Error bounds on complex floating-point multiplication
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Bibliographic record
Abstract
Given floating-point arithmetic with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -digit base- <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta"> <mml:semantics> <mml:mi> β </mml:mi> <mml:annotation encoding="application/x-tex">\beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> significands in which all arithmetic operations are performed as if calculated to infinite precision and rounded to a nearest representable value, we prove that the product of complex values <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z 0"> <mml:semantics> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">z_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z 1"> <mml:semantics> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">z_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be computed with maximum absolute error <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue z 0 double-vertical-bar z 1 EndAbsoluteValue one half beta Superscript 1 minus t Baseline StartRoot 5 EndRoot"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:msup> <mml:mi> β </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> <mml:msqrt> <mml:mn>5</mml:mn> </mml:msqrt> </mml:mrow> <mml:annotation encoding="application/x-tex">|z_0\|z_1| \frac {1}{2} \beta ^{1 - t} \sqrt {5}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In particular, this provides relative error bounds of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript negative 24 Baseline StartRoot 5 EndRoot"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>24</mml:mn> </mml:mrow> </mml:msup> <mml:msqrt> <mml:mn>5</mml:mn> </mml:msqrt> </mml:mrow> <mml:annotation encoding="application/x-tex">2^{-24} \sqrt {5}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript negative 53 Baseline StartRoot 5 EndRoot"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>53</mml:mn> </mml:mrow> </mml:msup> <mml:msqrt> <mml:mn>5</mml:mn> </mml:msqrt> </mml:mrow> <mml:annotation encoding="application/x-tex">2^{-53} \sqrt {5}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for IEEE 754 single and double precision arithmetic respectively, provided that overflow, underflow, and denormals do not occur. We also provide the numerical worst cases for IEEE 754 single and double precision arithmetic.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it