nfloat_complex_inv, nfloat_complex_div, nfloat_complex_sqrt, nfloat_complex_rsqrt

The inverse and the quotient are $\bar{y}/|y|^2$ and $x\bar{y}/|y|^2$ with a single reciprocal of $|y|^2$ shared by both parts (operands whose parts differ greatly in magnitude, or have extreme exponents, are rescaled first). At one word, $|y|^2$ is formed in registers and truncated to one word, and both parts of the numerator are divided by it with one shared inverse.

The square root of $z = a + bi$ is computed from $r = |z|$ (by nfloat_hypot at one and two words and for extreme exponents, otherwise as $\sqrt{a^2 + b^2}$) as $q = \sqrt{(r + |a|)/2}$ and $b/(2q)$, assigned to the real and imaginary parts according to the sign of $a$, so that nothing cancels; the reciprocal square root uses $v = 1/\sqrt{2r^2(r + |a|)}$ and the products $(r + |a|)v$ and $bv$. At one word on 64-bit machines, both are computed in registers.

@misc{flint-doc-nfloat-nfloat-complex-inv,
  author       = {{The FLINT team}},
  title        = {{nfloat\_complex\_inv, nfloat\_complex\_div, nfloat\_complex\_sqrt, nfloat\_complex\_rsqrt (nfloat.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/nfloat/nfloat_complex_inv.html},
  note         = {Generated 2026-11-02}
}