897634 - Fix Math.expm1 when !HAVE_EXPM1

Status: RESOLVED FIXED

(Core :: JavaScript Engine, defect)

Description

[Jason Orendorff [:jorendorff]]

Two issues:

1. Precision when the argument is > 0.00001 but still smallish

The current code computes exp(x)-1 when |x| >= 0.00001.

This loses some bits. The worst cases are:

    js> Math.expm1(1e-5)
    0.000010000050000166668   # system expm1
    0.000010000050000069649   # exp(x)-1
    js> Math.expm1(-1e-5)
    -0.000009999950000166666  # system expm1
    -0.000009999950000172397  # exp(x)-1

I'm pretty sure we can safely use that approximation when exp(x) is outside the range (1/2, 2), that is, |x| >= log(2) ~= 0.69314.

    js> Math.expm1(0.69315)
    1.0000056388880587        # system expm1
    1.0000056388880587        # exp(x) - 1

but that's a much bigger range where we'll need to use a series approximation.

2. Monotonicity. This one is a surprise to me.

In bug 717379 comment 76, 4esn0k notes:
> with current algorithm for expm1 (!HAVE_EXPM1), expm1 is not monotonic
> Math.expm1(-1e-2) === -0.009950166250831893 
> Math.expm1(-0.009999999999999998) === -0.009950166250831945 
> so
> Math.expm1(-1e-2) > Math.expm1(-0.009999999999999998)

These arguments are outside the ±0.00001 threshold, so the non-monotonicity is happening in the exp(x) - 1 part of the range. So... I guess this means exp() itself is not monotonic on 4esn0k's platform. It's hard to guard against that.

The Taylor series approximation we use near 0 is monotonic if the C++ stack provides monotonic multiplication and addition.

Comment 1

[4esn0k]

@Jason Orendorff, when i tested it, i used a code from https://bugzilla.mozilla.org/attachment.cgi?id=779974&action=diff 
where the threshold is 1e-2, not 0.00001

Comment 2

[4esn0k]

actually, the test should check, that:

Math.expm1(-threshold - Math.ulp(threshold)) <=
Math.expm1(-threshold) <=
Math.expm1(-threshold + Math.ulp(threshold)) <=

Math.expm1(threshold - Math.ulp(threshold)) <=
Math.expm1(threshold) <=
Math.expm1(threshold + Math.ulp(threshold))

Comment 3

[Jason Orendorff [:jorendorff]]

Ah. Thanks, 4esn0k! This makes more sense. The two issues are then related after all.

Comment 4

[André Bargull [:anba]]

Is it not possible to just use the fdlibm version of expm1? That's at least what's proposed in the ECMAScript spec.

Comment 5

[Tooru Fujisawa [:arai]]

Now Math.expm1 is using fdlibm expm1, so we just need to add testcase in comment #2 and make sure it satisfies the requirement.

Comment 6

[Tooru Fujisawa [:arai]]

it satisfies the comment #0 and comment #2 for several numbers, on OSX 64bit, and it should be same for other archs, as it's using same impl.
will post a patch to add testcase after testing on other archs.

Comment 7

[Tooru Fujisawa [:arai]]

Attachment: Add more testcase for Math.expm1.

about monotonicity (expm1-monotonicity.js), it passes all thresholds.

about precision (expm1-approx.js), it passes some more range, but result is still sloppy for large x.

Comment 8

[Jason Orendorff [:jorendorff]]

Comment on attachment 8767172 [details] [diff] [review]
Add more testcase for Math.expm1.

Review of attachment 8767172 [details] [diff] [review]:
-----------------------------------------------------------------

Great work! Thank you.

Comment 9

[Pulsebot]

Pushed by [e-mail]:
https://hg.mozilla.org/integration/mozilla-inbound/rev/ffc6d04b8cd7
Add more testcase for Math.expm1. r=jorendorff

Comment 10

[Carsten Book [:Tomcat]]

https://hg.mozilla.org/mozilla-central/rev/ffc6d04b8cd7