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Re: patch to bug #86829
> Just as an example, compare the results for
> x = 0x1.fffffffffffffp1023
Thank you for your answer and the counterexample. :-)
> If we had useful range info on floats we might conditionalize such
> transforms appropriately. Or we can enable it on floats and do
> the sqrt (x*x + 1) in double.
I think I managed to find a bound were the transformation can be done
without overflow harm, however I don't know about rounding problems,
Suppose we are handling double precision floats for now. The function
x/sqrt(1 + x*x) approaches 1 when x is big enough. How big must be x
for the function be 1?
Since sqrt(1 + x*x) > x when x > 1, then we must find a value to x
that x/sqrt(1 + x*x) < eps, where eps is the biggest double smaller
than 1. Such eps must be around 1 - 2^-53 in ieee double because the
mantissa has 52 bits. Solving for x yields that x must be somewhat
bigger than 6.7e7, so let's take 1e8. Therefore if abs(x) > 1e8, it is
enough to return copysign(1, x). Notice that this arguments is also
valid for x = +-inf (if target supports that) because sin(atan(+-inf))
= +-1, and it can be extended to other floating point formats.The
following test code illustrates my point:
This might still be faster than calculating sin(atan(x)) explicitly.
Please let me know if this is unfeasible. :-)
On Tue, Aug 21, 2018 at 11:27 AM, Jeff Law <email@example.com> wrote:
> On 08/21/2018 02:02 AM, Richard Biener wrote:
>> On Mon, Aug 20, 2018 at 9:40 PM Jeff Law <firstname.lastname@example.org> wrote:
>>> On 08/04/2018 07:22 AM, Giuliano Augusto Faulin Belinassi wrote:
>>>> Closes bug #86829
>>>> Description: Adds substitution rules for both sin(atan(x)) and
>>>> cos(atan(x)). These formulas are replaced by x / sqrt(x*x + 1) and 1 /
>>>> sqrt(x*x + 1) respectively, providing up to 10x speedup. This identity
>>>> can be proved mathematically.
>>>> 2018-08-03 Giuliano Belinassi <email@example.com>
>>>> * match.pd: add simplification rules to sin(atan(x)) and cos(atan(x)).
>>>> Bootstrap and Testing:
>>>> There were no unexpected failures in a proper testing in GCC 8.1.0
>>>> under a x86_64 running Ubuntu 18.04.
>>> I understand these are mathematical identities. But floating point
>>> arthmetic in a compiler isn't nearly that clean :-) We have to worry
>>> about overflows, underflows, rounding, and the simple fact that many
>>> floating point numbers can't be exactly represented.
>>> Just as an example, compare the results for
>>> x = 0x1.fffffffffffffp1023
>>> I think sin(atan (x)) is well defined in that case. But the x*x isn't
>>> because it overflows.
>>> So I think this has to be somewhere under the -ffast-math umbrella.
>>> And the testing requirements for that are painful -- you have to verify
>>> it doesn't break the spec benchmark.
>>> I know Richi acked in the PR, but that might have been premature.
>> It's under the flag_unsafe_math_optimizations umbrella, but sure,
>> a "proper" way to optimize this would be to further expand
>> sqrt (x*x + 1) to fabs(x) + ... (extra terms) that are precise enough
>> and not have this overflow issue.
>> But yes, I do not find (quickly skimming) other simplifications that
>> have this kind of overflow issue (in fact I do remember raising
>> overflow/underflow issues for other patches).
>> Thus approval withdrawn.
> At least until we can do some testing around spec. There's also a patch
> for logarithm addition/subtraction from MCC CS and another from Giuliano
> for hyperbolics that need testing with spec. I think that getting that
> testing done anytime between now and stage1 close is sufficient -- none
> of the 3 patches is particularly complex.
>> If we had useful range info on floats we might conditionalize such
>> transforms appropriately. Or we can enable it on floats and do
>> the sqrt (x*x + 1) in double.
> Yea. I keep thinking about what it might take to start doing some light
> VRP of floating point objects. I'd originally been thinking to just
> track 0.0 and exceptional value state. But the more I ponder the more I
> think we could use the range information to allow transformations that
> are currently guarded by the -ffast-math family of options.