Design

Bounded integers, normals and exponentials follow Appendix A of the specification. An empty bounded, Float32 normal or exponential fill leaves the position, and an empty uniform or Float64 normal fill aligns it.

Bounded integers

A bounded fill maps element i to draw i. A draw that Lemire rejects retries on split(g) of purpose(P_w) of the key, where g is the draw's index in the stream, so a fill cut at any element boundary equals the whole fill. A scalar bounded draw rejects onto the next draws of the stream instead.

Normals and exponentials

Float64 normals are the 1024-layer ziggurat of Appendix A, on the CPU. Element i takes 64-bit draw i. A draw that misses the inner rectangles, 0.43 % of them, continues on split(g) of purpose(0x4e524d3634) of the key, where g is the draw's index in the stream, and leaves the position alone. A fill cut at any element therefore equals the whole fill. tools/gen_zig_tables.swift writes Sources/Tandem/ZigTables.swift from the spec's tables/normal_f64_zig1024.json after it checks the file's SHA-256.

Float32 normal pair j is elements 2j and 2j + 1 from uniforms 2j and 2j + 1, so an odd count consumes one uniform more than it writes.

The normals and exponentials copy the arithmetic of tandem-c: a short series for log on the exponent-split argument, and series for cos and sin on an angle cut at the nearest quarter turn, with every multiply-add an explicit fma. The Float32 exponential takes tandem-c's two-part -log, within 0.58 ulp: the leading quotient carried with its residual and the exponent's ln 2 added by an exact two-sum. The ziggurat takes log only on a missed draw. The shader is compiled with fast math off and takes division and square root from precise::, which round correctly. The GPU f32 normals and exponentials and the CPU f32 and f64 ones are then bit for bit those of tandem-c.