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42 lines
2.0 KiB
Plaintext
42 lines
2.0 KiB
Plaintext
# Scripts to generate polynomial approximations for expm1f function using Sollya.
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#
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# To compute expm1f(x), for |x| > Ln(2), using expf(x) - 1.0f is accurate enough, since catastrophic
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# cancellation does not occur with the subtraction.
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#
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# For |x| <= Ln(2), we divide [-Ln2; Ln2] into 3 subintervals: [-Ln2; -1/8], [-1/8, 1/8], [1/8, Ln2],
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# and use a degree-6 polynomial to approximate expm1f in each interval.
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> f := expm1(x);
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# Polynomial approximation for e^(x) - 1 on [-Ln2, -1/8].
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> P1 := fpminmax(f, [|0, ..., 6|], [|24...], [-log(2), -1/8], relative);
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> log2(supnorm(P1, f, [-log(2), -1/8], relative, 2^(-50)));
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[-29.718757839645220560605567049447893449270454705067;-29.7187578396452193192777968211678241631166415833034]
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> P1;
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-6.899231408397099585272371768951416015625e-8 + x * (0.999998271465301513671875 + x * (0.4999825656414031982421875
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+ x * (0.16657467186450958251953125 + x * (4.1390590369701385498046875e-2 + x * (7.856394164264202117919921875e-3
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+ x * 9.380675037391483783721923828125e-4)))))
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# Polynomial approximation for e^(x) - 1 on [-1/8, 1/8].
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> P2 := fpminimax(f, [|1,...,6|], [|24...|], [-1/8, 1/8], relative);
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> log2(supnorm(P2, f, [-1/8, 1/8], relative, 2^(-50)));
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[-34.542864999883718873453825391741639571826398336605;-34.542864999883717632126055163461570285672585214842]
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> P2;
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x * (1 + x * (0.5 + x * (0.16666664183139801025390625 + x * (4.1666664183139801025390625e-2
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+ x * (8.3379410207271575927734375e-3 + x * 1.3894210569560527801513671875e-3)))))
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# Polynomial approximation for e^(x) - 1 on [1/8, Ln2].
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> P3 := fpminimax(f, [|0,...,6|], [|24...|], [1/8, log(2)], relative);
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> log2(supnorm(P3, f, [1/8, log(2)], relative, 2^(-50)));
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[-29.189438260653683379922869677995123967174571911561;-29.1894382606536821385950994497150546810207587897976]
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> P3;
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1.23142086749794543720781803131103515625e-7 + x * (0.9999969005584716796875 + x * (0.500031292438507080078125
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+ x * (0.16650259494781494140625 + x * (4.21491153538227081298828125e-2 + x * (7.53940828144550323486328125e-3
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+ x * 2.05591344274580478668212890625e-3)))))
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