floor_divide

floor_divide(x1: array, x2: array, /) → array

Rounds the result of dividing each element x1_i of the input array x1 by the respective element x2_i of the input array x2 to the greatest (i.e., closest to +infinity) integer-value number that is not greater than the division result.

Note

For input arrays which promote to an integer data type, the result of division by zero is unspecified and thus implementation-defined.

Parameters:

  • x1 (array) – dividend input array. Should have a real-valued data type.

  • x2 (array) – divisor input array. Must be compatible with x1 (see Broadcasting). Should have a real-valued data type.

Returns:

out (array) – an array containing the element-wise results. The returned array must have a data type determined by Type Promotion Rules.

Notes

Special cases

Note

Floor division was introduced in Python via PEP 238 with the goal to disambiguate “true division” (i.e., computing an approximation to the mathematical operation of division) from “floor division” (i.e., rounding the result of division toward negative infinity). The former was computed when one of the operands was a float, while the latter was computed when both operands were ints. Overloading the / operator to support both behaviors led to subtle numerical bugs when integers are possible, but not expected.

To resolve this ambiguity, / was designated for true division, and // was designated for floor division. Semantically, floor division was defined as equivalent to a // b == floor(a/b); however, special floating-point cases were left ill-defined.

Accordingly, floor division is not implemented consistently across array libraries for some of the special cases documented below. Namely, when one of the operands is infinity, libraries may diverge with some choosing to strictly follow floor(a/b) and others choosing to pair // with % according to the relation b = a % b + b * (a // b). The special cases leading to divergent behavior are documented below.

This specification prefers floor division to match floor(divide(x1, x2)) in order to avoid surprising and unexpected results; however, array libraries may choose to more strictly follow Python behavior.

For floating-point operands,

  • If either x1_i or x2_i is NaN, the result is NaN.

  • If x1_i is either +infinity or -infinity and x2_i is either +infinity or -infinity, the result is NaN.

  • If x1_i is either +0 or -0 and x2_i is either +0 or -0, the result is NaN.

  • If x1_i is +0 and x2_i is greater than 0, the result is +0.

  • If x1_i is -0 and x2_i is greater than 0, the result is -0.

  • If x1_i is +0 and x2_i is less than 0, the result is -0.

  • If x1_i is -0 and x2_i is less than 0, the result is +0.

  • If x1_i is greater than 0 and x2_i is +0, the result is +infinity.

  • If x1_i is greater than 0 and x2_i is -0, the result is -infinity.

  • If x1_i is less than 0 and x2_i is +0, the result is -infinity.

  • If x1_i is less than 0 and x2_i is -0, the result is +infinity.

  • If x1_i is +infinity and x2_i is a positive (i.e., greater than 0) finite number, the result is +infinity. (note: libraries may return NaN to match Python behavior.)

  • If x1_i is +infinity and x2_i is a negative (i.e., less than 0) finite number, the result is -infinity. (note: libraries may return NaN to match Python behavior.)

  • If x1_i is -infinity and x2_i is a positive (i.e., greater than 0) finite number, the result is -infinity. (note: libraries may return NaN to match Python behavior.)

  • If x1_i is -infinity and x2_i is a negative (i.e., less than 0) finite number, the result is +infinity. (note: libraries may return NaN to match Python behavior.)

  • If x1_i is a positive (i.e., greater than 0) finite number and x2_i is +infinity, the result is +0.

  • If x1_i is a positive (i.e., greater than 0) finite number and x2_i is -infinity, the result is -0. (note: libraries may return -1.0 to match Python behavior.)

  • If x1_i is a negative (i.e., less than 0) finite number and x2_i is +infinity, the result is -0. (note: libraries may return -1.0 to match Python behavior.)

  • If x1_i is a negative (i.e., less than 0) finite number and x2_i is -infinity, the result is +0.

  • If x1_i and x2_i have the same mathematical sign and are both nonzero finite numbers, the result has a positive mathematical sign.

  • If x1_i and x2_i have different mathematical signs and are both nonzero finite numbers, the result has a negative mathematical sign.

  • In the remaining cases, where neither -infinity, +0, -0, nor NaN is involved, the quotient must be computed and rounded to the greatest (i.e., closest to +infinity) representable integer-value number that is not greater than the division result. If the magnitude is too large to represent, the operation overflows and the result is an infinity of appropriate mathematical sign. If the magnitude is too small to represent, the operation underflows and the result is a zero of appropriate mathematical sign.