Standard Library - Calculus

Overview

Romasm's calculus library provides numerical differentiation and integration functions. All values use fixed-point scaling by 1000 (e.g., 5000 = 5.000).

Derivative Functions

derivative_x_squared

Calculates f'(x) = 2x for f(x) = x² using the central difference method.

Method: f'(x) ≈ (f(x+h) - f(x-h)) / (2h)

Input	Output
R0 = x (scaled by 1000)	R0 = 2x (scaled by 1000)

Example:

; Calculate f'(5) = 2*5 = 10
LOAD R0, 5000    ; x = 5.000
CALL derivative_x_squared
PRINT R0         ; Outputs ~10000 (10.000)

derivative_x_cubed

Calculates f'(x) = 3x² for f(x) = x³.

Input	Output
R0 = x (scaled by 1000)	R0 = 3x² (scaled by 1000)

derivative_sin

Calculates the derivative of sin(x) where x is in degrees: d/dx sin(x) = cos(x) × (π/180).

derivative_cos

Calculates the derivative of cos(x): d/dx cos(x) = -sin(x) × (π/180).

derivative_exp

Calculates the derivative of e^x: d/dx e^x = e^x (the derivative of e^x is itself!).

derivative_ln

Calculates the derivative of ln(x): d/dx ln(x) = 1/x.

Integral Functions

integral_x_squared

Calculates the definite integral ∫[a,b] x² dx analytically using the formula: (b³ - a³) / 3

Input	Output
R0 = a (lower bound, scaled) R1 = b (upper bound, scaled)	R0 = ∫[a,b] x² dx (scaled by 1000)

Example:

; Calculate ∫[0,5] x² dx = 125/3 ≈ 41.667
LOAD R0, 0       ; a = 0
LOAD R1, 5000    ; b = 5.000
CALL integral_x_squared
PRINT R0         ; Outputs ~41667 (41.667)

integral_trapezoidal_x_squared

Numerical integration using the trapezoidal rule (more accurate than rectangular rule).

Trapezoidal Rule: ∫[a,b] f(x) dx ≈ (b-a)/2 × [f(a) + f(b)] + Σ f(x_i)

Input: R0 = a, R1 = b, R2 = Δx (all scaled by 1000)

integral_simpson_x_squared

Numerical integration using Simpson's rule (most accurate).

Simpson's Rule: ∫[a,b] f(x) dx ≈ (b-a)/6 × [f(a) + 4f((a+b)/2) + f(b)]

Input: R0 = a, R1 = b, R2 = Δx (all scaled by 1000)

integral_sin

Definite integral of sin(x) from a to b (x in degrees).

integral_cos

Definite integral of cos(x) from a to b (x in degrees).

integral_exp

Definite integral of e^x: ∫[a,b] e^x dx = e^b - e^a

integral_ln

Definite integral of 1/x: ∫[a,b] 1/x dx = ln(b) - ln(a)

Numerical Methods

Romasm uses several numerical methods for calculus operations:

Central Difference (Derivatives)

f'(x) ≈ (f(x+h) - f(x-h)) / (2h)

More accurate than forward difference, uses values on both sides of x.

Rectangular Rule (Integration)

∫[a,b] f(x) dx ≈ Σ f(x_i) × Δx

Simplest method, approximates area as rectangles.

Trapezoidal Rule (Integration)

More accurate than rectangular rule, uses trapezoids instead of rectangles.

Simpson's Rule (Integration)

Most accurate numerical integration method, uses parabolic approximations.