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Another building block for a #Python floating point ninja toolset:

def veltkamp_split(x):
'Exact split into two 26-bit precision components'
t = x * 134217729.0
hi = t - (t - x)
lo = x - hi
return hi, lo…
Input: one signed 53-bit precision float

Output: two signed 26-bit precision floats

Invariant: x == hi + lo

Constant: 134217729.0 == 2.0 ** 27 + 1.0

>>> hi, lo = veltkamp_split(pi)
>>> hi + lo == pi
>>> hi.hex()
>>> lo.hex()

Note all the trailing zeros and the difference between the two exponents. Also both the lo and hi values are signed.
Use case: The 26-bit precision components can be multiplied losslessly (without rounding):

# Four exact components of e * pi:
>>> pi_hi*e_hi
>>> pi_hi*e_lo
>>> pi_lo*e_hi
>>> pi_lo*e_lo
The payoff for Veltkamp-Dekker splitting and piecewise multiplication is that you can build quad precision arithmetic out of everyday double precision arithmetic.
The part that is pure magic: How do you split a 53 bit number into two 26 bit numbers? Where did the extra bit go?

Answer: The extra bit is stored in the sign bit of the "lo" component.

53 bits + 1 sign = 26 bits + 1 sign + 26 bits + 1 sign

All bits accounted for.🧐

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