[AUD-014][High] Fix CORDIC periodic range reduction and atan2 zero vector

[gabrielsantosphilips]

Audit finding

• ID: AUD-014
• Status: Verified defect
• Severity: High
• Confidence: High
• Audited revision: 2f479320d805a1f9f35ebe4afaaeeded48913a94

Problem

SineCosine() moves an angle toward the convergence interval by only one pi shift. Inputs outside roughly one period remain outside the CORDIC convergence domain. Arctangent2(0,0) is not handled explicitly.

Source:

numerical-toolbox-cpp/numerical/math/Cordic.hpp

Lines 84 to 121 in 2f47932

	{
	const T pi{ std::numbers::pi_v<T> };
	const T halfPi{ pi / T(2) };
	T angle{ angleRadians };
	T sinSign{ T(1) };
	T cosSign{ T(1) };
	if (angle > halfPi)
	{
	angle -= pi;
	sinSign = T(-1);
	cosSign = T(-1);
	}
	else if (angle < -halfPi)
	{
	angle += pi;
	sinSign = T(-1);
	cosSign = T(-1);
	}
	T x{ K };
	T y{ T(0) };
	T z{ angle };
	for (std::size_t i = 0; i < Iterations; ++i)
	{
	T d{ (z >= T(0)) ? T(1) : T(-1) };
	T pow2i{ T(1) / static_cast<T>(std::size_t(1) << i) };
	T xNew{ x - d * y * pow2i };
	T yNew{ y + d * x * pow2i };
	z -= d * atanTable[i];
	x = xNew;
	y = yNew;
	}
	return SinCos{ sinSign * y, cosSign * x };
	}

Reproduction

• SineCosine(2*pi) returned approximately (-0.9851656, 0.1716062) instead of (0,1).
• Arctangent2(0,0) returned approximately 1.7432562, contradicting the documented zero convention.

Acceptance criteria

• Perform complete periodic range reduction before CORDIC iteration.
• Define and implement the (0,0) atan2 behavior.
• Constrain Iterations below the shift width.
• Add multi-period, axis, zero-vector, and iteration-bound tests.

[gabrielfrasantos]

Fixed and merged in #325 (d71b05f on main).

Both reported cases reproduced exactly: SineCosine(2*pi) returned (-0.9851656, 0.1716062) and Arctangent2(0,0) returned 1.7432562.

• SineCosine now reduces the angle over a full period with Fmod into [-pi, pi] before the existing half-period shift, so the CORDIC iteration always starts inside its convergence domain rather than only for inputs within roughly one period.
• Arctangent2(0, 0) returns 0 explicitly, matching the documented convention.
• Iterations is constrained below the shift width of std::size_t by static assertion, so std::size_t(1) << i cannot overflow.

After the fix, the worst absolute error across [-20, 20] radians is 3.0e-5, which is the expected resolution for 16 iterations:

SineCosine(0)        = ( 0.000000,  1.000000)
SineCosine(pi/2)     = ( 1.000000, -0.000018)
SineCosine(2*pi)     = (-0.000018,  1.000000)
SineCosine(4*pi)     = (-0.000018,  1.000000)
Arctangent2(0,0)     = 0
Arctangent2(-1,-1)   = -2.35618   (exact -2.35619)

Tests added: SineCosineOverMultiplePeriods, SineCosineAtAxesOutsideFirstPeriod, Arctangent2OfZeroVectorIsZero, plus EightIterationVariantSupportsFullPolarApi covering the eight-iteration configuration.

---

Generated by Claude Code