Although the IEEE 754 trick is computationally efficient, it can produce estimation errors up to 4 % of the true value. A method described in [3] restores accuracy by using all 64 bits and applying a correction function \(F\).
Begin by expressing the exponential function as:
where \(y_i\) and \(y_f\) are the integer and fractional parts of \(y/log(2)\), respectively. Define \(F = 1 + m - 2^{y_{f}}\). Given that \(m \equiv y_{f}\), rewrite \(F \left( y_{f} \right) = 1 + y_{f} - 2^{y_{f}}\). Model \(F \left( y_f \right)\) as a polynomial where \(y_f\) is defined over the range \([0, 1)\).
Estimate the exponential function by computing:
Interpret (I) as a 64‑bit signed integer, then reinterpret the result as a double‑precision floating‑point value. Using all 64 bits requires a factor \(2^{52}\) to align the exponent field with IEEE 754 format.