Mathematical functions

CENTL keeps exact expressions exact and approximates only when asked.

sqrt(4/9)                    = 2/3
circle_area(3)               = 9 * pi
approx(sqrt(2), 20)          ≈ [1.4142135623730950488, 1.4142135623730950489]

approx(expression) requests 20 significant decimal digits. approx(expression, digits) requests between 1 and 1000. The result is an outward-rounded interval containing the mathematical value, not a point estimate disguised as an exact answer. Because an enclosure is not an exact scalar, approx(...) must be the outermost evaluation request; embedding it in exact arithmetic, conditions, or symbolic transformations returns approximation_not_expression.

Trigonometry and elementary functions

Angles are radians unless converted explicitly with radians(degrees) or degrees(radians).

sin(x)   cos(x)   tan(x)
asin(x)  acos(x)  atan(x)  atan2(y, x)
sinh(x)  cosh(x)  tanh(x)
sqrt(x)  exp(x)   log(x)   abs(x)

The exact constants are pi, e, and tau. Trigonometric expressions remain symbolic until approximation is requested. Differentiation understands the one-argument functions above except abs.

Exact polynomial integration

CENTL 0.9.0 integrates exact rational-coefficient univariate polynomials:

integrate(3*x^2 + 2*x + 1, x) = x^3 + x^2 + x
integrate(x^2, x = 0, 1)      = 1/3

The two-argument form chooses the canonical antiderivative with integration constant zero. The bounded form requires exact rational bounds and returns the exact difference between endpoint values. Unsupported integrands, including transcendental and multivariate cases, remain visible as integrate(...) instead of being approximated. Accepted polynomial powers are positive and at most 64. Explicit zero powers also remain visible so CENTL does not erase the undefined 0^0 point. See symbolic calculus.

Geometry

The initial geometry vocabulary is deliberately direct:

square_area(side)
rectangle_area(width, height)
rectangle_perimeter(width, height)
triangle_area(base, height)
trapezoid_area(base1, base2, height)
circle_area(radius)
circumference(radius)
sphere_area(radius)
sphere_volume(radius)
cylinder_volume(radius, height)
hypot(a, b)
distance(x1, y1, x2, y2)
slope(x1, y1, x2, y2)

These are exact formula expansions. Rational inputs therefore produce exact rational results, while results involving pi or an irrational square root remain exact symbolic expressions. Wrap any result in approx for a rigorous real enclosure.

Concrete mathematics

CENTL currently includes exact unbounded-integer primitives for:

gcd(a, b)             lcm(a, b)
factorial(n)           fibonacci(n)
choose(n, k)           permutations(n, k)

Finite sums, products, recurrences, sequences, and generating functions form the concrete-mathematics layer. The implemented bounded exact forms are:

sum(expression, variable = lower, upper)
product(expression, variable = lower, upper)
sequence(expression, variable = lower, upper)
recurrence(initial, previous = step, index = lower, upper)

Bounds are inclusive exact integers. Empty sums are 0, empty products are 1, and empty sequences and recurrences are []. A recurrence's initial value occupies its lower index, and its step receives the preceding exact value and each later index. Sequence elements and recurrence terms remain exact scalar values within the active machine resource limits. See exact finite iteration and sequences. Generating functions remain a later extension and will use the same exact-first value model rather than a separate subsystem.

Current limits

• Approximation supports real arithmetic, integer powers, the constants above, and the listed elementary functions.
• Unresolved variables cannot be approximated until they are substituted.
• Complex values, units, geometric objects, limits, series, and integration outside the exact univariate-polynomial domain are planned but are not silently simulated by the evaluator.
• A failed domain proof or exhausted precision budget is a structured error.