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Mathematical Operations

The notation “_x_ modulo _y_” (_y_ must be finite and non-zero) computes a value _k_ of the same sign as _y_ (or zero) such that abs(_k_) < abs(_y_) and _x_ - _k_ = _q_ × _y_ for some integer _q_.

The phrase "the result of clamping _x_ between _lower_ and _upper_" (where _x_ is an extended mathematical value and _lower_ and _upper_ are mathematical values such that _lower_ ≤ _upper_) produces _lower_ if _x_ < _lower_, produces _upper_ if _x_ > _upper_, and otherwise produces _x_.

The mathematical function floor(_x_) produces the largest integer (closest to +∞) that is not larger than _x_.

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The mathematical function truncate(_x_) removes the fractional part of _x_ by rounding towards zero, producing -floor(-_x_) if _x_ < 0 and otherwise producing floor(_x_).

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Mathematical functions min, max, abs, floor, and truncate are not defined for Numbers and BigInts, and any usage of those methods that have non-mathematical value arguments would be an editorial error in this specification.

floor(_x_) = _x_ - (_x_ modulo 1).

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The mathematical function truncate(_x_) removes the fractional part of _x_ by rounding towards zero, producing -floor(-_x_) if _x_ < 0 and otherwise producing floor(_x_).

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Mathematical functions min, max, abs, floor, and truncate are not defined for Numbers and BigInts, and any usage of those methods that have non-mathematical value arguments would be an editorial error in this specification.

An interval from lower bound _a_ to upper bound _b_ is a possibly-infinite, possibly-empty set of numeric values of the same numeric type. Each bound will be described as either inclusive or exclusive, but not both. There are four kinds of intervals, as follows: