Exponentiation

Exponentiation – Exponentiation can be used by using the builtin pow-function or the ** operator. Learn More about it here.

Exponentiation using builtins: ** and pow()

Exponentiation can be used by using the builtin pow-function or the ** operator:

2**3 # 8
pow(2, 3) # 8

For most (all in Python 2.x) arithmetic operations the result’s type will be that of the wider operand. This is not true for **; the following cases are exceptions from this rule:

Base: int, exponent: int < 0:
2**-3

Out: 0.125 (result is a float)

This is also valid for Python 3.x.

Before Python 2.2.0, this raised a ValueError.

Base: int < 0 or float < 0, exponent: float != int

(-2) ** (0.5) # also (-2.) ** (0.5)

Out: (8.659560562354934e-17+1.4142135623730951j) (result is complex)

Before python 3.0.0, this raised a ValueError.

The operator module contains two functions that are equivalent to the **-operator:

import operator
operator.pow(4, 2) # 16
operator.pow(4, 3) # 64

or one could directly call the pow method:

val1, val2 = 4, 2
val1.pow(val2) # 16
val2.rpow(val1) # 16

in-place power operation isn’t supported by immutable classes like int, float, complex:

val1.ipow(val2)

Square root: math.sqrt() and cmath.sqrt

The math module contains the math.sqrt()-function that can compute the square root of any number (that can be converted to a float) and the result will always be a float:


import math
math.sqrt(9) # 3.0
math.sqrt(11.11) # 3.3331666624997918
math.sqrt(Decimal('6.25')) # 2.5

The math.sqrt() function raises a ValueError if the result would be complex:

math.sqrt(-10)
ValueError: math domain error

math.sqrt(x) is faster than math.pow(x, 0.5) or x ** 0.5 but the precision of the results is the same. The cmath module is extremely similar to the math module, except for the fact it can compute complex numbers and all of its results are in the form of a + bi. It can also use .sqrt():

import cmath
cmath.sqrt(4) # 2+0j
cmath.sqrt(-4) # 2j

What’s with the j? j is the equivalent to the square root of -1. All numbers can be put into the form a + bi, or in this case, a + bj. a is the real part of the number like the 2 in 2+0j. Since it has no imaginary part, b is 0. b represents part of the imaginary part of the number like the 2 in 2j. Since there is no real part in this, 2j can also be written as 0 + 2j.

Modular exponentiation: pow() with 3 arguments

Supplying pow() with 3 arguments pow(a, b, c) evaluates the modular exponentiation ab mod c:

pow(3, 4, 17) # 13

equivalent unoptimized expression:

3 ** 4 % 17 # 13

steps:

3 ** 4 # 81
81 % 17 # 13
For built-in types using modular exponentiation is only possible if:

First argument is an int

Second argument is an int >= 0

Third argument is an int != 0

These restrictions are also present in python 3.x

For example one can use the 3-argument form of pow to define a modular inverse function:

def modular_inverse(x, p):
"""Find a such as a·x ≡ 1 (mod p), assuming p is prime."""
return pow(x, p-2, p)

[modular_inverse(x, 13) for x in range(1,13)]

Out: [1, 7, 9, 10, 8, 11, 2, 5, 3, 4, 6, 12]

Exponentiation: Computing large integer roots

Even though Python natively supports big integers, taking the nth root of very large numbers can fail in Python.

x = 2 ** 100
cube = x ** 3
root = cube ** (1.0 / 3)

OverflowError: long int too large to convert to float

When dealing with such large integers, you will need to use a custom function to compute the nth root of a number.

def nth_root(x, n):
Start with some reasonable bounds around the nth root. upper_bound = 1
while upper_bound ** n <= x: upper_bound *= 2 lower_bound = upper_bound // 2 Keep searching for a better result as long as the bounds make sense. while lower_bound < upper_bound: mid = (lower_bound + upper_bound) // 2 mid_nth = mid ** n if lower_bound < mid and mid_nth < x: lower_bound = mid elif upper_bound > mid and mid_nth > x: upper_bound = mid
else:
Found perfect nth root. return mid
return mid + 1
x = 2 ** 100
cube = x ** 3
root = nth_root(cube, 3)
x == root

True

Exponentiation using the math module:
math.pow()

The math-module contains another math.pow() function. The difference to the builtin pow()-function or ** operator is that the result is always a float:

import math
math.pow(2, 2) # 4.0
math.pow(-2., 2) # 4.0

Which excludes computations with complex inputs:

math.pow(2, 2+0j)

TypeError: can’t convert complex to float

and computations that would lead to complex results:

math.pow(-2, 0.5)

ValueError: math domain error

Exponential function: math.exp() and cmath.exp()

Both the math and cmath-module contain the Euler number: e and using it with the builtin pow()-function or **-operator works mostly like math.exp():

import math
math.e ** 2 # 7.3890560989306495
math.exp(2) # 7.38905609893065
import cmath
cmath.e ** 2 # 7.3890560989306495
cmath.exp(2) # (7.38905609893065+0j)

However the result is different and using the exponential function directly is more reliable than builtin exponentiation with base math.e:

print(math.e ** 10) # 22026.465794806703
print(math.exp(10)) # 22026.465794806718
print(cmath.exp(10).real) # 22026.465794806718
difference starts here ---------------^

Exponential function minus 1: math.expm1()

The math module contains the expm1()-function that can compute the expression math.e ** x – 1 for very small x with higher precision than math.exp(x) or cmath.exp(x) would allow:

import math
print(math.e ** 1e-3 - 1) # 0.0010005001667083846
print(math.exp(1e-3) - 1) # 0.0010005001667083846
print(math.expm1(1e-3)) # 0.0010005001667083417

——————^

For very small x the difference gets bigger:

print(math.e ** 1e-15 - 1) # 1.1102230246251565e-15
print(math.exp(1e-15) - 1) # 1.1102230246251565e-15
print(math.expm1(1e-15)) # 1.0000000000000007e-15

^——————-

The improvement is significant in scientific computing. For example the Planck’s law contains an exponential function minus 1:

def planks_law(lambda_, T):
from scipy.constants import h, k, c # If no scipy installed hardcode these!
return 2 * h * c ** 2 / (lambda_ ** 5 * math.expm1(h * c / (lambda_ * k * T)))
def planks_law_naive(lambda_, T):
from scipy.constants import h, k, c # If no scipy installed hardcode these!
return 2 * h * c ** 2 / (lambda_ ** 5 * (math.e ** (h * c / (lambda_ * k * T)) - 1))
planks_law(100, 5000) # 4.139080074896474e-19
planks_law_naive(100, 5000) # 4.139080073488451e-19

^———-

planks_law(1000, 5000) # 4.139080128493406e-23
planks_law_naive(1000, 5000) # 4.139080233183142e-23

^————

Magic methods and exponentiation: builtin, math and cmath

Supposing you have a class that stores purely integer values:

class Integer(object):
def init(self, value):
self.value = int(value) # Cast to an integer
def repr(self):
return '{cls}({val})'.format(cls=self.class.name, val=self.value)
def pow(self, other, modulo=None):
if modulo is None:
print('Using pow')
return self.class(self.value ** other)
else:
print('Using pow with modulo')
return self.class(pow(self.value, other, modulo))
def float(self):
print('Using float')
return float(self.value)
def complex(self):
print('Using complex')
return complex(self.value, 0)

Using the builtin pow function or ** operator always calls pow:

Integer(2) ** 2 # Integer(4)

Prints: Using pow

Integer(2) ** 2.5 # Integer(5)

Prints: Using pow

pow(Integer(2), 0.5) # Integer(1)

Prints: Using pow

operator.pow(Integer(2), 3) # Integer(8)
Prints: Using pow
operator.pow(Integer(3), 3) # Integer(27)

Prints: Using pow

The second argument of the pow() method can only be supplied by using the builtin-pow() or by directly calling the method:

pow(Integer(2), 3, 4) # Integer(0)
Prints: Using pow with modulo
Integer(2).pow(3, 4) # Integer(0)
Prints: Using pow with modulo

While the math-functions always convert it to a float and use the float-computation:

import math
math.pow(Integer(2), 0.5) # 1.4142135623730951
Prints: Using float

cmath-functions try to convert it to complex but can also fallback to float if there is no explicit conversion to complex:

import cmath
cath.exp(Integer(2)) # (7.38905609893065+0j)
Prints: Using complex
del Integer.complex # Deleting complex method - instances cannot be cast to complex
cmath.exp(Integer(2)) # (7.38905609893065+0j)
Prints: Using float

Neither math nor cmath will work if also the float()-method is missing:

del Integer.float # Deleting complex method
math.sqrt(Integer(2)) # also cmath.exp(Integer(2))

TypeError: a float is required

Exponentiation: Roots: nth-root with fractional exponents

While the math.sqrt function is provided for the specific case of square roots, it’s often convenient to use the exponentiation operator (**) with fractional exponents to perform nth-root operations, like cube roots.

The inverse of an exponentiation is exponentiation by the exponent’s reciprocal. So, if you can cube a number by putting it to the exponent of 3, you can find the cube root of a number by putting it to the exponent of 1/3.

x = 3
y = x ** 3
y
27
z = y ** (1.0 / 3)
z
3.0
z == x
True

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