We sometimes need to compute the most significant digits of the product of small integers with a multiplier requiring much storage: e.g., a large integer (e.g., $5^{100}$) or an irrational number ($\pi$). We only need to access the most significant digits of the multiplier-as long as the integers are sufficiently small. We provide an efficient algorithm to compute the range of integers given a truncated multiplier and a desired number of digits.

翻译：我们有时需要计算小整数与需要大量存储的乘法器的乘积的最高有效位数字：例如，大整数（如$5^{100}$）或无理数（$\pi$）。只需访问乘法器的最高有效位数字——只要整数足够小即可。我们提供了一种高效算法，用于在给定截断乘法器和所需位数时计算整数的取值范围。