Product uncertainty underflows to zero with many factors #105

Problem

Since #85 (#86), the first-order uncertainty of a product of signals is computed from the mantissas and exponents of the factors (np.frexp), so that no intermediate product overflows or underflows whatever the order and magnitudes of a few factors. However, the products of the mantissas before and after each factor are computed with np.cumprod: each mantissa lies in [0.5, 1), so that their product decreases geometrically with the number of factors and finally underflows, even when the product of the signals itself is perfectly representable.

Reproduced with factors all equal to 1 and an uncertainty of 0.01 on each (expected: 0.01 × √n):

The products of the mantissas first lose precision (subnormal numbers), then vanish: the uncertainty of a product of many signals is wrong, then artificially zero, a scientifically misleading result. sigima.proc.signal.product of 200 float32 signals returns dy = 0.

Proposal

• Renormalize the running products of the mantissas at each step (np.frexp of the current product, exponents accumulated as integers), for the products before and after each factor: they stay in [0.5, 1) whatever the number of factors. Scaling by powers of two is exact, so that the results are bit-identical wherever no intermediate product underflowed (existing tests unchanged).
• The result changes for many factors: the contract of sigima.signal.product goes to version 2 (version 1 was published with Sigima 1.3.0; the one-off exception of 1.4 does not apply to later changes), with its qualification.
• Tests with several hundred to several thousand factors, in float32 and float64, including factors different from 1.

Reported by an independent review of Sigima 1.4 (unreleased).