ASINH is LN(|x|+SQRT(x*x+1)) with every step stored, so small inputs lose their digits

FN-ASINH-001: ASINH is LN(|x|+SQRT(x*x+1)) with every step stored, so small inputs lose their digits

Excel differs from the mathematical answer · measured · recorded 2026-11-04

ASINH is not the correctly rounded inverse hyperbolic sine. Its answers are those of SIGN(x)*LN(|x|+SQRT(x*x+1)), with x*x, the +1, the SQRT and the |x|+root each rounded to 64 bits and then to 53 bits before the next step, and the LN rounded to binary64. That route loses most of the digits of a small input: once x*x+1 rounds to 1, the answer is LN of |x|+1 after rounding, so it moves in steps of about 2.2E-16. ASINH(-1.4901631254755754E-16) is -2.22E-16, about 1.5 times the true value, and in the recorded rows every |x| up to 1.05E-16, including every subnormal, gives 0 on either sign. At the other end the x*x overflows: answers are finite up to the largest recorded input below it, 9.23E153, and #NUM! from the next recorded input, 2.38E154, although the true value there is about 356. In Excel build 20430 this route reproduces all 4,534 recorded rows (3,017 distinct inputs): 3,427 numbers and 1,107 #NUM!. The correctly rounded asinh matches only 1,224 of the 3,427.

Correct value: mpmath asinh at 2000 bits, rounded to the nearest binary64

Correct value: mpmath asinh at 2000 bits, rounded to the nearest binary64

Correct value: mpmath asinh at 2000 bits, rounded to the nearest binary64

Correct value: mpmath asinh at 2000 bits, rounded to the nearest binary64

Correct value: mpmath asinh at 2000 bits, rounded to the nearest binary64

Correct value: mpmath asinh at 2000 bits, rounded to the nearest binary64

Basis: All 4,534 rows of two recorded sets, 3,017 and 1,517 rows, from Excel build 20430. The route was recomputed independently in exact rational arithmetic with a correctly rounded LN, and matched every row. The stored steps matter on 5 of the rows (3 distinct inputs): the same formula in plain binary64 misses exactly those 5.

Inferred: That Excel's code is this expression, in this order, is inferred; what is measured is that the route reproduces every recorded row. The overflow boundary at |x| = 2^512, about 1.34E154, is inferred from x*x overflowing there: the recorded rows only bracket it, between 9.23E153 and 2.38E154. Excel reads a subnormal written to a cell back as 0, so the 0 at subnormal inputs may come from that step and not from this route.

Open: Only one Excel build, 20430, has been read over this range.