## 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
Available since 2.0, vectorized in 2.13

real fma(real x, real y, real z)
Return z plus the result of x multiplied by y. $\text{fma}(x,y,z) = (x \times y) + z$
Available since 2.0

real ldexp(real x, int y)
Return the product of x and two raised to the y power. $\text{ldexp}(x,y) = x 2^y$
Available since 2.25

R ldexp(T1 x, T2 y)
Vectorized implementation of the ldexp function
Available since 2.25

real lmultiply(real x, real y)
Return the product of x and the natural logarithm of y. $\text{lmultiply}(x,y) = \begin{cases} 0 & \text{if } x = y = 0 \\ x \log y & \text{if } x, y \neq 0 \\ \text{NaN} & \text{otherwise} \end{cases}$
Available since 2.10

R lmultiply(T1 x, T2 y)
Vectorized implementation of the lmultiply function
Available since 2.25

R log1p(T x)
natural logarithm of 1 plus x
Available since 2.0, vectorized in 2.13

R log1m(T x)
natural logarithm of 1 minus x
Available since 2.0, vectorized in 2.13

R log1p_exp(T x)
natural logarithm of one plus the natural exponentiation of x
Available since 2.0, vectorized in 2.13

R log1m_exp(T x)
logarithm of one minus the natural exponentiation of x
Available since 2.0, vectorized in 2.13

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. $\mathrm{log\_diff\_exp}(x,y) = \begin{cases} \log(\exp(x)-\exp(y)) & \text{if } x > y \\[6pt] \textrm{NaN} & \text{otherwise} \end{cases}$
Available since 2.0

R log_diff_exp(T1 x, T2 y)
Vectorized implementation of the log_diff_exp function
Available since 2.25

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 $\begin{eqnarray*} \mathrm{log\_mix}(\theta, \lambda_1, \lambda_2) & = & \log \!\left( \theta \exp(\lambda_1) + \left( 1 - \theta \right) \exp(\lambda_2) \right) \\[3pt] & = & \mathrm{log\_sum\_exp}\!\left(\log(\theta) + \lambda_1, \ \log(1 - \theta) + \lambda_2\right). \end{eqnarray*}$
Available since 2.6

R log_mix(T1 theta, T2 lp1, T3 lp2)
Vectorized implementation of the log_mix function
Available since 2.26

R log_sum_exp(T1 x, T2 y)
Return the natural logarithm of the sum of the natural exponentiation of x and the natural exponentiation of y. $\mathrm{log\_sum\_exp}(x,y) = \log(\exp(x)+\exp(y))$
Available since 2.0, vectorized in 2.33

R log_inv_logit(T x)
natural logarithm of the inverse logit function of x
Available since 2.0, vectorized in 2.13

R log_inv_logit_diff(T1 x, T2 y)
natural logarithm of the difference of the inverse logit function of x and the inverse logit function of y
Available since 2.25

R log1m_inv_logit(T x)
natural logarithm of 1 minus the inverse logit function of x
Available since 2.0, vectorized in 2.13