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A function of a small argument evaluates to 0 #2074
Description
A function of a small argument evaluates to 0 at the default hundred digits. Measured on 2.6.0:
| Input | Evaluates to | Value |
|---|---|---|
sin(10^(-60)) |
0 |
1e-60 |
tan(10^(-60)) |
0 |
1e-60 |
arcsin(10^(-60)) |
0 |
1e-60 |
arctan(10^(-60)) |
0 |
1e-60 |
erf(10^(-60)) |
0 |
1.128…e-60 |
sin(pi) |
0 |
0, which is right |
Cause. Each kernel creates its value with the floor at zero at half the working digits, absolute. That floor exists for the residual a function leaves at its own zeros: sin(pi) evaluates to about 1e-100. But that residual is relative to the argument, about 1e-100 * pi, and a function of an exact small argument cancels nothing. #2072 makes the floor relative to what cancelled for arithmetic. The same, with the argument's size as the scale, would separate these two cases for the functions.
sinh(10^(-60)) is a different case. It is evaluated from e^(10^(-60)), which is 1 within half the digits.
Part of #1944's class; found while working on it.
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Activity
Measure a function's residual against its argument (#2076)