Design

Bounded integers

below draws on [0, n) by Lemire's multiply-and-reject over uint32 or uint64 draws (dtype, default uint64). With size or out it draws element i from stream draw i and retries a rejected draw on a fallback generator keyed by the draw's global index, so it uses exactly one draw per element, the position advances by that, and a fill cut at any element boundary equals the whole fill. Without them it is one scalar draw, which takes as many draws as the rejection loop needs. n = 0 returns 0.

Normals

normal with float64 is the 1024-layer ziggurat of Appendix A. Element i comes from 64-bit draw i, so a fill equals the scalar draws and a fill cut at any element equals the whole fill. A draw outside the inner rectangles, about 0.4 % of them, continues on a fallback stream keyed by its global draw index. An empty fill aligns the position to 64.

normal with float32 is Box-Muller on pairs of float32 uniforms a, b: with r = sqrt(-2 ln(1 - a)) the pair is (r cos 2 pi b, r sin 2 pi b). Pair j gives elements 2j (cos half) and 2j + 1 (sin half) from uniforms 2j and 2j + 1. An odd size keeps the cos half of its last pair and still consumes both uniforms. A scalar draw is the cos half and consumes two uniforms, so it equals element 0 of a fill.

The values of both are the bits of tandem-c's normal fills on every compiler.

Exponentials

exponential draws -ln(1 - u) from one uniform u of the dtype per element, float64 from float64 uniforms in double and float32 from float32 uniforms in single, with the polynomial logarithm of the normals and no libm call. Element i comes from uniform i, so a fill equals the scalar draws, a fill cut at any element boundary equals the whole fill, and n = 0 leaves the position unchanged. The bits are the same in every Tandem port and on every compiler.