Scalar

Create

Action	Code	Details
NaN	not_a_number()
π	pi()
e	e()
Square root of 2	sqrt2()
Positive infinity	positive_infinity()
Negative infinity	negative_infinity()
Float epsilon (smallest representable difference)	?

Create

Action	Code	Details
Declare unconstrainted integer variable	int x;
Declare constrained integer variable	int<lower=1, upper=10> n;
Declare unconstrained float variable	real x;
Declare unconstrained float with value	real x = 0.5;
Declare unconstrained float variable as NaN	real x = not_a_number();
Declare constrained float variable	real<lower=0, upper=1> prob;

Test

Action	Code	Details
Is integer equal	x == 5	Not recommended for float types
Approximately equal with proportional tolerance tol	?
Approximately equal with absolute tolerance tol	?
Is NaN	is_nan(x)
Is NaN	is_not_a_number(x)
Is finite	!is_inf(x) && !is_nan(x)
Is infinite	is_inf(x)
Is positive infinity	x > 0 && is_inf(x)
Is negative infinity	x < 0 && is_inf(x)

Derive

Action	Code	Details
Absolute value	fabs(x)
Threshold (step)	step(x)	0 if x < 0, else 1
Addition	x + y
Subtraction (difference)	x - y
Positive difference	fdim(x, y)	x - y for x >= y, else 0
Absolute difference	fabs(x - y)
Multiplication	x * y
Division	x / y
Reciprocal	inv(x)	1 / x
Multiplication followed by addition of z	fma(x, y, z)	x * y + z
Square (power of 2)	square(x)	x ^ 2
Inverse square (power of -2)	inv_square(x)	x ^ -2
Square root	sqrt(x)
Inverse square root	inv_sqrt(x)	1 / sqrt(x)
Cube root	cbrt(x)	pow(x, 1/3)
Raise to power p	x ^ p
Raise to power p	pow(x, p)
Natural logarithm	log(x)
Base-2 logarithm	log2(x)
Base-10 logarithm	log10(x)
Natural logarithm of reciprocal	-log(x)	log(1/x)
Natural logarithm of (1 - x)	log1m(x)	log(1 - x)
Natural logarithm of 1 - exp(x)	log1m_exp(x)	log(1 - e^x)
Natural logarithm of (1 + x)	log1p(x)	log(1 + x)
Natural logarithm of 1 + exp(x)	log1p_exp(x)	log(1 + e^x)
Multiplication with log	lmultiply(x , p)	p * log(x) for x > 0
Natural exponential	exp(x)
Natural exponential of x - 1	expm1(x)	(e^x) - 1
Min	fmin(x, y)
Max	fmax(x, y)
Modulus	fmod(x, y)	x - floor(x / y) * y
Floor	floor(x)
Ceil	ceil(x)
Round	round(x)	Output is real
Truncate to nearest integer	trunc(x)	Output is real
Log-sum of exponentials	log_sum_exp(x, y)	log(e^x + e^y)
Log-diff of exponentials	log_diff_exp(x, y)	log(e^x - e^y) for x > y
Proportional log-sum of exponentials (used for mixtures)	log_mix(theta, x, y)	log(theta * e^x + (1 - theta) * e^y)
Proportional log-sum of exponentials (used for mixtures)	log_sum_exp(theta + [x, y])
Proportional log-sum of exponentials (used for mixtures)	log_sum_exp(log(theta) + x, log1m(theta) + y)
Logit transform	logit(x)
Inverse logit transform	inv_logit(x)
Natural logarithm of inverse logit	log_inv_logit(x)
Natural logarithm of 1 minus inverse logit	log1m_inv_logit(x)
Binomial coefficient (ncr)	choose(m, k)