fix possible precision loss in xentropy and fair loss objectives

[jameslamb]

cppcheck shows the following warnings (#4539 (comment))

src/metric/dcg_calculator.cpp:110:0: style: The function 'CalDCGAtK' is never used. [unusedFunction]

src/metric/regression_metric.hpp:215:33: style: Expression 'log(1 + x)' can be replaced by 'log1p(x)' to avoid loss of precision. [unpreciseMathCall]
    return c * x - c * c * std::log(1.0f + x / c);
                                ^
src/metric/xentropy_metric.hpp:197:30: style: Expression 'log(1 + x)' can be replaced by 'log1p(x)' to avoid loss of precision. [unpreciseMathCall]
          double hhat = std::log(1.0f + std::exp(score[i]));  // auto-convert
                             ^
src/metric/xentropy_metric.hpp:203:30: style: Expression 'log(1 + x)' can be replaced by 'log1p(x)' to avoid loss of precision. [unpreciseMathCall]
          double hhat = std::log(1.0f + std::exp(score[i]));  // auto-convert
                             ^
src/objective/xentropy_objective.hpp:206:34: style: Expression 'log(1 + x)' can be replaced by 'log1p(x)' to avoid loss of precision. [unpreciseMathCall]
        const double hhat = std::log(1.0f + epf);
                                 ^
src/objective/xentropy_objective.hpp:234:22: style: Expression 'log(1 + x)' can be replaced by 'log1p(x)' to avoid loss of precision. [unpreciseMathCall]
    output[0] = std::log(1.0f + std::exp(input[0]));
                     ^
src/objective/xentropy_objective.hpp:262:38: style: Expression 'exp(x) - 1' can be replaced by 'expm1(x)' to avoid loss of precision. [unpreciseMathCall]
    double initscore = std::log(std::exp(havg) - 1.0f);
                                     ^

I think taking these suggestions could reduce the risk of precision loss in the calculation of these metrics.

From https://www.cplusplus.com/reference/cmath/log1p/

Returns the natural logarithm of one plus x.

For small magnitude values of x, log1p may be more accurate than log(1+x).

[github-actions]

This pull request has been automatically locked since there has not been any recent activity since it was closed. To start a new related discussion, open a new issue at https://github.com/microsoft/LightGBM/issues including a reference to this.