# Accuracy

These exceptions apply to the four-ulp accuracy target:

- `lgamma` near its zeros at 1 and 2 is within 4 ulps of 2, not of the
  result. `digamma` near its zeros is within 4 ulps of 1, and below -20
  within 4 ulps of ln(1 - x).
- `binom` and `binom_dn` of a non-whole n with a whole k up to 64 are
  within 2k ulps.
- `binom_dk` has no relative bound near n = 2k, where its two ψ terms
  cancel.
- Normal and log-normal evaluation of `f32` inputs uses `f32`
  arithmetic, with parameters and precomputed coefficients rounded to
  `f32`. Their densities are within 12 ulps and their cdfs within 14
  where |z| \< 3, z being the standard score of x or of ln x. They lose
  accuracy as z² grows, to about 80 and 170 ulps at |z| = 12. Their
  quantiles are within 13 ulps, except near a quantile of 0, where
  cancellation leaves an absolute error of up to 8 ulps of the standard
  deviation. These figures exclude additional error from parameter
  rounding and ill-conditioned location/scale transformations.
- Complex functions count ulps of the result’s magnitude, so a much
  smaller part can carry most of the error. Complex powers, Γ and
  binomial coefficients allow ulps of their magnitude times one more
  than the exponent’s magnitude. `exp` gives an exact direction at the
  float nearest a whole number of quarter turns, so e^(iπ/2) is exactly
  i.