stillwater-sc / universal / 37160643253

87%

master: 84%

Build Type

push

github

Committed by web-flow

Commit Message

fix(takum): elementary functions past double's range for rbits 4-5 (#1629)

* fix(takum): elementary functions past double's range for rbits 4-5

At rbits = 4 and 5 a takum spans 2^+/-65535 or 2^+/-2^32 against double's
2^+/-1024, and the math functions evaluate in a double. Three things went
wrong:
- the result leaves double's range: exp(800) is an ordinary takum<32,4>,
  but std::exp overflowed and the takum was NaR;
- the argument leaves double's range: double(2^2000) is infinity, so
  log(2^2000), whose true value is ~1386, came back NaR;
- the argument is tiny: double(2^-2000) is 0, so sin(2^-2000) was 0 and
  floor(-2^-2000) was 0 instead of -1.

New header math/wide_range.hpp builds on two primitives that never leave
double's range. ln_abs(x) is ln|x| as a double-double, taken from the
decoded fields: c ln2 + log1p(f) linear, l/2 logarithmic. from_ln(L) is
the takum whose magnitude is e^L: an integer power of two (or integer
characteristic) plus a bounded remainder a double evaluates, assembled by
the codec from an int64_t characteristic, saturating where the true value
is past the takum's own range.

Covered, for takum and takum_log (71 call sites):
  exp exp2 exp10 expm1, sinh cosh asinh acosh, pow (all overloads)
  log log2 log10 log1p, erf erfc (asymptotic series past x = 26.5),
  hypot and atan2 (common power-of-two scaling; atan2 falls back to y/x
  or +/-pi when the ratio is below 2^-30), trunc round floor ceil frac,
  tiny-argument identities (sin tan asin atan tanh atanh -> x,
  cos sec cosh -> 1, cot csc -> 1/x),
  fmod and remainder, exactly for the linear takum: its values are
  dyadic, so x mod y is reduced with modular exponentiation and
  2^65535 mod 3 costs ~16 squarings.

Each wrapper checks range_fits_double at compile time, so rbits <= 3 is
untouched, and the new code returns nullopt whenever the plain double
evaluation is sound, so results inside double's range do not change.

Deliberately left as NaR: sin/cos/tan/sec/csc/co... (continued)

Coverage Stats

307 of 321 new or added lines in 10 files covered. (95.64%)

33 existing lines in 9 files now uncovered.

46106 of 53551 relevant lines covered (86.1%)

9734940.32 hits per line