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This fix comes from glibc, from files which originated from the same place as the newlib files. Those files in glibc carry the same license as the newlib files. Bug 14155 is spurious underflow exceptions from Bessel functions for large arguments. (The correct results for large x are roughly constant * sin or cos (x + constant) / sqrt (x), so no underflow exceptions should occur based on the final result.) There are various places underflows may occur in the intermediate calculations that cause the failures listed in that bug. This patch fixes problems for the double version where underflows occur in calculating the intermediate functions P and Q (in particular, x**-12 gets computed while calculating Q). Appropriate approximations are used for P and Q for arguments at least 0x1p28 and above to avoid the underflows. For sufficiently large x - 0x1p129 and above - the code already has a cut-off to avoid calculating P and Q at all, which means the approximations -0.125 / x and 0.375 / x can't themselves cause underflows calculating Q. This cut-off is heuristically reasonable for the point beyond which Q can be neglected (based on expecting around 0x1p-64 to be the least absolute value of sin or cos for large arguments representable in double). The float versions use a cut-off 0x1p17, which is less heuristically justifiable but should still only affect values near zeroes of the Bessel functions where these implementations are intrinsically inaccurate anyway (bugs 14469-14472), and should serve to avoid underflows (the float underflow for jn in bug 14155 probably comes from the recurrence to compute jn). ldbl-96 uses 0x1p129, which may not really be enough heuristically (0x1p143 or so might be safer - 143 = 64 + 79, number of mantissa bits plus total number of significant bits in representation) but again should avoid underflows and only affect values where the code is substantially inaccurate anyway. ldbl-128 and ldbl-128ibm share a completely different implementation with no such cut-off, which I propose to fix separately. Signed-off-by: Keith Packard <keithp@keithp.com>
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