1.2.0

FlameMath 1.2.0

New Functions

Language Improvements

• Apply(f, list) — Splat a list as arguments to a function. Apply(Add, [1, 2, 3]) → 6
• Substitute(expr, symbol, value) — Substitute a symbol with a value in an expression. Substitute(x^2 + x, x, 3) → 12
• Hold(expr) — Prevent evaluation of an expression, returning it unevaluated

Number Theory

• GCD(a, b) / LCM(a, b) — Greatest common divisor and least common multiple
• IsPrime(n) — Primality testing via deterministic Miller-Rabin with 12 fixed bases; hybridly uses a global Sieve of Eratosthenes for small primes
• PrimesInRange(m, n) — Returns all primes in [m, n] using a segmented Sieve of Eratosthenes
• PowerMod(base, exp, mod) — Modular exponentiation via BigInteger.modPow
• Binomial(n, k) — Binomial coefficient using a multiplicative formula (avoids full factorials)
• Multinomial(n, k1, k2, ...) — Multinomial coefficient
• WieferichPrime(n) — Finds the smallest Wieferich prime up to n
• Prime(n) — Returns the n-th prime number using Meissel-Lehmer prime counting with binary search for large n, sieve for small n. Prime(4) → 7

Inverse Trigonometric Functions

• ArcSin(x) — Inverse sine with exact symbolic values for well-known angles (0, Pi/6, Pi/4, Pi/3, Pi/2)
• ArcCos(x) — Inverse cosine with exact symbolic values across [0, Pi], including negative arguments
• ArcTan(x) — Inverse tangent with exact symbolic values for 0, 1, Sqrt(3), 1/Sqrt(3)
• ArcTan2(y, x) — Two-argument (quadrant-aware) arctangent

Hyperbolic Functions

• Sinh(x) — Hyperbolic sine
• Cosh(x) — Hyperbolic cosine
• Tanh(x) — Hyperbolic tangent

Numeric Utilities

• Sign(n) — Returns the sign of an integer: -1, 0, or 1
• Clamp(n, low, hi) — Restricts a numeric value to a given range [low, hi]

List Operations

• Join(list1, list2, ...) — Concatenate multiple lists into one
• Take(list, n) — Return the first n elements (positive) or last |n| elements (negative)
• Drop(list, n) — Remove the first n elements (positive) or last |n| elements (negative)
• First(list) — Return the first element, or Null if empty
• Last(list) — Return the last element, or Null if empty
• Count(list, value) — Count occurrences of a value in a list
• Tally(list) — Frequency count of elements, returning [[element, count], ...] pairs in first-occurrence order. Works with any element type including nested lists
• Union(list1, list2, ...) — Set union across multiple lists, preserving first-occurrence order
• Intersection(list1, list2, ...) — Set intersection across multiple lists
• FoldScan(f, start, list) — Cumulative fold (scan) over a list, returning all intermediate accumulator values. FoldScan(Add, 0, [1,2,3,4]) → [0,1,3,6,10]

String Functions

• StrHas(str, substring) — Tests whether a string contains a given substring. Case-sensitive
• StrReplace(str, target, replacement) — Replaces all occurrences of a substring with a replacement string

Bug Fixes

• Flat functions not flattened when called via symbol alias — Built-in functions with the isFlat attribute (e.g., Add, Mul) were not being flattened when invoked indirectly through a symbol reference (e.g., passing Add to FoldScan). This caused incorrect canonical ordering in the result, such as FoldScan(Add, 0, [a, b, c, d]) producing c + a + b instead of a + b + c

Improvements

• Sqrt() simplifies radicals — Sqrt now extracts the largest perfect-square factor from integer arguments and numeric factors in products. Sqrt(12) → 2*√3, Sqrt(4*x^2) → 2*√(x^2). Symbolic parts are left under the radical (no assumptions about variable signs)
• Square root display — Pow(n, (1/2)) and Sqrt(...) expressions now render with the √ symbol instead of function-call notation
• Zip() generalized — Now accepts any number of lists (variadic), not just two. Zip([1,2], [3,4], [5,6]) → [[1,3,5], [2,4,5]]
• Zip() bug fix — Fixed incorrect element indexing in the inner loop
• Documentation reorganized — Function reference docs are now organized into category subfolders (math, list, ntheory, general, etc.)
• Mathematical functions return integers where applicable — Functions like Sin(0) now return 0 instead of 0.0
• Version tracking — Added version.properties for runtime version identification

FlameInt: Arbitrary-Precision Integer Migration

IntegerAtom has been migrated from long to FlameInt, a custom arbitrary-precision integer type using base-2^32 limb arrays. This removes the 64-bit ceiling on all integer arithmetic — functions like Binomial(100, 50), Factorial(50), and large Pow expressions now produce exact results instead of silently overflowing.

Arithmetic

• Addition, subtraction, multiplication — Schoolbook algorithms operating on int[] magnitude arrays with unsigned limb arithmetic
• Division (Knuth Algorithm D) — Multi-limb long division with normalization, replacing a previous repeated-subtraction implementation. O(m·n) per division
• Exponentiation (FlameInt.pow) — Binary exponentiation (repeated squaring), replacing Math.pow/long casts in PowFunc that silently overflowed to Long.MAX_VALUE
• Modular remainder — mod() now extracts the remainder directly from the division algorithm instead of recomputing via separate divide + multiply + subtract

Bug Fixes

• Knuth D unsigned comparison — The quotient refinement loop used signed > where Long.compareUnsigned was needed, producing wrong results for large multi-limb divisions
• Knuth D normalization overflow — Bits were lost during left-shift normalization; fixed by prepending a zero limb before shifting
• leadingZeros computed popcount — The helper counted set bits instead of leading zeros; replaced with Integer.numberOfLeadingZeros

Internal

• Number theory utilities (PrimeSieve, NumberTheoryUtils) added as shared infrastructure
• PrimeSieve now supports Meissel-Lehmer π(x) computation, cached prime list, and prefix count array for O(1) lookups
• Function references compartmentalized into separate folders by category