Vocabulary/hat
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^ y Exponential
Rank 0 -- operates on individual atoms of y, producing a result of the same shape -- WHY IS THIS IMPORTANT?
ey, the y-th power of the mathematical constant: e .
The antilogarithm of a natural logarithm.
^1 2.71828 ^2 7.38906 ^ ^. 123.45 123.45
Related Primitives
Logarithm (^. y)
Use These Combinations
Combinations using ^ y that have exceptionally good performance include:
What it does Type;
Precisions;
RanksSyntax Variants;
Restrictions
Benefits;
Bug Warnings
eπy ^@o. y handles large values of y
x ^ y Power
Rank 0 0 -- operates on individual atoms of x and y, producing a result of the same shape -- WHY IS THIS IMPORTANT?
xy, the y-th power of numeric noun: x .
2^8 256
Related Primitives
Logarithm (x ^. y)
Details
1. Some results are defined in J that are often left undefined:
- 0 ^ 0 = 1
- _ ^ 0 = 1
- 1 ^ _ = 1
2. Taking a rational power of an extended integer produces a floating-point result whenever the denominator of the power is not 1.
Use These Combinations
Combinations using x ^ y that have exceptionally good performance include:
What it does Type;
Precisions;
RanksSyntax Variants;
Restrictions
Benefits;
Bug Warnings
Integer powers non-complex x, integer y x ^ y Uses repeated multiplication (avoids log) Powers mod(m) integer, extended integer x m&|@^ y
m&|@(n&^) y
Avoids the large result of exponentiation
x ^!.p y Stope Function
Rank 0 0 -- operates on individual atoms of x and y, producing a result of the same shape -- WHY IS THIS IMPORTANT?
x ^!.p y is x*(x+p)*(x+2*p)*... for y terms.
5 ^!.1 (3) NB. 5 * 6 * 7 210
Common Uses
1. Calculate the number of permutations of x things taken y at a time, xPy
5 ^!._1 (3) NB. 5 * 4 * 3 60
2. Calculate the rising Pochhammer symbol (x)y as used in the hypergeometric function
5.1 ^!.1 (3) NB. 5.1 * 6.1 * 7.1 220.881
More Information
1. y must be nonnegative and tolerantly equal to an integer.
2. x ^!.p y is the same as */x + p*i. y .