Feel++ expressions

1. Using Mathematical expressions

Mathematical expressions are described as strings in C++, python or json files. An expression is composed of symbols and operations acting on them.

Available standard operations and functions are :

+, -, *, / addition, substraction, multiplication and division respectively
^ given two real numbers x and y, x^y compute x` to the power y
log
exp
sin
cos
tan
asin
acos
atan
sinh
cosh
tanh
asinh
acosh
atanh
atan2
sqrt
mod: given integers x and y, compute mod(x,y)
min, max: given 2 real numbers x and y, min(x,y) and max(x,y) compute the minimum and maximym respectively.
lgamma: Evaluation of lgamma(x), the natural logarithm of the Gamma function.
tgamma: Evaluation of tgamma(x), the true Gamma function.

2. Additional Mathematical functions

2.1. ceil

Description: ceil(v) computes the smallest integer value not less than arg.
Keyword: ceil
Math: Given real number $x$ , integer $m$ and the set of integers $\mathbb{Z}$ , ceil can be defined by the equation

$\lceil x \rceil=\min \{m\in\mathbb{Z}\mid m\ge x\}.$

C++ example

auto e = expr("ceil(u):u");

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json example

"expr":"ceil(u):u";

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2.2. floor

Description: floor(v) computes the largest integer value not greater than arg.
Keyword: floor
Math: Given real number $x$ , integer $m$ and the set of integers $\mathbb{Z}$ , floor can be defined by the equation

$\lfloor x \rfloor=\max \{m\in\mathbb{Z}\mid m\le x\},$

C++ example

auto e = expr("floor(u):u");

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json example

"expr":"floor(u):u";

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2.3. fract

Description: fract(v) compute the fractional part of the argument. This is calculated as $v - floor(v)$ .
Keyword: fract
Math: The fractional part is the sawtooth function, denoted by $\{x\}$ for real $x$ and defined by the formula

$\{x\} = x - \lfloor x \rfloor$

For all $x$ , we have

$0 \leq \{ x \} < 1$

C++ example

auto e = expr("fract(u):u");

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json example

"expr":"fract(u):u";

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2.4. clamp

Description: clamp(v,lo,hi) If v compares less than lo, returns lo; otherwise if hi compares less than v, returns hi; otherwise returns v.
Keyword: clamp

C++ example

auto e = expr("clamp(u,0,1):u");

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json example

"expr":"clamp(u,0,1):u";

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2.5. Modulo

Description: mod(x,y) compute the modulo operation: it returns the remainder or signed remainder of the division $x/y$ .
Keyword: mod

auto e = expr("mod(u,v):u:v");
e.setParameterValues( { { "u", 2 }, { "v", 1 }} );
CHECK( std::abs( e.evaluate()(0,0) ) < 1e-12 ) << "Error in modulo";
e.setParameterValues( { { "u", 3 }, { "v", 6 }} );
CHECK( std::abs( e.evaluate()(0,0) - 3) < 1e-12 ) << "Error in modulo";
e.setParameterValues( { { "u", 6.1 }, { "v", 3 }} );
CHECK( std::abs( e.evaluate()(0,0) - 0.1) < 1e-12 ) << "Error in modulo";
}

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2.6. sign

Description: sign(v) returns 1 if v positive, -1 if negative, 0 if v is zero. This is calculated as $(0 < v)- (v < 0)$ .
Keyword: sign

auto e = expr("sign(u):u");

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3. Mappings

3.1. mapabcd

mapabcd is the function $f$ that allows to map $[a,b]$ to $[c,d]$ , it is defines as follows

$f(t) = c + \left(\frac{d-c}{b-a}\right)(t-a)$

with $f(a)=c$ and $f(b)=d$ .

auto e = expr("mapabcd(t,1,2,-1,1):t");
e.setParameterValues( { { "t", 1 } } );
CHECK( std::abs(e.evaluate()(0,0)- (-1)) < 1e-12 ) << "Invalid mapping";
e.setParameterValues( { { "t", 2 } } );
CHECK( std::abs( e.evaluate()(0,0)- 1 ) < 1e-12 ) << "Invalid mapping";
e.setParameterValues( { { "t", 1.5 } } );
CHECK( std::abs( e.evaluate()(0,0) ) < 1e-12 ) << "Invalid mapping";

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4. Step functions

A function $f\colon \mathbb{R} \rightarrow \mathbb{R}$ is called a step function if it can be written as

$f(x) = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}(x), \mbox{ for all real numbers } x.$

where $n\ge 0$ , $\alpha_i$ are real numbers, $A_i$ are intervals, and $\chi_A$ is the indicator function of $A$ :

$\chi_A(x) = \begin{cases} 1 & \text{if } x \in A \\ 0 & \text{if } x \notin A \\ \end{cases}$

4.1. step1

Description: step1(u,edge) return 0 is u is strictly less than edge, 1 otherwise.
Keyword: step1

C++ example

auto e = expr("step1(u,edge):u:edge");

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json example

"expr":"step1(u,edge):u:edge";

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4.2. smoothstep

Description: smoothstep(u,lo,hi) is calculated as follows:

$\mbox{smoothstep}(x,lo,hi)=\left\{\begin{array}{ll}lo & x \leq lo \\ 3 x^{2}-2 x^{3} & lo \leq x \leq hi \\ hi & hi \leq x\end{array}\right.$

smoothstep uses the clamp function

Keyword: smoothstep

C++ example

auto e = expr("smoothstep(u,lo,hi):u:lo:hi");

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json example

"expr":"smoothstep(u,lo,hi):u:lo:hi"

Feel++ expressions

1. Using Mathematical expressions

2. Additional Mathematical functions

2.1. ceil

2.2. floor

2.3. fract

2.4. clamp

2.5. Modulo

2.6. sign

3. Mappings

3.1. mapabcd

4. Step functions

4.1. step1

4.2. smoothstep

4.3. triangle

4.4. rectangle

4.5. pulse

4.6. sinwave