Scientific Computing · Cluster
Floating Point
Representation, rounding, and error.
- 01 IEEE 754 Binary Formats: Sign, Exponent, and Fraction Layouts How binary16, binary32, binary64, and binary128 divide a fixed-width word into three fields, why the exponent is biased, and what the implicit leading bit buys.
- 02 Round-to-Nearest-Even and the Mechanics of Correct Rounding How an adder decides where to cut a result, what guard, round, and sticky bits encode, and why the tie case resolves toward an even last bit.
- 03 Subnormals, Signed Zero, and the Cost of Gradual Underflow Why the exponent field needed two reserved encodings, what a subnormal costs in precision, and how signed zero and NaN payloads change the meaning of equality.
- 04 Catastrophic Cancellation and the Cases Where Subtraction Is Exact Why subtracting two nearly equal numbers amplifies existing error rather than creating it, when the difference is exact by Sterbenz, and how formulas are rewritten to avoid the loss.
- 05 Compensated Summation and How Summation Order Bounds Error The worst-case error of a naive sum grows with the number of terms, pairwise reduction turns that into a logarithm, and compensation removes the growth at the cost of a serial dependency.
- 06 Fused Multiply-Add, Single Rounding, and the Cost of Contraction What a fused multiply-add computes internally, why one rounding instead of two halves the error bound, and how compiler contraction makes identical source produce different bits.
- 07 Correctly Rounded Decimal Conversion and Shortest Round-Trip Printing Why parsing a decimal string to the nearest double needs exact arithmetic in the hard cases, and how the shortest string that maps back is found without searching.
- 08 Forward Error, Backward Error, and Condition Numbers Separating what the algorithm got wrong from what the problem amplifies, why the rounding model hands over a backward error for free, and how a small residual can still mean a wrong answer.
- 09 Error-Free Transformations and Double-Double Arithmetic How TwoSum and TwoProduct recover the exact rounding error of an operation, what a double-double pair really represents, and where the transformations stop being exact.
- 10 Reassociation, Excess Precision, and the Limits of Bitwise Reproducibility Why identical source can produce different bits on two machines, which compiler transformations are responsible, and how ordering or exactness restores a stable result.