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## 3.14 Composed Functions

The functions in this section are equivalent in theory to combinations of other functions. In practice, they are implemented to be more efficient and more numerically stable than defining them directly using more basic Stan functions.

R expm1(T x)
natural exponential of x minus 1

real fma(real x, real y, real z)
Return z plus the result of x multiplied by y. $\text{fma}(x,y,z) = (x \times y) + z$

real multiply_log(real x, real y)
Warning: This function is deprecated and should be replaced with lmultiply. Return the product of x and the natural logarithm of y. $\mathrm{multiply\_log}(x,y) = \begin{cases} 0 & \text{if } x = y = 0 \\ x \log y & \text{if } x, y \neq 0 \\ \text{NaN} & \text{otherwise} \end{cases}$

real lmultiply(real x, real y)
Return the product of x and the natural logarithm of y. $\text{lmultiply}(x,y) = \begin{cases} 0 & \text{if } x = y = 0 \\ x \log y & \text{if } x, y \neq 0 \\ \text{NaN} & \text{otherwise} \end{cases}$

R log1p(T x)
natural logarithm of 1 plus x

R log1m(T x)
natural logarithm of 1 minus x

R log1p_exp(T x)
natural logarithm of one plus the natural exponentiation of x

R log1m_exp(T x)
logarithm of one minus the natural exponentiation of x

real log_diff_exp(real x, real y)
Return the natural logarithm of the difference of the natural exponentiation of x and the natural exponentiation of y. $\mathrm{log\_diff\_exp}(x,y) = \begin{cases} \log(\exp(x)-\exp(y)) & \text{if } x > y \\[6pt] \textrm{NaN} & \text{otherwise} \end{cases}$

real log_mix(real theta, real lp1, real lp2)
Return the log mixture of the log densities lp1 and lp2 with mixing proportion theta, defined by $\begin{eqnarray*} \mathrm{log\_mix}(\theta, \lambda_1, \lambda_2) & = & \log \!\left( \theta \exp(\lambda_1) + \left( 1 - \theta \right) \exp(\lambda_2) \right) \\[3pt] & = & \mathrm{log\_sum\_exp}\!\left(\log(\theta) + \lambda_1, \ \log(1 - \theta) + \lambda_2\right). \end{eqnarray*}$

real log_sum_exp(real x, real y)
Return the natural logarithm of the sum of the natural exponentiation of x and the natural exponentiation of y. $\mathrm{log\_sum\_exp}(x,y) = \log(\exp(x)+\exp(y))$

R log_inv_logit(T x)
natural logarithm of the inverse logit function of x

R log1m_inv_logit(T x)
natural logarithm of 1 minus the inverse logit function of x