2)) *
                 sinh (0.354490770181104e2 * sqrt (omega2))) *
                sqrt (omega2) / (0.348294222590750e3 *
                         cosh (0.354490770181104e2 *
                           sqrt (omega2)) *
                         sinh (0.354490770181104e2 *
                           sqrt (omega2)) *
                         pow (sin
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) * sqrt (omega2) -
                         0.348294222590750e3 *
                         cos (0.354490770181104e2 *
                          sqrt (omega2)) *
                         pow (sinh
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) *
                         sin (0.354490770181104e2 *
                          sqrt (omega2)) *
                         sqrt (omega2) +
                         0.348294222590750e3 *
                         pow (cosh
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) *
                         cos (0.354490770181104e2 *
                          sqrt (omega2)) *
                         sin (0.354490770181104e2 *
                          sqrt (omega2)) *
                         sqrt (omega2) +
                         0.348294222590750e3 *
                         cosh (0.354490770181104e2 *
                           sqrt (omega2)) *
                         pow (cos
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) *
                         sinh (0.354490770181104e2 *
                           sqrt (omega2)) *
                         sqrt (omega2) +
                         0.120668159897616e2 *
                         pow (cosh
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) *
                         pow (sin
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) +
                         0.120668159897616e2 *
                         pow (cos
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) *
                         pow (sinh
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) +
                         0.502654824574368e4 *
                         pow (sinh
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) *
                         pow (sin
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) * omega2 +
                         0.502654824574368e4 *
                         pow (cosh
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) *
                         pow (cos
                          (0.354490770181104e2 *
                           sqrt (omega2)),
                          0.2e1) * omega2) +
                0.708981540362208e2 * cos (0.354490770181104e2 *
                               sqrt (omega2)) *
                sinh (0.354490770181104e2 * sqrt (omega2)) /
                (0.348294222590750e3 *
                 cosh (0.354490770181104e2 * sqrt (omega2)) *
                 sinh (0.354490770181104e2 * sqrt (omega2)) *
                 pow (sin (0.354490770181104e2 * sqrt (omega2)),
                  0.2e1) * sqrt (omega2) -
                 0.348294222590750e3 * cos (0.354490770181104e2 *
                            sqrt (omega2)) *
                 pow (sinh (0.354490770181104e2 * sqrt (omega2)),
                  0.2e1) * sin (0.354490770181104e2 *
                        sqrt (omega2)) *
                 sqrt (omega2) +
                 0.348294222590750e3 *
                 pow (cosh (0.354490770181104e2 * sqrt (omega2)),
                  0.2e1) * cos (0.354490770181104e2 *
                        sqrt (omega2)) *
                 sin (0.354490770181104e2 * sqrt (omega2)) *
                 sqrt (omega2) +
                 0.348294222590750e3 * cosh (0.354490770181104e2 *
                             sqrt (omega2)) *
                 pow (cos (0.354490770181104e2 * sqrt (omega2)),
                  0.2e1) * sinh (0.354490770181104e2 *
                         sqrt (omega2)) *
                 sqrt (omega2) +
                 0.120668159897616e2 *
                 pow (cosh (0.354490770181104e2 * sqrt (omega2)),
                  0.2e1) * pow (sin (0.354490770181104e2 *
                             sqrt (omega2)),
                        0.2e1) +
                 0.120668159897616e2 *
                 pow (cos (0.354490770181104e2 * sqrt (omega2)),
                  0.2e1) * pow (sinh (0.354490770181104e2 *
                              sqrt (omega2)),
                        0.2e1) +
                 0.502654824574368e4 *
                 pow (sinh (0.354490770181104e2 * sqrt (omega2)),
                  0.2e1) * pow (sin (0.354490770181104e2 *
                             sqrt (omega2)),
                        0.2e1) * omega2 +
                 0.502654824574368e4 *
                 pow (cosh (0.354490770181104e2 * sqrt (omega2)),
                  0.2e1) * pow (cos (0.354490770181104e2 *
                             sqrt (omega2)),
                        0.2e1) * omega2) *
                sqrt (omega2) * (-0.209935630488671e8 *
                         sinh (0.354490770181104e2 *
                           sqrt (omega2)) *
                         sin (0.354490770181104e2 *
                          sqrt (omega2)) *
                         sqrt (omega2) -
                         0.727331795502673e6 *
                         cosh (0.354490770181104e2 *
                           sqrt (omega2)) *
                         sin (0.354490770181104e2 *
                          sqrt (omega2)) +
                         0.727331795502673e6 *
                         cos (0.354490770181104e2 *
                          sqrt (omega2)) *
                         sinh (0.354490770181104e2 *
                           sqrt (omega2)))) *
    pow (omega2, -0.1e1 / 0.2e1) - 0.833338826540431e3;
}
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