Complete ULTRAMEGA Expression Parser Manual

Complete Syntax Guide

Operator Precedence (Highest to Lowest)

Precedence	Operator	Description	Associativity	Example
1	( )	Parentheses (grouping)	N/A	(2 + 3) * 4
2	^, **	Exponentiation	Right	2^3^2 = 2^(3^2) = 512
3	+x, -x, !, not	Unary operators	Right	-5, !true
4	*, /, %	Multiplication, Division, Modulo	Left	6 * 4 / 2
5	+, -	Addition, Subtraction	Left	5 + 3 - 2
6	<, >, <=, >=	Comparison	Left	5 > 3
7	==, !=, <>	Equality	Left	x == 5
8	&&, and	Logical AND	Left	a && b
9	\|\|, or	Logical OR	Left	a \|\| b

Function Call Syntax

// Basic function calls
function_name(argument1, argument2, ...)

// Examples
sin(pi/4)                    // Single argument
pow(2, 8)                    // Two arguments  
mean(1, 2, 3, 4, 5)         // Multiple arguments
kinetic_energy(mass, vel)    // Variables as arguments
if(x > 0, x, -x)            // Conditional function
            

Variable Assignment and Usage

// Set variables in C#
evaluator.SetVariable("mass", 1000);
evaluator.SetVariable("velocity", 30);
evaluator.SetVariable("pi_over_4", Math.PI / 4);

// Use variables in expressions
evaluator.Evaluate("kinetic_energy(mass, velocity)");
evaluator.Evaluate("sin(pi_over_4)");
evaluator.Evaluate("mass * velocity^2 / 2");
            

Numeric Formats

Supported Number Formats

Integers: 42, -17, 0
Decimals: 3.14159, -2.5, 0.001
Scientific: 1.23e-4, 6.022E23, -9.81e0
Leading decimal: .5 (equals 0.5)

Special Operators and Keywords

Operator/Keyword	Description	Example	Result
and	Logical AND (alternative to &&)	true and false	0 (false)
or	Logical OR (alternative to \|\|)	true or false	1 (true)
not	Logical NOT (alternative to !)	not true	0 (false)
**	Exponentiation (alternative to ^)	2**8	256
<>	Not equal (alternative to !=)	5 <> 3	1 (true)

⚠️ Important Syntax Rules

Case Sensitivity: Variable names and function names are case-sensitive
Whitespace: Spaces are ignored in expressions
Parentheses: Always match opening and closing parentheses
Decimal Separator: Use period (.) for decimal point, not comma
Function Arguments: Separate with commas, not semicolons

📊 Mathematical Functions 120+ Functions

Basic Operations (15)

Function	Description	Formula	Example	Result
abs(x)	Absolute value	\|x\|	abs(-5.7)	5.7
sqrt(x)	Square root	√x	sqrt(25)	5
cbrt(x)	Cube root	∛x	cbrt(27)	3
pow(x, y)	x raised to power y	x^y	pow(2, 10)	1024
root(n, x)	nth root of x	x^(1/n)	root(4, 16)	2
square(x)	x squared	x²	square(7)	49
cube(x)	x cubed	x³	cube(4)	64
reciprocal(x)	Reciprocal of x	1/x	reciprocal(8)	0.125
sign(x)	Sign of x	sgn(x)	sign(-3.5)	-1
min(...)	Minimum of arguments	min(a,b,c,...)	min(3,1,4,2)	1
max(...)	Maximum of arguments	max(a,b,c,...)	max(3,1,4,2)	4
clamp(x,min,max)	Clamp x between min and max	max(min, min(x, max))	clamp(15,0,10)	10

Exponential and Logarithmic Functions (12)

Function	Description	Formula	Example	Result
exp(x)	Exponential function	e^x	exp(1)	2.718
exp2(x)	Base-2 exponential	2^x	exp2(10)	1024
exp10(x)	Base-10 exponential	10^x	exp10(3)	1000
expm1(x)	e^x - 1 (accurate for small x)	e^x - 1	expm1(0.001)	0.001001
ln(x)	Natural logarithm	ln(x)	ln(e)	1
log(x)	Base-10 logarithm	log₁₀(x)	log(100)	2
log(base,x)	Logarithm base b	log_b(x)	log(2,8)	3
log2(x)	Base-2 logarithm	log₂(x)	log2(1024)	10
log10(x)	Base-10 logarithm	log₁₀(x)	log10(1000)	3
log1p(x)	ln(1 + x) (accurate for small x)	ln(1 + x)	log1p(0.001)	0.0009995

Trigonometric Functions (18)

Function	Description	Formula	Example	Result
sin(x)	Sine (radians)	sin(x)	sin(pi/2)	1
cos(x)	Cosine (radians)	cos(x)	cos(0)	1
tan(x)	Tangent (radians)	tan(x)	tan(pi/4)	1
csc(x)	Cosecant	1/sin(x)	csc(pi/2)	1
sec(x)	Secant	1/cos(x)	sec(0)	1
cot(x)	Cotangent	1/tan(x)	cot(pi/4)	1
asin(x)	Arcsine	sin⁻¹(x)	asin(1)	π/2
acos(x)	Arccosine	cos⁻¹(x)	acos(0)	π/2
atan(x)	Arctangent	tan⁻¹(x)	atan(1)	π/4
atan2(y,x)	Two-argument arctangent	atan2(y,x)	atan2(1,1)	π/4
acsc(x)	Arccosecant	csc⁻¹(x)	acsc(1)	π/2
asec(x)	Arcsecant	sec⁻¹(x)	asec(1)	0
acot(x)	Arccotangent	cot⁻¹(x)	acot(1)	π/4
deg(x)	Radians to degrees	x × 180/π	deg(pi)	180
rad(x)	Degrees to radians	x × π/180	rad(180)	π

Hyperbolic Functions (12)

Function	Description	Formula	Example	Result
sinh(x)	Hyperbolic sine	(e^x - e^(-x))/2	sinh(1)	1.175
cosh(x)	Hyperbolic cosine	(e^x + e^(-x))/2	cosh(0)	1
tanh(x)	Hyperbolic tangent	sinh(x)/cosh(x)	tanh(1)	0.762
csch(x)	Hyperbolic cosecant	1/sinh(x)	csch(1)	0.851
sech(x)	Hyperbolic secant	1/cosh(x)	sech(0)	1
coth(x)	Hyperbolic cotangent	cosh(x)/sinh(x)	coth(1)	1.313
asinh(x)	Inverse hyperbolic sine	ln(x + √(x² + 1))	asinh(1)	0.881
acosh(x)	Inverse hyperbolic cosine	ln(x + √(x² - 1))	acosh(2)	1.317
atanh(x)	Inverse hyperbolic tangent	½ln((1+x)/(1-x))	atanh(0.5)	0.549
acsch(x)	Inverse hyperbolic cosecant	asinh(1/x)	acsch(1)	0.881
asech(x)	Inverse hyperbolic secant	acosh(1/x)	asech(0.5)	1.317
acoth(x)	Inverse hyperbolic cotangent	atanh(1/x)	acoth(2)	0.549

Statistical Functions (15)

Function	Description	Formula	Example	Result
sum(...)	Sum of all arguments	Σxᵢ	sum(1,2,3,4,5)	15
mean(...)	Arithmetic mean	Σxᵢ/n	mean(1,2,3,4,5)	3
median(...)	Middle value	x₍ₙ₊₁₎/₂	median(1,2,3,4,5)	3
mode(...)	Most frequent value	argmax(frequency)	mode(1,2,2,3)	2
range(...)	Maximum minus minimum	max - min	range(1,2,3,4,5)	4
variance(...)	Population variance	Σ(xᵢ-μ)²/n	variance(1,2,3,4,5)	2
stdev(...)	Standard deviation	√(variance)	stdev(1,2,3,4,5)	1.581
skewness(...)	Distribution skewness	E[(X-μ)³]/σ³	skewness(1,2,3,4,5)	0
kurtosis(...)	Distribution kurtosis	E[(X-μ)⁴]/σ⁴ - 3	kurtosis(1,2,3,4,5)	-1.2

Number Theory Functions (15)

Function	Description	Formula	Example	Result
factorial(n)	Factorial	n!	factorial(5)	120
double_factorial(n)	Double factorial	n!!	double_factorial(5)	15
fibonacci(n)	Fibonacci number	F(n)	fibonacci(10)	55
lucas(n)	Lucas number	L(n)	lucas(5)	11
gcd(a,b)	Greatest common divisor	gcd(a,b)	gcd(48,18)	6
lcm(a,b)	Least common multiple	ab/gcd(a,b)	lcm(12,8)	24
isprime(n)	Check if prime	prime test	isprime(17)	1
nextprime(n)	Next prime after n	min{p > n : prime(p)}	nextprime(10)	11
totient(n)	Euler's totient function	φ(n)	totient(12)	4
mobius(n)	Möbius function	μ(n)	mobius(6)	1
divisor_sigma(n,k)	Sum of k-th powers of divisors	σₖ(n)	divisor_sigma(6,1)	12

Combinatorial Functions (8)

Function	Description	Formula	Example	Result
permutation(n,r)	Permutations	n!/(n-r)!	permutation(5,3)	60
combination(n,r)	Combinations	n!/(r!(n-r)!)	combination(5,3)	10
binomial(n,k)	Binomial coefficient	C(n,k)	binomial(5,2)	10
stirling1(n,k)	Stirling numbers 1st kind	s(n,k)	stirling1(4,2)	11
stirling2(n,k)	Stirling numbers 2nd kind	S(n,k)	stirling2(4,2)	7
bell(n)	Bell numbers	B(n)	bell(4)	15
catalan(n)	Catalan numbers	C(n)	catalan(4)	14

Special Functions (20)

Function	Description	Formula	Example	Result
gamma(x)	Gamma function	Γ(x)	gamma(5)	24
loggamma(x)	Log of gamma function	ln(Γ(x))	loggamma(5)	3.178
digamma(x)	Digamma function	ψ(x)	digamma(2)	0.423
polygamma(n,x)	Polygamma function	ψ⁽ⁿ⁾(x)	polygamma(1,2)	0.645
beta(a,b)	Beta function	B(a,b)	beta(2,3)	0.083
erf(x)	Error function	erf(x)	erf(1)	0.843
erfc(x)	Complementary error function	1 - erf(x)	erfc(1)	0.157
erfinv(x)	Inverse error function	erf⁻¹(x)	erfinv(0.5)	0.477
besselj0(x)	Bessel function J₀	J₀(x)	besselj0(0)	1
besselj1(x)	Bessel function J₁	J₁(x)	besselj1(1)	0.440
bessely0(x)	Bessel function Y₀	Y₀(x)	bessely0(1)	0.088
bessely1(x)	Bessel function Y₁	Y₁(x)	bessely1(1)	-0.781
riemann_zeta(s)	Riemann zeta function	ζ(s)	riemann_zeta(2)	1.645
elliptic_k(k)	Complete elliptic integral K	K(k)	elliptic_k(0.5)	1.854
elliptic_e(k)	Complete elliptic integral E	E(k)	elliptic_e(0.5)	1.351

Rounding Functions (5)

Function	Description	Formula	Example	Result
floor(x)	Largest integer ≤ x	⌊x⌋	floor(3.7)	3
ceil(x)	Smallest integer ≥ x	⌈x⌉	ceil(3.2)	4
round(x)	Round to nearest integer	round(x)	round(3.6)	4
round(x,n)	Round to n decimal places	round(x,n)	round(3.14159,2)	3.14
trunc(x)	Truncate to integer	trunc(x)	trunc(3.9)	3
frac(x)	Fractional part	x - trunc(x)	frac(3.7)	0.7

🔬 Physics Functions 50+ Functions

Classical Mechanics (15)

Function	Description	Formula	Example	Result
velocity(d,t)	Velocity from distance and time	v = d/t	velocity(100,5)	20 m/s
acceleration(v,t)	Acceleration from velocity and time	a = v/t	acceleration(20,4)	5 m/s²
force(m,a)	Force from mass and acceleration	F = ma	force(10,9.8)	98 N
momentum(m,v)	Linear momentum	p = mv	momentum(5,20)	100 kg⋅m/s
kinetic_energy(m,v)	Kinetic energy	KE = ½mv²	kinetic_energy(10,20)	2000 J
potential_energy(m,g,h)	Gravitational potential energy	PE = mgh	potential_energy(10,9.8,5)	490 J
work(F,d,θ)	Work done	W = Fd cos θ	work(100,5,0)	500 J
power_mech(W,t)	Mechanical power	P = W/t	power_mech(1000,10)	100 W
impulse(F,t)	Impulse	J = Ft	impulse(100,2)	200 N⋅s
angular_velocity(θ,t)	Angular velocity	ω = θ/t	angular_velocity(pi,2)	π/2 rad/s
angular_acceleration(ω,t)	Angular acceleration	α = ω/t	angular_acceleration(10,2)	5 rad/s²
centripetal_force(m,v,r)	Centripetal force	Fc = mv²/r	centripetal_force(5,10,2)	250 N
moment_of_inertia(m,r)	Moment of inertia (point mass)	I = mr²	moment_of_inertia(2,3)	18 kg⋅m²
torque(F,r)	Torque	τ = Fr	torque(50,0.5)	25 N⋅m
angular_momentum(I,ω)	Angular momentum	L = Iω	angular_momentum(10,5)	50 kg⋅m²/s

Oscillations and Waves (8)

Function	Description	Formula	Example	Result
period(f)	Period from frequency	T = 1/f	period(50)	0.02 s
frequency(T)	Frequency from period	f = 1/T	frequency(0.01)	100 Hz
wave_speed(f,λ)	Wave speed	v = fλ	wave_speed(440,0.77)	339 m/s
wavelength(v,f)	Wavelength	λ = v/f	wavelength(340,440)	0.77 m
simple_harmonic_motion(A,ω,t,φ)	Simple harmonic motion	x = A cos(ωt + φ)	simple_harmonic_motion(5,2,1,0)	-2.08
pendulum_period(L,g)	Simple pendulum period	T = 2π√(L/g)	pendulum_period(1,9.8)	2.01 s
spring_period(m,k)	Mass-spring system period	T = 2π√(m/k)	spring_period(1,100)	0.628 s
resonant_frequency(L,C)	LC circuit resonant frequency	f = 1/(2π√(LC))	resonant_frequency(1e-3,1e-6)	159 Hz

Thermodynamics (7)

Function	Description	Formula	Example	Result
ideal_gas_pressure(n,R,T,V)	Ideal gas pressure	P = nRT/V	ideal_gas_pressure(1,8.314,300,0.024)	103,925 Pa
ideal_gas_volume(n,R,T,P)	Ideal gas volume	V = nRT/P	ideal_gas_volume(1,8.314,300,101325)	0.0246 m³
ideal_gas_temperature(P,V,n,R)	Ideal gas temperature	T = PV/(nR)	ideal_gas_temperature(101325,0.024,1,8.314)	293 K
heat_capacity(Q,ΔT)	Heat capacity	C = Q/ΔT	heat_capacity(1000,20)	50 J/K
thermal_expansion(L₀,α,ΔT)	Linear thermal expansion	ΔL = L₀αΔT	thermal_expansion(10,1.2e-5,50)	0.006 m
stefan_boltzmann(σ,A,T)	Stefan-Boltzmann law	P = σAT⁴	stefan_boltzmann(5.67e-8,1,300)	459 W
carnot_efficiency(T_hot,T_cold)	Carnot efficiency	η = 1 - T_c/T_h	carnot_efficiency(500,300)	0.4

Electromagnetism (8)

Function	Description	Formula	Example	Result
coulomb_force(q1,q2,r)	Coulomb force	F = kq₁q₂/r²	coulomb_force(1e-6,1e-6,0.1)	0.899 N
electric_field(F,q)	Electric field	E = F/q	electric_field(100,1e-6)	1e8 N/C
electric_potential(E,d)	Electric potential	V = Ed	electric_potential(1000,0.01)	10 V
capacitance(Q,V)	Capacitance	C = Q/V	capacitance(1e-6,12)	8.33e-8 F
magnetic_force(q,v,B,θ)	Magnetic force	F = qvB sin θ	magnetic_force(1e-6,1e5,0.1,pi/2)	1e-2 N
lorentz_force(q,E,v,B)	Lorentz force	F = q(E + v×B)	lorentz_force(1e-6,1000,1e5,0.1)	0.011 N
magnetic_flux(B,A,θ)	Magnetic flux	Φ = BA cos θ	magnetic_flux(0.1,0.01,0)	0.001 Wb
faraday_law(dΦ_dt)	Faraday's law	ε = -dΦ/dt	faraday_law(-0.1)	0.1 V

Quantum Mechanics (6)

Function	Description	Formula	Example	Result
planck_energy(f)	Photon energy	E = hf	planck_energy(1e15)	6.63e-19 J
de_broglie_wavelength(m,v)	Matter wave wavelength	λ = h/(mv)	de_broglie_wavelength(9.1e-31,1e6)	7.28e-10 m
photon_energy(λ)	Photon energy from wavelength	E = hc/λ	photon_energy(500e-9)	3.97e-19 J
photon_momentum(λ)	Photon momentum	p = h/λ	photon_momentum(500e-9)	1.33e-27 kg⋅m/s
bohr_radius(Z)	Bohr radius for Z protons	a₀/Z²	bohr_radius(1)	5.29e-11 m
rydberg_energy(Z)	Rydberg energy	Ry × Z²	rydberg_energy(1)	13.6 eV

Special Relativity (5)

Function	Description	Formula	Example	Result
lorentz_factor(v)	Lorentz factor	γ = 1/√(1-v²/c²)	lorentz_factor(0.9*c)	2.29
time_dilation(t₀,v)	Time dilation	t = γt₀	time_dilation(10,0.9*c)	22.9 s
length_contraction(L₀,v)	Length contraction	L = L₀/γ	length_contraction(10,0.9*c)	4.36 m
relativistic_energy(m,v)	Relativistic energy	E = γmc²	relativistic_energy(1,0.9*c)	2.06e17 J
relativistic_momentum(m,v)	Relativistic momentum	p = γmv	relativistic_momentum(1,0.9*c)	6.18e8 kg⋅m/s

Optics (6)

Function	Description	Formula	Example	Result
snell_law(n1,θ1,n2)	Snell's law refraction	sin θ₂ = n₁sin θ₁/n₂	snell_law(1,pi/4,1.5)	0.471
lens_equation(f,d)	Thin lens equation	1/f = 1/d₀ + 1/dᵢ	lens_equation(0.1,0.15)	0.3 m
magnification(dᵢ,d₀)	Linear magnification	m = dᵢ/d₀	magnification(0.3,0.15)	2
brewster_angle(n1,n2)	Brewster angle	θB = arctan(n₂/n₁)	brewster_angle(1,1.5)	0.983 rad
critical_angle(n1,n2)	Critical angle for total internal reflection	θc = arcsin(n₂/n₁)	critical_angle(1.5,1)	0.7297 rad
rayleigh_scattering(λ)	Rayleigh scattering intensity	I ∝ 1/λ⁴	rayleigh_scattering(500e-9)	1.6e31

⚡ Electronics Functions 35+ Functions

Basic Circuit Laws (5)

Function	Description	Formula	Example	Result
ohms_law_voltage(I,R)	Voltage from current and resistance	V = IR	ohms_law_voltage(2,100)	200 V
ohms_law_current(V,R)	Current from voltage and resistance	I = V/R	ohms_law_current(12,1000)	0.012 A
ohms_law_resistance(V,I)	Resistance from voltage and current	R = V/I	ohms_law_resistance(12,0.1)	120 Ω
power_electrical(V,I)	Electrical power	P = VI	power_electrical(12,2)	24 W
energy_electrical(P,t)	Electrical energy	E = Pt	energy_electrical(100,3600)	360000 J

Resistor Networks (4)

Function	Description	Formula	Example	Result
resistors_series(...)	Series resistance	R = R₁ + R₂ + R₃ + ...	resistors_series(100,200,300)	600 Ω
resistors_parallel(...)	Parallel resistance	1/R = 1/R₁ + 1/R₂ + ...	resistors_parallel(100,100)	50 Ω
voltage_divider(Vin,R1,R2)	Voltage divider output	Vout = Vin × R₂/(R₁+R₂)	voltage_divider(12,1000,2000)	8 V
current_divider(Iin,R1,R2)	Current divider output	I₁ = Iin × R₂/(R₁+R₂)	current_divider(0.1,1000,2000)	0.067 A

Capacitor Functions (5)

Function	Description	Formula	Example	Result
capacitors_series(...)	Series capacitance	1/C = 1/C₁ + 1/C₂ + ...	capacitors_series(100e-6,100e-6)	5e-5 F
capacitors_parallel(...)	Parallel capacitance	C = C₁ + C₂ + ...	capacitors_parallel(100e-6,200e-6)	3e-4 F
capacitor_energy(C,V)	Energy stored in capacitor	E = ½CV²	capacitor_energy(100e-6,12)	0.0072 J
capacitor_charge(C,V)	Charge on capacitor	Q = CV	capacitor_charge(100e-6,12)	0.0012 C
rc_time_constant(R,C)	RC time constant	τ = RC	rc_time_constant(1000,100e-6)	0.1 s
capacitor_impedance(f,C)	Capacitive reactance	Xc = 1/(2πfC)	capacitor_impedance(60,100e-6)	26.5 Ω

Inductor Functions (4)

Function	Description	Formula	Example	Result
inductors_series(...)	Series inductance	L = L₁ + L₂ + ...	inductors_series(1e-3,2e-3)	0.003 H
inductors_parallel(...)	Parallel inductance	1/L = 1/L₁ + 1/L₂ + ...	inductors_parallel(1e-3,1e-3)	5e-4 H
inductor_energy(L,I)	Energy stored in inductor	E = ½LI²	inductor_energy(0.01,2)	0.02 J
rl_time_constant(L,R)	RL time constant	τ = L/R	rl_time_constant(0.01,100)	1e-4 s
inductor_impedance(f,L)	Inductive reactance	XL = 2πfL	inductor_impedance(60,0.01)	3.77 Ω

AC Circuit Analysis (8)

Function	Description	Formula	Example	Result
impedance_magnitude(R,X)	Impedance magnitude	\|Z\| = √(R² + X²)	impedance_magnitude(100,100)	141.4 Ω
impedance_phase(R,X)	Impedance phase angle	φ = arctan(X/R)	impedance_phase(100,100)	0.785 rad
ac_power_apparent(V,I)	Apparent power	S = VI	ac_power_apparent(120,10)	1200 VA
ac_power_real(V,I,φ)	Real power	P = VI cos φ	ac_power_real(120,10,0)	1200 W
ac_power_reactive(V,I,φ)	Reactive power	Q = VI sin φ	ac_power_reactive(120,10,pi/4)	849 VAR
power_factor(φ)	Power factor	PF = cos φ	power_factor(pi/3)	0.5
quality_factor(ωL,R)	Q factor of circuit	Q = ωL/R	quality_factor(1000,10)	100

Resonance and Filters (3)

Function	Description	Formula	Example	Result
resonant_freq_lc(L,C)	LC resonant frequency	f = 1/(2π√(LC))	resonant_freq_lc(1e-3,1e-6)	159.2 Hz
bandwidth(f0,Q)	Bandwidth of resonant circuit	BW = f₀/Q	bandwidth(1000,50)	20 Hz
cutoff_frequency_rc(R,C)	RC filter cutoff frequency	fc = 1/(2πRC)	cutoff_frequency_rc(1000,1e-6)	159.2 Hz
cutoff_frequency_rl(L,R)	RL filter cutoff frequency	fc = R/(2πL)	cutoff_frequency_rl(0.01,1000)	15915 Hz

Decibel Conversions (5)

Function	Description	Formula	Example	Result
db_voltage(V1,V2)	Voltage ratio in dB	dB = 20 log₁₀(V₁/V₂)	db_voltage(10,1)	20 dB
db_power(P1,P2)	Power ratio in dB	dB = 10 log₁₀(P₁/P₂)	db_power(100,1)	20 dB
db_to_ratio(dB)	dB to voltage ratio	ratio = 10^(dB/20)	db_to_ratio(6)	1.995
dbm_to_watts(dBm)	dBm to watts	W = 10^((dBm-30)/10)	dbm_to_watts(30)	1 W
watts_to_dbm(W)	Watts to dBm	dBm = 10 log₁₀(W) + 30	watts_to_dbm(0.001)	0 dBm

Digital Electronics (3)

Function	Description	Formula	Example	Result
binary_to_decimal(binary)	Binary to decimal conversion	Σ bᵢ × 2ⁱ	binary_to_decimal(1010)	10
decimal_to_binary(decimal)	Decimal to binary conversion	Convert to base 2	decimal_to_binary(10)	1010
bit_count(number)	Count set bits	Population count	bit_count(15)	4

Transmission Lines (4)

Function	Description	Formula	Example	Result
characteristic_impedance(L,C)	Characteristic impedance	Z₀ = √(L/C)	characteristic_impedance(1e-6,1e-12)	1000 Ω
propagation_delay(L,C)	Propagation delay	tpd = √(LC)	propagation_delay(1e-6,1e-12)	1e-9 s
reflection_coefficient(ZL,Z0)	Reflection coefficient	Γ = (ZL-Z₀)/(ZL+Z₀)	reflection_coefficient(75,50)	0.2
vswr(gamma)	Voltage standing wave ratio	VSWR = (1+\|Γ\|)/(1-\|Γ\|)	vswr(0.2)	1.5

🔧 Logic & Boolean Functions 45+ Functions

Basic Boolean Operations (8)

Function	Description	Formula	Example	Result
bool_and(...)	Boolean AND operation	a ∧ b ∧ c ∧ ...	bool_and(1,1,0)	0
bool_or(...)	Boolean OR operation	a ∨ b ∨ c ∨ ...	bool_or(0,0,1)	1
bool_not(x)	Boolean NOT operation	¬x	bool_not(1)	0
bool_xor(...)	Boolean XOR operation	a ⊕ b ⊕ c ⊕ ...	bool_xor(1,0,1)	0
bool_nand(...)	Boolean NAND operation	¬(a ∧ b)	bool_nand(1,1)	0
bool_nor(...)	Boolean NOR operation	¬(a ∨ b)	bool_nor(0,0)	1
bool_implies(a,b)	Boolean implication	a → b	bool_implies(1,0)	0
bool_equiv(a,b)	Boolean equivalence	a ↔ b	bool_equiv(1,1)	1

Bitwise Operations (15)

Function	Description	Formula	Example	Result
bit_and(a,b)	Bitwise AND	a & b	bit_and(12,10)	8
bit_or(a,b)	Bitwise OR	a \| b	bit_or(12,10)	14
bit_xor(a,b)	Bitwise XOR	a ^ b	bit_xor(12,10)	6
bit_not(a)	Bitwise NOT	~a	bit_not(12)	-13
bit_shift_left(a,n)	Left bit shift	a << n	bit_shift_left(5,2)	20
bit_shift_right(a,n)	Right bit shift	a >> n	bit_shift_right(20,2)	5
bit_rotate_left(a,n)	Left bit rotation	Rotate left n bits	bit_rotate_left(5,1)	10
bit_rotate_right(a,n)	Right bit rotation	Rotate right n bits	bit_rotate_right(10,1)	5
bit_set(a,pos)	Set bit at position	a \| (1 << pos)	bit_set(8,0)	9
bit_clear(a,pos)	Clear bit at position	a & ~(1 << pos)	bit_clear(9,0)	8
bit_flip(a,pos)	Flip bit at position	a ^ (1 << pos)	bit_flip(8,0)	9
bit_test(a,pos)	Test bit at position	(a >> pos) & 1	bit_test(9,0)	1
bit_mask(n)	Create n-bit mask	(1 << n) - 1	bit_mask(4)	15
bit_population(a)	Count set bits	Population count	bit_population(15)	4
bit_trailing_zeros(a)	Count trailing zeros	Trailing zero count	bit_trailing_zeros(8)	3

Set Operations (8)

Function	Description	Formula	Example	Result
set_union(A,B)	Set union (bitset)	A ∪ B	set_union(12,10)	14
set_intersection(A,B)	Set intersection (bitset)	A ∩ B	set_intersection(12,10)	8
set_difference(A,B)	Set difference (bitset)	A \ B	set_difference(12,10)	4
set_symmetric_difference(A,B)	Symmetric difference (bitset)	A △ B	set_symmetric_difference(12,10)	6
set_complement(A)	Set complement (bitset)	Ā	set_complement(12)	-13
set_subset(A,B)	Check if A ⊆ B (bitset)	A ⊆ B	set_subset(8,12)	1
set_superset(A,B)	Check if A ⊇ B (bitset)	A ⊇ B	set_superset(12,8)	1
set_cardinality(A)	Set cardinality (bitset)	\|A\|	set_cardinality(15)	4

Fuzzy Logic (8)

Function	Description	Formula	Example	Result
fuzzy_and(...)	Fuzzy AND (minimum)	min(a,b,c,...)	fuzzy_and(0.7,0.3,0.9)	0.3
fuzzy_or(...)	Fuzzy OR (maximum)	max(a,b,c,...)	fuzzy_or(0.7,0.3,0.9)	0.9
fuzzy_not(x)	Fuzzy NOT	1 - x	fuzzy_not(0.7)	0.3
fuzzy_triangular_norm(a,b)	Product t-norm	a × b	fuzzy_triangular_norm(0.7,0.8)	0.56
membership_triangle(x,a,b,c)	Triangular membership	Triangle function	membership_triangle(5,0,5,10)	1
membership_trapezoid(x,a,b,c,d)	Trapezoidal membership	Trapezoid function	membership_trapezoid(5,0,3,7,10)	1
membership_gaussian(x,μ,σ)	Gaussian membership	e^(-½((x-μ)/σ)²)	membership_gaussian(5,5,1)	1

Sequential Logic (6)

Function	Description	Formula	Example	Result
sr_latch(S,R,Q_prev)	SR latch	Set-Reset latch	sr_latch(1,0,0)	1
d_flip_flop(D,CLK,Q_prev)	D flip-flop	Data flip-flop	d_flip_flop(1,1,0)	1
jk_flip_flop(J,K,CLK,Q_prev)	JK flip-flop	JK flip-flop	jk_flip_flop(1,1,1,0)	1
counter_up(count,CLK,modulus)	Up counter	(count + 1) mod n	counter_up(7,1,8)	0
counter_down(count,CLK,modulus)	Down counter	(count - 1) mod n	counter_down(0,1,8)	7

📐 Symbolic Math Functions 30+ Functions

Polynomial Operations (3)

Function	Description	Formula	Example	Result
poly_eval(x,a0,a1,a2,...)	Evaluate polynomial	a₀ + a₁x + a₂x² + ...	poly_eval(2,1,2,3)	17
poly_derivative(n,coeff)	Polynomial derivative coefficient	n × aₙ	poly_derivative(3,5)	15
horner_method(x,a0,a1,a2,...)	Horner's method evaluation	Efficient polynomial eval	horner_method(2,1,2,3)	17

Numerical Differentiation (6)

Function	Description	Formula	Example	Result
derivative_forward(f_x,f_x_h,h)	Forward difference	(f(x+h) - f(x))/h	derivative_forward(1,1.1,0.1)	1
derivative_backward(f_x,f_x_h,h)	Backward difference	(f(x) - f(x-h))/h	derivative_backward(1.1,1,0.1)	1
derivative_central(f_x_h,f_x_neg_h,h)	Central difference	(f(x+h) - f(x-h))/(2h)	derivative_central(1.1,0.9,0.1)	1

Numerical Integration (5)

Function	Description	Formula	Example	Result
trapezoid_rule(f_a,f_b,h)	Trapezoidal rule	h(f(a) + f(b))/2	trapezoid_rule(1,4,1)	2.5
simpson_rule(f_a,f_m,f_b,h)	Simpson's rule	h(f(a) + 4f(m) + f(b))/6	simpson_rule(1,2,4,0.5)	1.25
simpson_38_rule(f0,f1,f2,f3,h)	Simpson's 3/8 rule	3h(f₀ + 3f₁ + 3f₂ + f₃)/8	simpson_38_rule(1,1.5,2.5,4,0.33)	1.78

Matrix Operations (5)

Function	Description	Formula	Example	Result
matrix_det_2x2(a,b,c,d)	2×2 matrix determinant	ad - bc	matrix_det_2x2(1,2,3,4)	-2
matrix_trace_2x2(a,d)	2×2 matrix trace	a + d	matrix_trace_2x2(1,4)	5
matrix_mult_2x2(...)	2×2 matrix multiplication element	Matrix multiplication	matrix_mult_2x2(1,2,3,4,5,6,7,8)	19

Vector Operations (6)

Function	Description	Formula	Example	Result
vector_magnitude(...)	Vector magnitude	√(x² + y² + z² + ...)	vector_magnitude(3,4)	5
vector_dot_product(...)	Dot product	v₁·v₂	vector_dot_product(1,2,3,4)	11
vector_cross_product_2d(x1,y1,x2,y2)	2D cross product	x₁y₂ - y₁x₂	vector_cross_product_2d(1,2,3,4)	-2
vector_angle(x1,y1,x2,y2)	Angle between vectors	arccos(v₁·v₂/(\|v₁\|\|v₂\|))	vector_angle(1,0,0,1)	π/2
vector_projection(...)	Vector projection	Scalar projection	vector_projection(1,2,3,4)	0.44

Complex Number Operations (8)

Function	Description	Formula	Example	Result
complex_magnitude(a,b)	Complex magnitude	\|a + bi\| = √(a² + b²)	complex_magnitude(3,4)	5
complex_argument(a,b)	Complex argument	arg(a + bi) = atan2(b,a)	complex_argument(1,1)	π/4
complex_add(a,b,c,d)	Complex addition (real part)	(a+bi) + (c+di) = (a+c)	complex_add(1,2,3,4)	4
complex_multiply_real(a,b,c,d)	Complex multiplication (real)	Re[(a+bi)(c+di)] = ac-bd	complex_multiply_real(1,2,3,4)	-5
complex_multiply_imag(a,b,c,d)	Complex multiplication (imag)	Im[(a+bi)(c+di)] = ad+bc	complex_multiply_imag(1,2,3,4)	10
complex_divide_real(a,b,c,d)	Complex division (real)	Real part of division	complex_divide_real(1,2,3,4)	0.44
complex_exp_real(a,b)	Complex exponential (real)	Re[e^(a+bi)] = e^a cos(b)	complex_exp_real(1,0)	2.718
complex_exp_imag(a,b)	Complex exponential (imag)	Im[e^(a+bi)] = e^a sin(b)	complex_exp_imag(0,pi/2)	1

🔄 Unit Conversion Functions 30+ Functions

Length Conversions (8)

Function	Description	Formula	Example	Result
meters_to_feet(m)	Meters to feet	m / 0.3048	meters_to_feet(10)	32.81 ft
feet_to_meters(ft)	Feet to meters	ft × 0.3048	feet_to_meters(10)	3.048 m
inches_to_cm(in)	Inches to centimeters	in × 2.54	inches_to_cm(12)	30.48 cm
cm_to_inches(cm)	Centimeters to inches	cm / 2.54	cm_to_inches(25.4)	10 in
miles_to_km(mi)	Miles to kilometers	mi × 1.609344	miles_to_km(10)	16.09 km
km_to_miles(km)	Kilometers to miles	km / 1.609344	km_to_miles(16.09)	10 mi
nautical_miles_to_km(nmi)	Nautical miles to km	nmi × 1.852	nautical_miles_to_km(10)	18.52 km

Mass Conversions (4)

Function	Description	Formula	Example	Result
kg_to_pounds(kg)	Kilograms to pounds	kg × 2.20462	kg_to_pounds(10)	22.05 lb
pounds_to_kg(lb)	Pounds to kilograms	lb / 2.20462	pounds_to_kg(22)	9.98 kg
grams_to_ounces(g)	Grams to ounces	g × 0.035274	grams_to_ounces(100)	3.53 oz
ounces_to_grams(oz)	Ounces to grams	oz / 0.035274	ounces_to_grams(3.53)	100 g

Temperature Conversions (6)

Function	Description	Formula	Example	Result
celsius_to_fahrenheit(C)	Celsius to Fahrenheit	C × 9/5 + 32	celsius_to_fahrenheit(20)	68°F
fahrenheit_to_celsius(F)	Fahrenheit to Celsius	(F - 32) × 5/9	fahrenheit_to_celsius(68)	20°C
celsius_to_kelvin(C)	Celsius to Kelvin	C + 273.15	celsius_to_kelvin(20)	293.15 K
kelvin_to_celsius(K)	Kelvin to Celsius	K - 273.15	kelvin_to_celsius(293.15)	20°C
degrees_to_radians(deg)	Degrees to radians	deg × π/180	degrees_to_radians(180)	π rad
radians_to_degrees(rad)	Radians to degrees	rad × 180/π	radians_to_degrees(pi)	180°

Energy Conversions (6)

Function	Description	Formula	Example	Result
joules_to_calories(J)	Joules to calories	J / 4.184	joules_to_calories(1000)	239.0 cal
calories_to_joules(cal)	Calories to joules	cal × 4.184	calories_to_joules(100)	418.4 J
joules_to_btu(J)	Joules to BTU	J / 1055.06	joules_to_btu(10000)	9.48 BTU
btu_to_joules(BTU)	BTU to joules	BTU × 1055.06	btu_to_joules(1)	1055.06 J
ev_to_joules(eV)	Electron volts to joules	eV × 1.602177e-19	ev_to_joules(1)	1.602e-19 J
joules_to_ev(J)	Joules to electron volts	J / 1.602177e-19	joules_to_ev(1.602e-19)	1 eV

Pressure Conversions (6)

Function	Description	Formula	Example	Result
pascal_to_psi(Pa)	Pascals to PSI	Pa / 6894.76	pascal_to_psi(101325)	14.7 psi
psi_to_pascal(psi)	PSI to Pascals	psi × 6894.76	psi_to_pascal(14.7)	101325 Pa
pascal_to_atm(Pa)	Pascals to atmospheres	Pa / 101325	pascal_to_atm(101325)	1 atm
atm_to_pascal(atm)	Atmospheres to Pascals	atm × 101325	atm_to_pascal(1)	101325 Pa
pascal_to_mmhg(Pa)	Pascals to mmHg	Pa / 133.322	pascal_to_mmhg(101325)	760 mmHg
mmhg_to_pascal(mmHg)	mmHg to Pascals	mmHg × 133.322	mmhg_to_pascal(760)	101325 Pa

🔢 Mathematical & Physical Constants 47 Constants

Mathematical Constants (15)

Constant	Value	Description	Usage Example
pi	3.141592653589793	Pi (π)	2 * pi * r
e	2.718281828459045	Euler's number	exp(1) = e
tau	6.283185307179586	Tau (2π)	tau * r
phi	1.618033988749895	Golden ratio	(1 + sqrt(5))/2
sqrt2	1.4142135623730951	√2	sqrt2 * side
sqrt3	1.7320508075688772	√3	sqrt3 / 2
sqrt5	2.23606797749979	√5	sqrt5 + 1
ln2	0.6931471805599453	ln(2)	ln2 * bits
ln10	2.302585092994046	ln(10)	ln10 * decades
log2e	1.4426950408889634	log₂(e)	log2e * ln_value
log10e	0.4342944819032518	log₁₀(e)	log10e * ln_value
euler	0.5772156649015329	Euler-Mascheroni constant	Gamma function
catalan	0.9159655941772190	Catalan's constant	Series analysis
apery	1.2020569031595943	Apéry's constant ζ(3)	Zeta function
khinchin	2.6854520010653064	Khinchin's constant	Continued fractions

Physical Constants (22)

Constant	Value	Units	Description
c	299792458	m/s	Speed of light in vacuum
h	6.62607015e-34	J⋅Hz⁻¹	Planck constant
hbar	1.054571817e-34	J⋅s	Reduced Planck constant ℏ = h/2π
g	9.80665	m/s²	Standard gravity
G	6.67430e-11	m³⋅kg⁻¹⋅s⁻²	Gravitational constant
k	1.380649e-23	J/K	Boltzmann constant
Na	6.02214076e23	mol⁻¹	Avogadro constant
R	8.314462618	J⋅mol⁻¹⋅K⁻¹	Gas constant
sigma	5.670374419e-8	W⋅m⁻²⋅K⁻⁴	Stefan-Boltzmann constant
me	9.1093837015e-31	kg	Electron mass
mp	1.67262192369e-27	kg	Proton mass
mn	1.67492749804e-27	kg	Neutron mass
u	1.66053906660e-27	kg	Atomic mass unit
e_charge	1.602176634e-19	C	Elementary charge
epsilon0	8.8541878128e-12	F/m	Vacuum permittivity
mu0	1.25663706212e-6	H/m	Vacuum permeability
ke	8.9875517923e9	N⋅m²/C²	Coulomb constant
alpha	7.2973525693e-3	dimensionless	Fine structure constant
rydberg	1.0973731568160e7	m⁻¹	Rydberg constant
bohr	5.29177210903e-11	m	Bohr radius
wien	2.897771955e-3	m⋅K	Wien displacement constant

Electronics Constants (7)

Constant	Value	Units	Description
kT_300K	4.14e-21	J	Thermal energy at 300K
kT_300K_eV	0.0259	eV	Thermal energy at 300K in eV
impedance_vacuum	376.730313668	Ω	Impedance of free space
conductance_quantum	7.748091729e-5	S	Conductance quantum
resistance_quantum	12906.40372	Ω	Resistance quantum
magnetic_flux_quantum	2.067833848e-15	Wb	Magnetic flux quantum
josephson_constant	483597.8484e9	Hz/V	Josephson constant

Special Values (3)

Constant	Value	Description	Usage
inf	∞	Positive infinity	Limit calculations
infinity	∞	Positive infinity (alias)	Mathematical analysis
nan	NaN	Not a Number	Error handling

📏 Physical Units System 80+ Units

Length Units (12)

Unit	Name	Conversion Factor	Relative to Meter	Usage
m	meter	1.0	Base unit	Standard length
cm	centimeter	0.01	10⁻² m	Small measurements
mm	millimeter	0.001	10⁻³ m	Precision engineering
km	kilometer	1000	10³ m	Long distances
in	inch	0.0254	2.54 cm	Imperial system
ft	foot	0.3048	12 inches	Architecture, aviation
yd	yard	0.9144	3 feet	Fabric, sports fields
mi	mile	1609.344	5280 feet	Road distances
nm	nanometer	1e-9	10⁻⁹ m	Light wavelengths
μm	micrometer	1e-6	10⁻⁶ m	Microscopy
angstrom	angstrom	1e-10	10⁻¹⁰ m	Atomic dimensions

Mass Units (7)

Unit	Name	Conversion Factor	Relative to Kilogram	Usage
kg	kilogram	1.0	Base unit	Standard mass
g	gram	0.001	10⁻³ kg	Laboratory measurements
mg	milligram	1e-6	10⁻⁶ kg	Pharmaceuticals
lb	pound	0.453592	≈ 454 g	Imperial system
oz	ounce	0.0283495	1/16 pound	Food portions
ton	metric ton	1000	10³ kg	Large masses
u	atomic mass unit	1.66054e-27	≈ proton mass	Atomic physics

Time Units (8)

Unit	Name	Conversion Factor	Relative to Second	Usage
s	second	1.0	Base unit	Standard time
ms	millisecond	0.001	10⁻³ s	Computing, reaction times
μs	microsecond	1e-6	10⁻⁶ s	Electronics
ns	nanosecond	1e-9	10⁻⁹ s	High-speed electronics
min	minute	60	60 s	Daily activities
hr	hour	3600	60 min	Work schedules
day	day	86400	24 hr	Calendar time
year	year	31557600	365.25 days	Long-term planning

Energy Units (8)

Unit	Name	Conversion Factor	Relative to Joule	Usage
J	joule	1.0	Base unit	Standard energy
eV	electron volt	1.602177e-19	≈ 1.6×10⁻¹⁹ J	Atomic physics
keV	kiloelectron volt	1.602177e-16	10³ eV	X-ray energies
MeV	megaelectron volt	1.602177e-13	10⁶ eV	Nuclear physics
cal	calorie	4.184	≈ 4.18 J	Chemistry
kcal	kilocalorie	4184	10³ cal	Food energy
BTU	british thermal unit	1055.06	≈ 1055 J	HVAC systems

SI Derived Units (13)

Unit	Name	Quantity	SI Base Units	Usage
Hz	hertz	Frequency	s⁻¹	Oscillations, radio
N	newton	Force	kg⋅m⋅s⁻²	Mechanical force
Pa	pascal	Pressure	kg⋅m⁻¹⋅s⁻²	Pressure, stress
W	watt	Power	kg⋅m²⋅s⁻³	Electrical power
V	volt	Electric potential	kg⋅m²⋅s⁻³⋅A⁻¹	Electrical voltage
Ω	ohm	Electrical resistance	kg⋅m²⋅s⁻³⋅A⁻²	Electrical resistance
F	farad	Capacitance	kg⁻¹⋅m⁻²⋅s⁴⋅A²	Electrical capacitance
H	henry	Inductance	kg⋅m²⋅s⁻²⋅A⁻²	Electrical inductance
C	coulomb	Electric charge	A⋅s	Electrical charge
T	tesla	Magnetic field	kg⋅s⁻²⋅A⁻¹	Magnetic field strength
Wb	weber	Magnetic flux	kg⋅m²⋅s⁻²⋅A⁻¹	Magnetic flux

Additional Units (25+)

Category	Units	Description	Examples
Temperature	K, celsius, fahrenheit	Temperature scales	Thermal calculations
Current	A, mA, μA	Electric current	Electronics design
Pressure	atm, bar, mmHg, psi	Pressure units	Engineering applications
Angles	rad, deg, arcmin, arcsec	Angular measurements	Navigation, astronomy
Volume	L, mL, gal, qt	Volume measurements	Fluid mechanics
Area	hectare, acre	Large area measurements	Land surveying