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    "title": "Volder: CORDIC shift-add coordinate rotation and vectoring",
    "summary": "Volder presents signed binary shift-add coordinate operations for real-time trigonometric rotation and vectoring, with a fixed gain correction and finite-word navigation implementation.\n\nA computer can turn a coordinate vector through many small angles whose tangents are powers of two. Each step then uses shifts and additions instead of a general multiplier. Running these steps in one mode follows a requested angle; another mode drives the vertical coordinate toward zero and accumulates direction and magnitude. Every pseudo-rotation also stretches the vector, so a scale correction is essential. Volder connects this practical method to a navigation computer; its finite precision, correction routines and hardware schedule remain part of the result.",
    "boundary": "Exact 1959 IRE Trans. Electron. Comput. EC-8(3),330-334, authorized reprint PDF 1-5 all technical text read; reprint pagination differs from original. First quadrant-handling step swaps/signs coordinates by 90 degrees; subsequent signed shift-add pseudo-rotations provide rotation/vectoring with constant gain only when scheduled steps are retained. Gain compensation and finite word/truncation error remain; navigation example includes two multiplication routines, not an entirely multiplication-free application. Own ideal 23-step gain 1.64676025812105 differs from printed 1.646760255 by 3.12 e-9: initial 1 e-9 equality check failed and is preserved, original finite-precision coefficient/table glyphs unreconstructed. No binary-table/hardware-time replication. Similar multiply/divide/hyperbolic/log routines are mentioned but not derived here; later unified methods remain separate. No all-domain optimality or independent certification.",
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        "description": "Exact 1959 IRE Trans. Electron. Comput. EC-8(3),330-334, authorized reprint PDF 1-5 all technical text read; reprint pagination differs from original. First quadrant-handling step swaps/signs coordinates by 90 degrees; subsequent signed shift-add pseudo-rotations provide rotation/vectoring with constant gain only when scheduled steps are retained. Gain compensation and finite word/truncation error remain; navigation example includes two multiplication routines, not an entirely multiplication-free application. Own ideal 23-step gain 1.64676025812105 differs from printed 1.646760255 by 3.12 e-9: initial 1 e-9 equality check failed and is preserved, original finite-precision coefficient/table glyphs unreconstructed. No binary-table/hardware-time replication. Similar multiply/divide/hyperbolic/log routines are mentioned but not derived here; later unified methods remain separate. No all-domain optimality or independent certification.",
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      "statement": "# Volder: CORDIC shift-add coordinate rotation and vectoring\n\nVolder presents signed binary shift-add coordinate operations for real-time trigonometric rotation and vectoring, with a fixed gain correction and finite-word navigation implementation.\n\nA computer can turn a coordinate vector through many small angles whose tangents are powers of two. Each step then uses shifts and additions instead of a general multiplier. Running these steps in one mode follows a requested angle; another mode drives the vertical coordinate toward zero and accumulates direction and magnitude. Every pseudo-rotation also stretches the vector, so a scale correction is essential. Volder connects this practical method to a navigation computer; its finite precision, correction routines and hardware schedule remain part of the result.\n\n## Scientific scope\n\nExact 1959 IRE Trans. Electron. Comput. EC-8(3),330-334, authorized reprint PDF 1-5 all technical text read; reprint pagination differs from original. First quadrant-handling step swaps/signs coordinates by 90 degrees; subsequent signed shift-add pseudo-rotations provide rotation/vectoring with constant gain only when scheduled steps are retained. Gain compensation and finite word/truncation error remain; navigation example includes two multiplication routines, not an entirely multiplication-free application. Own ideal 23-step gain 1.64676025812105 differs from printed 1.646760255 by 3.12 e-9: initial 1 e-9 equality check failed and is preserved, original finite-precision coefficient/table glyphs unreconstructed. No binary-table/hardware-time replication. Similar multiply/divide/hyperbolic/log routines are mentioned but not derived here; later unified methods remain separate. No all-domain optimality or independent certification.\n\n## People and systems\n\nJack E. Volder\n\n\n\n## Original sources\n\n- [Volder: CORDIC shift-add coordinate rotation and vectoring; exact original contribution text](https://www.ece.ucdavis.edu/~vojin/CLASSES/EPFL/Papers/3-Volder_CORDIC.pdf)\n- [IEEE-deposited exact CORDIC journal identity](https://api.crossref.org/works/10.1109%2FTEC.1959.5222693)\n",
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