def factorial(argument: int) -> int:
"""Returns the product of all non-negative integers less than or equal to some non-negative integer.
Args:
argument (int): Non-negative integer used as input to factorial function
Returns:
int: Product of all non-negative integers less than or equal to 'argument'
Raises:
InputError: If 'argument' is not a non-negative integer
"""
# Ensure that input is an integer.
if int(argument) != argument:
raise exceptions.InputError(
argument, "Input to factorial must be a positive integer.")
argument = int(argument)
# Ensure that input is non-negative.
if argument < 0:
raise exceptions.InputError(
argument, "Input to factorial must be a positive integer.")
# Calculate factorial using repeated multiplication.
result = 1
for i in range(1, argument + 1):
result *= i
return result
def decimal_to_binary_fraction(value: float) -> str:
"""Returns a string that contains the binary equivalent of a decimal fraction.
Args:
value (float): Value in decimal
Returns:
str: Value in binary
Raises:
InputError: If input is not a valid positive number.
"""
if common.is_negative(value):
raise exceptions.InputError(value, "decimal_to_binary_fraction does not accept negative inputs.")
bits = list()
while value != 0 and (len(bits) <= 15):
new_value = float(value * 2)
number_to_store = int(new_value)
value = new_value % 1
bits.append(number_to_store)
binary_string = "0."
for x in bits:
binary_string = binary_string + str(x)
return binary_string
def decimal_to_binary_integer(value: int) -> str:
"""Returns a string that contains the binary equivalent of a decimal integer.
Args:
value (int): Value in decimal
Returns:
str: Value in binary
Raises:
InputError: If input is not a valid positive number.
"""
if common.is_negative(value):
raise exceptions.InputError(value, "decimal_to_binary_integer does not accept negative inputs.")
bits = list()
if value == 0:
return ""
while value != 0:
number_to_store = value % 2
value = value // 2
bits.append(number_to_store)
binary_string = ""
for x in reversed(bits):
binary_string = binary_string + str(x)
return binary_string
def ln_taylor(argument: float, term_count: int = 50) -> float:
"""Returns an approximation of the value of the natural logarithm of some argument using the Maclaurin series expansion of ln(1-x).
Maclaurin series used only converges when 'argument' is a real value between 0 and 2.
Args:
argument (float): Argument to natural logarithm being approximated
term_count (int): Number of terms used in approximation of natural logarithm
Returns:
float: Approximation of natural logarithm of 'argument'
Raises:
InputError: If 'argument' is not within interval of convergence for Taylor series used.
"""
# Ensure that series will converge for argument provided.
if argument <= 0 or argument >= 2:
raise exceptions.InputError(
argument,
"This expansion of natural logarithms only converges for arguments between 0 and 2."
)
# Perform calculation.
x = 1 - argument
result = 0
numerator = 1
for denominator in range(1, term_count):
numerator *= x
result -= numerator / denominator
return result
def pow(base: float, exponent: float, root_used: int = int(1E10)) -> float:
"""Returns an approximation of some power with base 'base' and exponent 'exponent'.
Function implementation does not accept powers with negative bases and real exponents.
Args:
base (float): Base of power being approximated
exponent (float): Exponent of power being approximated
root_used (float): Root used to approximate real part of exponent. Larger roots yield greater accuracy but risk overflow problems for large bases.
Returns:
float: Approximation of power with base 'base' and exponent 'exponent'
Raises:
InputError: If 'base' is negative and 'exponent' is not an integer.
"""
# Get integer and decimal parts of exponent.
integer_part_of_exponent = int(exponent)
fractional_part_of_exponent = exponent - integer_part_of_exponent
# Ensure that base is not negative if exponent is not a integer.
if common.is_negative(base) and fractional_part_of_exponent != 0:
raise exceptions.InputError(None,
"Negative base with fractional exponent.")
# Calculate integer part using exponentiation by squaring
result = pow_int(float(base), integer_part_of_exponent)
# If fractional part remains, approximate it using Newton's method for the denominator and exponentiation by
# squaring for the numerator
if fractional_part_of_exponent != 0:
result *= pow_int(radical(float(base), root_used),
int(root_used * fractional_part_of_exponent))
return result
def binary_to_decimal_fraction(value: str) -> float:
"""Returns a float that contains the decimal equivalent of a binary fraction.
Args:
value (str): Value in binary
Returns:
float: Value in decimal
Raises:
InputError: If input is not a valid positive binary number.
"""
if value[0] == '-':
raise exceptions.InputError(value, "binary_to_decimal_fraction does not accept negative inputs.")
if not is_binary(value):
raise exceptions.InputError(value, "Not in binary.")
the_number = str(value)[2:]
position = -1
value = 0
for char in the_number:
value += int(char) * exponents.pow(2,position)
position -= 1
return value
def binary_to_decimal_integer(value: str) -> int:
"""Returns an integer that contains the decimal equivalent of a binary integer.
Args:
value (str): Value in binary
Returns:
int: Value in decimal
Raises:
InputError: If input is not a valid positive binary number.
"""
if value[0] == '-':
raise exceptions.InputError(value, "binary_to_decimal_integer does not accept negative inputs.")
if not is_binary(value):
raise exceptions.InputError(value, "Not in binary.")
the_number = str(value)
position = 0
value = 0
for char in reversed(the_number):
value += int(char) * exponents.pow(2,position)
position += 1
return value
def log(argument: float, base: float = 10) -> float:
"""Returns an approximation of the value of some logarithm with base 'base' and argument 'argument'.
Args:
argument (float): Argument to logarithm being approximated
base (float): Base of logarithm being approximated
Returns:
float: Approximation of value of logarithm of with base 'base' and argument 'argument'
Raises:
InputError: If 'argument' and/or 'base' are outside of domain of logarithm.
"""
# Ensure that base is not equal to 1.
if base == 1:
raise exceptions.InputError(base,
"Logarithm with base of 1 is undefined.")
# Ensure that base is positive.
if not common.is_positive(base):
raise exceptions.InputError(
base, "Logarithm with non-positive base is undefined.")
# Calculate logarithm with arbitrary base using conversion of base property of logarithms.
return ln(argument) / ln(base)
def inverse(argument: float) -> float:
"""Returns inverse of input.
Args:
argument (float): Non-zero input to inverse function
Returns:
float: Inverse of 'argument'.
Raises:
InputError: If 'argument' is equal to 0
"""
# Ensure that input is not equal to 0.
if argument == 0:
raise exceptions.InputError(argument, "Inverse of 0 is undefined.")
# Calculate inverse of argument.
return 1 / argument
def is_odd(argument: int) -> bool:
"""Returns true is input is odd and false otherwise.
Args:
argument (int): Value to be checked
Returns:
bool: True if 'argument' is odd and false otherwise
Raises:
InputError: If 'argument' is not an integer
"""
# Ensure that argument is an integer.
if not isinstance(argument, int):
raise exceptions.InputError(argument,
"Parity of non-integers is undefined.")
return argument & 1
def ln(argument: float) -> float:
"""Returns an approximation of the value of the natural logarithm of some argument.
Args:
argument (float): Argument to natural logarithm being approximated
Returns:
float: Approximation of natural logarithm of 'argument'
Raises:
InputError: If 'argument' is not within domain of natural logarithm (set of positive real numbers).
"""
# Ensure that argument is within domain of natural logarithm.
if not common.is_positive(argument):
raise exceptions.InputError(argument, "Not in domain of log function.")
# We use frexp to fetch mantissa and exponent of argument in a platform agnostic way.
mantissa, exponent = math.frexp(argument)
# It follows from the properties of logarithms that ln(m*2^p)=ln(m)+p*ln(2).
ln_2 = 0.6931471805599453
ln_mantissa = ln_taylor(mantissa)
return ln_mantissa + exponent * ln_2
def pow_int(base: float, exponent: int) -> float:
"""Returns value of 'base' to the power of 'exponent' using exponentiation by squaring where the latter is an integer.
Args:
base (float): Base of power to be returned
exponent (int): Exponent of power to be returned
Returns:
float: Value of 'base' to the power of 'exponent'
Raises:
InputError: If 'exponent' is not an integer
"""
# Ensure that exponent is an integer.
if not isinstance(exponent, int):
raise exceptions.InputError(
exponent,
"Exponentiation by squaring does not work with non-integer exponents."
)
# If exponent is negative, use the property x^y = (1/x)^-y.
if common.is_negative(exponent):
base = common.inverse(base)
exponent *= -1
# Iterate over bits in exponent
result = 1
while common.is_positive(exponent):
# If exponent is odd, we can use the fact that x^y = x(x^2)^((n-1)/2)
if common.is_odd(exponent):
result *= base
# Normally, we'd subtract 1 from y here but we don't actually need to since it will be lost in the coming bit-shift.
# Now that y is even, we can use the fact that x^y = (x^2)^(n/2) when y is even
base *= base
exponent >>= 1
return result
def radical(radicand: float, index: int, delta: float = 1E-10) -> float:
"""Returns an approximation of the real value of some radical using Newton's method for finding roots of functions if such a value exists.
Value returned is positive if a positive radical exists and negative otherwise. If no real root exists, an exception is raised.
Args:
radicand (float): Radicand of radical to be determined
index (int): Index of radical to be determined
delta (float): Error tolerance
Returns:
float: Approximation of radical defined by arguments
Raises:
InputError: If no real root exists for provided arguments
"""
# Ensure that real value of root exists.
if common.is_negative(radicand) and common.is_even(index):
raise exceptions.InputError(None, "Not in domain of radical function.")
# Approximate radicand using Newton's method.
approx = 1.0
old_mantissa, old_exp = math.frexp(approx)
while True:
# Incrementally improve approximation.
power = pow_int(approx, index)
approx = (index - 1 + radicand / power) * (approx / index)
# Return when approximation stops improving
crrt_mantissa, crrt_exp = math.frexp(approx)
diff = abs(old_mantissa - crrt_mantissa)
# We use 1E-15 here because a double precision floating point mantissa can hold no more than 52 bits of precision.
if crrt_exp == old_exp and diff < 1E-15:
return approx
old_mantissa = crrt_mantissa
old_exp = crrt_exp
