# 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)))))