Algorithmic Details

Deep technical write-ups of selected algorithms used inside Universal. These docs are for readers who want the math, the transfer functions, and the trade-off space rather than the API-level usage.

Available Documents

• LNS log-domain add/sub algorithms – the seven shipped algorithms for evaluating log2(1 + 2^d) and log2(1 - 2^d), with transfer-function plots, error envelopes, and a picker decision tree.
• Multi-component floating-point arithmetic – Priest’s error-free transformations, Bailey/Hida’s hand-crafted fixed-precision dd / qd, Shewchuk’s adaptive expansions, and how Universal’s floatcascade<N> building block ties them together.
• Multi-limb arithmetic – how the fixed-size types store integers and significands in blocks, the 32-bit-limb kernels that add, multiply (schoolbook) and divide (Knuth’s Algorithm D) them under blocktriple, integer and blockbinary, how those changes were verified bit-identical, and what they gained.
• efloat: correctly rounded arithmetic and math library – how efloat rounds its arithmetic and math library correctly at any precision (Ziv’s strategy over stated error bounds), the algorithm behind each function, the alternatives tried and dropped, how it is verified against MPFR, measured accuracy and speed, and the open work.

Companion sections

• Number Systems – API-level reference for each arithmetic type.
• Design – design rationale and architectural decisions.