XMScalarACos documentation for ULP errors

Question

XMScalarACos documentation for ULP errors

John 76

Hi all,

I saw this page: https://learn.microsoft.com/en-us/windows/win32/api/directxmath/nf-directxmath-xmscalaracos

but I cannot find the documentation for ULP error. Does it never be documented?

Thanks.

Junjie Zhu - MSFT 21,746 Reputation points

2022-04-27T10:08:44.733+00:00

It's been a few days, have you solved the problem?
John 76 Reputation points

2022-04-28T23:53:04.327+00:00

Yes, thanks.

1 answer

Your answer

Junjie Zhu - MSFT 21,746 Reputation points

2022-04-27T10:08:44.733+00:00

It's been a few days, have you solved the problem?
John 76 Reputation points

2022-04-28T23:53:04.327+00:00

Yes, thanks.

Answer 1

Junjie Zhu - MSFT 21,746

Hello,
Welcome to Microsoft Q&A!

XMScalarACos uses a 7-degree minimax approximation.

According to the documentation Floating-point rules (Direct3D 11)

IEEE-754 requires floating-point operations to produce a result that is the nearest representable value to an infinitely-precise result, known as round-to-nearest-even. Direct3D 11 defines the same requirement:32-bit floating-point operations produce a result that is within 0.5 unit-last-place (ULP) of the infinitely-precise result.

Thank you.

If the answer is the right solution, please click "Accept Answer" and kindly upvote it. If you have extra questions about this answer, please click "Comment".
Note: Please follow the steps in our documentation to enable e-mail notifications if you want to receive the related email notification for this thread.

John 76 Reputation points

2022-04-28T23:08:00.697+00:00

Hi, I tried, looks like 0.5 ULP is not true for arccosine, but I solved my problem by another way.

Thank you.

John 76

My test code, please correct me if my understanding is wrong.
Implementation was copied from https://github.com/microsoft/DirectXMath (DirectXMath.h and DirectXMathMisc.inl)

constexpr float XM_PI = 3.141592654f;  
  
inline float XMScalarACos(float Value) noexcept  
{  
    // Clamp input to [-1,1].  
    bool nonnegative = (Value >= 0.0f);  
    float x = fabsf(Value);  
    float omx = 1.0f - x;  
    if (omx < 0.0f)  
    {  
        omx = 0.0f;  
    }  
    float root = sqrtf(omx);  
  
    // 7-degree minimax approximation  
    float result = ((((((-0.0012624911f * x + 0.0066700901f) * x - 0.0170881256f) * x + 0.0308918810f) * x - 0.0501743046f) * x + 0.0889789874f) * x - 0.2145988016f) * x + 1.5707963050f;  
    result *= root;  
  
    // acos(x) = pi - acos(-x) when x < 0  
    return (nonnegative ? result : XM_PI - result);  
}  
  
void test() {  
    mt19937 gen(random_device{}());  
    uniform_real_distribution<float> dis(-1.f, 1.f);  
  
    while (true) {  
        const float in = dis(gen);  
        const float dx = XMScalarACos(in);  
        const float std = (float)std::acos((double)in);  
  
        const auto i_dx  = std::bit_cast<int32_t>(dx);  
        const auto i_std = std::bit_cast<int32_t>(std);  
  
        assert(abs(i_dx - i_std) <= 2);  
    }  
}

when in = -0.128314734

Share via

XMScalarACos documentation for ULP errors

1 answer

Your answer