complex: Fix expm1 precision loss for small arguments#1634
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complex: Fix expm1 precision loss for small arguments#1634
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Use cancellation-free formula that separates real and imaginary parts: real: expm1(a)*cos(b) - 2*sin(b/2)^2 imag: (expm1(a)+1)*sin(b) The previous exp(z)-1 approach lost digits when z was small due to catastrophic cancellation. The new formula delegates to the real expm1() for the real component and uses a half-angle identity for the cosine term, preserving full precision down to 1E-40 at 200 digits. The imaginary part reuses expm1(a) instead of calling exp(a) separately, eliminating one full Taylor series evaluation.
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Summary
expm1(z)to use a cancellation-free formula that separates real and imaginary parts:expm1(a)*cos(b) - 2*sin(b/2)^2(expm1(a)+1)*sin(b)exp(z)-1approach lost significant digits for small z due to catastrophic cancellationexpm1(a)instead of callingexp(a)separately, eliminating one full Taylor series evaluationTest plan