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