Expression Evaluation
*********************

When evaluating an arithmetic expression, FFmpeg uses an internal formula evaluator, implemented through the libavutil/eval.h interface.

An expression may contain unary, binary operators, constants, and functions.

Two expressions EXPR1 and EXPR2 can be combined to form another expression "EXPR1;EXPR2". EXPR1 and EXPR2 are evaluated in turn, and the new expression evaluates to the value of EXPR2.

The following binary operators are available: +, -, *, /, ^.

The following unary operators are available: +, -.

The following functions are available:

abs(x)
     Compute absolute value of X.

acos(x)
     Compute arccosine of X.

asin(x)
     Compute arcsine of X.

atan(x)
     Compute arctangent of X.

ceil(expr)
     Round the value of expression EXPR upwards to the nearest integer. For example, "ceil(1.5)" is "2.0".

cos(x)
     Compute cosine of X.

cosh(x)
     Compute hyperbolic cosine of X.

eq(x, y)
     Return 1 if X and Y are equivalent, 0 otherwise.

exp(x)
     Compute exponential of X (with base e, the Euler's number).

floor(expr)
     Round the value of expression EXPR downwards to the nearest integer. For example, "floor(-1.5)" is "-2.0".

gauss(x)
     Compute Gauss function of X, corresponding to exp(-x*x/2) / sqrt(2*PI).

gcd(x, y)
     Return the greatest common divisor of X and Y. If both X and Y are 0 or either or both are less than zero then behavior is undefined.

gt(x, y)
     Return 1 if X is greater than Y, 0 otherwise.

gte(x, y)
     Return 1 if X is greater than or equal to Y, 0 otherwise.

hypot(x, y)
     This function is similar to the C function with the same name; it returns "sqrt(X*X + Y*Y)", the length of the hypotenuse of a right triangle with sides of length X and Y, or the distance of the point (X, Y) from the origin.

if(x, y)
     Evaluate X, and if the result is non-zero return the result of the evaluation of Y, return 0 otherwise.

if(x, y, z)
     Evaluate X, and if the result is non-zero return the evaluation result of Y, otherwise the evaluation result of Z.

ifnot(x, y)
     Evaluate X, and if the result is zero return the result of the evaluation of Y, return 0 otherwise.

ifnot(x, y, z)
     Evaluate X, and if the result is zero return the evaluation result of Y, otherwise the evaluation result of Z.

isinf(x)
     Return 1.0 if X is +/-INFINITY, 0.0 otherwise.

isnan(x)
     Return 1.0 if X is NAN, 0.0 otherwise.

ld(var)
     Allow to load the value of the internal variable with number VAR, which was previously stored with st(VAR, EXPR). The function returns the loaded value.

log(x)
     Compute natural logarithm of X.

lt(x, y)
     Return 1 if X is lesser than Y, 0 otherwise.

lte(x, y)
     Return 1 if X is lesser than or equal to Y, 0 otherwise.

max(x, y)
     Return the maximum between X and Y.

min(x, y)
     Return the maximum between X and Y.

mod(x, y)
     Compute the remainder of division of X by Y.

not(expr)
     Return 1.0 if EXPR is zero, 0.0 otherwise.

pow(x, y)
     Compute the power of X elevated Y, it is equivalent to "(X)^(Y)".

random(x)
     Return a pseudo random value between 0.0 and 1.0. X is the index of the internal variable which will be used to save the seed/state.

root(expr, max)
     Find an input value for which the function represented by EXPR with argument LD(0) is 0 in the interval 0..MAX.

     The expression in EXPR must denote a continuous function or the result is undefined.

     LD(0) is used to represent the function input value, which means that the given expression will be evaluated multiple times with various input values that the expression can access through ld(0). When the expression evaluates to 0 then the corresponding input value will be returned.

sin(x)
     Compute sine of X.

sinh(x)
     Compute hyperbolic sine of X.

sqrt(expr)
     Compute the square root of EXPR. This is equivalent to "(EXPR)^.5".

squish(x)
     Compute expression 1/(1 + exp(4*x)).

st(var, expr)
     Allow to store the value of the expression EXPR in an internal variable. VAR specifies the number of the variable where to store the value, and it is a value ranging from 0 to 9. The function returns the value stored in the internal variable. Note, Variables are currently not shared between expressions.

tan(x)
     Compute tangent of X.

tanh(x)
     Compute hyperbolic tangent of X.

taylor(expr, x)

taylor(expr, x, id)
     Evaluate a Taylor series at X, given an expression representing the ld(id)-th derivative of a function at 0.

     When the series does not converge the result is undefined.

     LD(ID) is used to represent the derivative order in EXPR, which means that the given expression will be evaluated multiple times with various input values that the expression can access through ld(id). If ID is not specified then 0 is assumed.

     Note, when you have the derivatives at y instead of 0, taylor(expr, x-y) can be used.

time(0)
     Return the current (wallclock) time in seconds.

trunc(expr)
     Round the value of expression EXPR towards zero to the nearest integer. For example, "trunc(-1.5)" is "-1.0".

while(cond, expr)
     Evaluate expression EXPR while the expression COND is non-zero, and returns the value of the last EXPR evaluation, or NAN if COND was always false.

The following constants are available:

PI
     area of the unit disc, approximately 3.14

E
     exp(1) (Euler's number), approximately 2.718

PHI
     golden ratio (1+sqrt(5))/2, approximately 1.618

Assuming that an expression is considered "true" if it has a non-zero value, note that:

* works like AND

+ works like OR

For example the construct:

     if (A AND B) then C

is equivalent to:

     if(A*B, C)

In your C code, you can extend the list of unary and binary functions, and define recognized constants, so that they are available for your expressions.

The evaluator also recognizes the International System unit prefixes. If 'i' is appended after the prefix, binary prefixes are used, which are based on powers of 1024 instead of powers of 1000. The 'B' postfix multiplies the value by 8, and can be appended after a unit prefix or used alone. This allows using for example 'KB', 'MiB', 'G' and 'B' as number postfix.

The list of available International System prefixes follows, with indication of the corresponding powers of 10 and of 2.

y
     10^-24 / 2^-80

z
     10^-21 / 2^-70

a
     10^-18 / 2^-60

f
     10^-15 / 2^-50

p
     10^-12 / 2^-40

n
     10^-9 / 2^-30

u
     10^-6 / 2^-20

m
     10^-3 / 2^-10

c
     10^-2

d
     10^-1

h
     10^2

k
     10^3 / 2^10

K
     10^3 / 2^10

M
     10^6 / 2^20

G
     10^9 / 2^30

T
     10^12 / 2^40

P
     10^15 / 2^40

E
     10^18 / 2^50

Z
     10^21 / 2^60

Y
     10^24 / 2^70