"""
Classes and functions to support unit conversion.
The module provides a basic set of predefined physical quantities
in its built-in library; however, it also supports generation of
personal libararies which can be saved and reused.
This module is based on the PhysicalQuantities module
in Scientific Python, by Konrad Hinsen. Modifications by
Justin Gray.
"""
import sys
import re
import os.path
from collections import OrderedDict
from configparser import RawConfigParser as ConfigParser
# pylint: disable=E0611, F0401
from math import floor, pi
import numpy as np
####################################
# Class Definitions
####################################
[docs]
class NumberDict(OrderedDict):
"""
Dictionary storing numerical values.
An instance of this class acts like an array of numbers with
generalized (non-integer) indices. A value of zero is assumed
for undefined entries. NumberDict instances support addition
and subtraction with other NumberDict instances, and multiplication
and division by scalars.
"""
[docs]
def __getitem__(self, item):
"""
Get the item, or 0.
Parameters
----------
item : key
key to get the item
Returns
-------
int
value of the given key
"""
try:
return dict.__getitem__(self, item)
except KeyError:
return 0
def __coerce__(self, other):
"""
Change other dict to NumberDicts.
Parameters
----------
other : dict
the dict instance to be coerced
Returns
-------
NumberDict
new NumberDict with keys/values from original
"""
if isinstance(other, dict):
other = NumberDict(other)
return self, other
def __add__(self, other):
"""
Add another NumberDict to myself.
Parameters
----------
other : NumberDict
the other NumberDict Instance
Returns
-------
NumberDict
new NumberDict with self+other values
"""
sum_dict = NumberDict()
for k, v in self.items():
sum_dict[k] = v
for k, v in other.items():
sum_dict[k] = sum_dict[k] + v
return sum_dict
def __sub__(self, other):
"""
Add another NumberDict from myself.
Parameters
----------
other : NumberDict
the other NumberDict Instance
Returns
-------
NumberDict
new NumberDict instance, with self-other values
"""
sum_dict = NumberDict()
for k, v in self.items():
sum_dict[k] = v
for k, v in other.items():
sum_dict[k] = sum_dict[k] - v
return sum_dict
def __rsub__(self, other):
"""
Add subtract myself from another NumberDict.
Parameters
----------
other : NumberDict
the other NumberDict Instance
Returns
-------
NumberDict
new NumberDict instance, with other-self values
"""
sum_dict = NumberDict()
for k, v in other.items():
sum_dict[k] = v
for k, v in self.items():
sum_dict[k] = sum_dict[k] - v
return sum_dict
def __mul__(self, other):
"""
Multiply myself by another NumberDict.
Parameters
----------
other : NumberDict
the other NumberDict Instance
Returns
-------
NumberDict
new NumberDict instance, with other*self values
"""
new = NumberDict()
for key, value in self.items():
new[key] = other * value
return new
__rmul__ = __mul__
def __div__(self, other):
"""
Divide myself by another NumberDict.
Parameters
----------
other : int
value to divide by
Returns
-------
NumberDict
new NumberDict instance, with self/other values
"""
new = NumberDict()
for key, value in self.items():
new[key] = value / other
return new
__truediv__ = __div__ # for python 3
def __repr__(self):
"""
Return a string deceleration of myself.
Parameters
----------
other : NumberDict
the other NumberDict Instance
Returns
-------
str
str representation for the creation of this NumberDict
"""
return repr(dict(self))
[docs]
class PhysicalUnit(object):
"""
Physical unit.
A physical unit is defined by a name (possibly composite), a scaling
factor, and the exponentials of each of the SI base units that enter into
it. Units can be multiplied, divided, and raised to integer powers.
Parameters
----------
names : dict or str
A dictionary mapping each name component to its
associated integer power (e.g., C{{'m': 1, 's': -1}})
for M{m/s}). As a shorthand, a string may be passed
which is assigned an implicit power 1.
factor : float
A scaling factor.
powers : list of int
The integer powers for each of the nine base units.
offset : float
An additive offset to the base unit (used only for temperatures).
Attributes
----------
_names : dict or str
A dictionary mapping each name component to its
associated integer power (e.g., C{{'m': 1, 's': -1}})
for M{m/s}). As a shorthand, a string may be passed
which is assigned an implicit power 1.
_factor : float
A scaling factor.
_powers : list of int
The integer powers for each of the nine base units.
_offset : float
An additive offset to the base unit (used only for temperatures)
"""
[docs]
def __init__(self, names, factor, powers, offset=0):
"""
Initialize all attributes.
"""
if isinstance(names, str):
self._names = NumberDict(((names, 1),))
else:
self._names = names
self._factor = float(factor)
self._offset = float(offset)
self._powers = powers
def __repr__(self):
"""
Get the string representation of this unit.
Returns
-------
str
str representation of how to instantiate this PhysicalUnit
"""
return 'PhysicalUnit(%s,%s,%s,%s)' % (self._names, self._factor,
self._powers, self._offset)
def __str__(self):
"""
Convert myself to string.
Returns
-------
str
str representation of a PhysicalUnit
"""
return '<PhysicalUnit ' + self.name() + '>'
def __lt__(self, other):
"""
Compare myself to other.
Parameters
----------
other : PhysicalUnit
The other physical unit to be compared to
Returns
-------
bool
self._factor < other._factor
"""
if self._powers != other._powers or self._offset != other._offset:
raise TypeError(f"Units '{self.name()}' and '{other.name()}' are incompatible.")
return self._factor < other._factor
def __gt__(self, other):
"""
Compare myself to other.
Parameters
----------
other : PhysicalUnit
The other physical unit to be compared to
Returns
-------
bool
self._factor > other._factor
"""
if self._powers != other._powers:
raise TypeError(f"Units '{self.name()}' and '{other.name()}' are incompatible.")
return self._factor > other._factor
def __eq__(self, other):
"""
Test for equality.
Parameters
----------
other : PhysicalUnit
The other physical unit to be compared to
Returns
-------
bool
true if _factor, _offset, and _powers all match
"""
return (self._factor == other._factor and
self._offset == other._offset and
self._powers == other._powers)
def __mul__(self, other):
"""
Multiply myself by other.
Parameters
----------
other : PhysicalUnit
The other physical unit to be compared to
Returns
-------
PhysicalUnit
new PhysicalUnit instance representing the product of two units
"""
if self._offset != 0 or (isinstance(other, PhysicalUnit) and
other._offset != 0):
raise TypeError(f"Can't multiply units: either '{self.name()}' or '{other.name()}' "
"has a non-zero offset.")
if isinstance(other, PhysicalUnit):
return PhysicalUnit(self._names + other._names,
self._factor * other._factor,
[a + b for a, b in zip(self._powers, other._powers)])
else:
return PhysicalUnit(self._names + {str(other): 1},
self._factor * other,
self._powers,
self._offset * other)
__rmul__ = __mul__
def __div__(self, other):
"""
Divide myself by other.
Parameters
----------
other : PhysicalUnit
The other physical unit to be operated on
Returns
-------
PhysicalUnit
new PhysicalUnit instance representing the self/other
"""
if self._offset != 0 or (isinstance(other, PhysicalUnit) and
other._offset != 0):
raise TypeError(f"Can't divide units: either '{self.name()}' or '{other.name()}' "
"has a non-zero offset.")
if isinstance(other, PhysicalUnit):
return PhysicalUnit(self._names - other._names,
self._factor / other._factor,
[a - b for (a, b) in zip(self._powers,
other._powers)])
else:
return PhysicalUnit(self._names + {str(other): -1},
self._factor / float(other), self._powers)
__truediv__ = __div__ # for python 3
def __rdiv__(self, other):
"""
Divide other by myself.
Parameters
----------
other : PhysicalUnit
The other physical unit to be operated on
Returns
-------
PhysicalUnit
new PhysicalUnit instance representing the other/self
"""
return PhysicalUnit({str(other): 1} - self._names,
float(other) / self._factor,
[-x for x in self._powers])
__rtruediv__ = __rdiv__
def __pow__(self, power):
"""
Raise myself to a power.
Parameters
----------
power : float or int
power to raise self by
Returns
-------
PhysicalUnit
new PhysicalUnit of self^power
"""
if self._offset != 0:
raise TypeError(f"Can't exponentiate unit '{self.name()}' because it "
"has a non-zero offset.")
if isinstance(power, int):
return PhysicalUnit(power * self._names, pow(self._factor, power),
[x * power for x in self._powers])
if isinstance(power, float):
inv_exp = 1. / power
rounded = int(floor(inv_exp + 0.5))
if abs(inv_exp - rounded) < 1.e-10:
if all([x % rounded == 0 for x in self._powers]):
f = self._factor**power
p = [x / rounded for x in self._powers]
if all([x % rounded == 0 for x in self._names.values()]):
names = self._names / rounded
else:
names = NumberDict()
if f != 1.:
names[str(f)] = 1
for x, name in zip(p, _UNIT_LIB.base_names):
names[name] = x
return PhysicalUnit(names, f, p)
raise TypeError(f"Can't exponentiate unit '{self.name()}': "
"only integer and inverse integer exponents are allowed.")
[docs]
def in_base_units(self):
"""
Return the base unit equivalent of this unit.
Returns
-------
PhysicalUnit
The equivalent base unit.
"""
num = ''
denom = ''
for unit, power in zip(_UNIT_LIB.base_names, self._powers):
if power < 0:
denom = denom + '/' + unit
if power < -1:
denom = denom + '**' + str(-power)
elif power > 0:
num = num + '*' + unit
if power > 1:
num = num + '**' + str(power)
if len(num) == 0:
num = '1'
else:
num = num[1:]
return _find_unit(num + denom)
[docs]
def conversion_tuple_to(self, other):
"""
Compute the tuple of (factor, offset) for conversion.
Parameters
----------
other : PhysicalUnit
Another unit.
Returns
-------
Tuple with two floats
The conversion factor and offset from this unit to another unit.
"""
if self._powers != other._powers:
raise TypeError(f"Units '{self.name()}' and '{other.name()}' are incompatible.")
# let (s1,d1) be the conversion tuple from 'self' to base units
# (ie. (x+d1)*s1 converts a value x from 'self' to base units,
# and (x/s1)-d1 converts x from base to 'self' units)
# and (s2,d2) be the conversion tuple from 'other' to base units
# then we want to compute the conversion tuple (S,D) from
# 'self' to 'other' such that (x+D)*S converts x from 'self'
# units to 'other' units
# the formula to convert x from 'self' to 'other' units via the
# base units is (by definition of the conversion tuples):
# ( ((x+d1)*s1) / s2 ) - d2
# = ( (x+d1) * s1/s2) - d2
# = ( (x+d1) * s1/s2 ) - (d2*s2/s1) * s1/s2
# = ( (x+d1) - (d1*s2/s1) ) * s1/s2
# = (x + d1 - d2*s2/s1) * s1/s2
# thus, D = d1 - d2*s2/s1 and S = s1/s2
factor = self._factor / other._factor
offset = self._offset - (other._offset * other._factor / self._factor)
return (factor, offset)
[docs]
def is_compatible(self, other):
"""
Check for compatibility with another unit.
Parameters
----------
other : PhysicalUnit
Another unit.
Returns
-------
bool
Indicates if two units are compatible.
"""
return self._powers == other._powers
[docs]
def is_dimensionless(self):
"""
Dimensionless PQ.
Returns
-------
bool
Indicates if this is dimensionless.
"""
return not any(self._powers)
[docs]
def is_angle(self):
"""
Check if this PQ is an Angle.
Returns
-------
bool
Indicates if this an angle type.
"""
return (self._powers[_UNIT_LIB.base_types['angle']] == 1 and
sum(self._powers) == 1)
[docs]
def set_name(self, name):
"""
Set the name.
Parameters
----------
name : str
The name.
"""
self._names = NumberDict()
self._names[name] = 1
[docs]
def name(self):
"""
Compute the name of this unit.
Returns
-------
str
String representation of the unit.
"""
num = ''
denom = ''
for unit, power in self._names.items():
if power < 0:
denom = denom + '/' + unit
if power < -1:
denom = denom + '**' + str(-power)
elif power > 0:
num = num + '*' + unit
if power > 1:
num = num + '**' + str(power)
if len(num) == 0:
num = '1'
else:
num = num[1:]
return num + denom
####################################
# Module Functions
####################################
def _new_unit(name, factor, powers):
"""
Create new Unit.
Parameters
----------
name : str
The name of the new unit
factor : float
conversion factor to base units
powers : [int, ...]
power of base units
"""
_UNIT_LIB.unit_table[name] = PhysicalUnit(name, factor, powers)
[docs]
def add_offset_unit(name, baseunit, factor, offset, comment=''):
"""
Adding Offset Unit.
Parameters
----------
name : str
The name of the unit.
baseunit : str or instance of PhysicalUnit
The unit upon which this offset unit is based.
factor : str
The scaling factor used to define the new unit w.r.t. base unit.
offset : float
Zero offset for new unit.
comment : str
Optional comment to describe unit.
"""
if isinstance(baseunit, str):
baseunit = _find_unit(baseunit)
# else, baseunit should be a instance of PhysicalUnit
# names, factor, powers, offset=0
unit = PhysicalUnit(baseunit._names, baseunit._factor * factor,
baseunit._powers, offset)
unit.set_name(name)
if name in _UNIT_LIB.unit_table:
if (_UNIT_LIB.unit_table[name]._factor != unit._factor or
_UNIT_LIB.unit_table[name]._powers != unit._powers):
raise KeyError(f"Unit '{name}' already defined with different factor or powers.")
_UNIT_LIB.unit_table[name] = unit
_UNIT_LIB.set('units', name, unit)
if comment:
_UNIT_LIB.help.append((name, comment, unit))
[docs]
def add_unit(name, unit, comment=''):
"""
Adding Unit.
Parameters
----------
name : str
The name of the unit being added. For example: 'Hz'.
unit : str
Definition of the unit w.r.t. some other unit. For example: '1/s'.
comment : str
Optional comment to describe unit.
"""
if comment:
_UNIT_LIB.help.append((name, comment, unit))
if isinstance(unit, str):
unit = eval(unit, {'__builtins__': None, 'pi': pi}, # nosec: scope limited
_UNIT_LIB.unit_table)
unit.set_name(name)
if name in _UNIT_LIB.unit_table:
if (_UNIT_LIB.unit_table[name]._factor != unit._factor or
_UNIT_LIB.unit_table[name]._powers != unit._powers):
raise KeyError(f"Unit '{name}' already defined with different factor or powers.")
_UNIT_LIB.unit_table[name] = unit
_UNIT_LIB.set('units', name, unit)
_UNIT_LIB = ConfigParser()
def _do_nothing(string):
"""
Make the ConfigParser case sensitive.
Defines an optionxform for the units configparser that
does nothing, resulting in a case-sensitive parser.
Parameters
----------
string : str
The string to be transformed for the ConfigParser
Returns
-------
str
The same string that was given as a parameter.
"""
return string
_UNIT_LIB.optionxform = _do_nothing
[docs]
def import_library(libfilepointer):
"""
Import a units library, replacing any existing definitions.
Parameters
----------
libfilepointer : file
New library file to work with.
Returns
-------
ConfigParser
Newly updated units library for the module.
"""
global _UNIT_LIB
global _UNIT_CACHE
_UNIT_CACHE = {}
_UNIT_LIB = ConfigParser()
_UNIT_LIB.optionxform = _do_nothing
# New in Python 3.2: read_file() replaces readfp().
if sys.version_info >= (3, 2):
_UNIT_LIB.read_file(libfilepointer)
else:
_UNIT_LIB.readfp(libfilepointer)
required_base_types = ['length', 'mass', 'time', 'temperature', 'angle']
_UNIT_LIB.base_names = list()
# used to is_angle() and other base type checking
_UNIT_LIB.base_types = dict()
_UNIT_LIB.unit_table = dict()
_UNIT_LIB.prefixes = dict()
_UNIT_LIB.help = list()
for prefix, factor in _UNIT_LIB.items('prefixes'):
factor, comma, comment = factor.partition(',')
_UNIT_LIB.prefixes[prefix] = float(factor)
base_list = [0] * len(_UNIT_LIB.items('base_units'))
for i, (unit_type, name) in enumerate(_UNIT_LIB.items('base_units')):
_UNIT_LIB.base_types[unit_type] = i
powers = list(base_list)
powers[i] = 1
# print '%20s'%unit_type, powers
# cant use add_unit because no base units exist yet
_new_unit(name, 1, powers)
_UNIT_LIB.base_names.append(name)
# test for required base types
missing = [utype for utype in required_base_types
if utype not in _UNIT_LIB.base_types]
if missing:
raise ValueError("Not all required base types were present in the config file. missing: "
f"{missing}, at least {required_base_types} required.")
_update_library(_UNIT_LIB)
return _UNIT_LIB
[docs]
def update_library(filename):
"""
Update units in current library from `filename`.
Parameters
----------
filename : str or file
Source of units configuration data.
"""
if isinstance(filename, str):
inp = open(filename, 'rU')
else:
inp = filename
try:
cfg = ConfigParser()
cfg.optionxform = _do_nothing
# New in Python 3.2: read_file() replaces readfp().
if sys.version_info >= (3, 2):
cfg.read_file(inp)
else:
cfg.readfp(inp)
_update_library(cfg)
finally:
inp.close()
def _update_library(cfg):
"""
Update library from :class:`ConfigParser` `cfg`.
Parameters
----------
cfg : ConfigParser
ConfigParser loaded with unit_lib.ini data
"""
retry1 = set()
for name, unit in cfg.items('units'):
data = [item.strip() for item in unit.split(',')]
if len(data) == 2:
unit, comment = data
try:
add_unit(name, unit, comment)
except NameError:
retry1.add((name, unit, comment))
elif len(data) == 4:
factor, baseunit, offset, comment = data
try:
add_offset_unit(name, baseunit, float(factor), float(offset),
comment)
except NameError:
retry1.add((name, baseunit, float(factor), float(offset),
comment))
else:
raise ValueError(f"Unit '{name}' definition {unit} has invalid format.")
retry_count = 0
last_retry_count = -1
while last_retry_count != retry_count and retry1:
last_retry_count = retry_count
retry_count = 0
retry2 = retry1.copy()
for data in retry2:
if len(data) == 3:
name, unit, comment = data
try:
add_unit(name, unit, comment)
retry1.remove(data)
except NameError:
retry_count += 1
else:
try:
name, factor, baseunit, offset, comment = data
add_offset_unit(name, factor, baseunit, offset, comment)
retry1.remove(data)
except NameError:
retry_count += 1
if retry1:
raise ValueError('The following units were not defined because they'
' could not be resolved as a function of any other'
' defined units: %s.' % [x[0] for x in retry1])
_UNIT_CACHE = {}
def _is_unitless(units):
if units is None:
return True
unit_meta = _find_unit(units)
return unit_meta is not None and unit_meta.is_dimensionless()
def _find_unit(unit, error=False):
"""
Find unit helper function.
Parameters
----------
unit : str
str representing the desired unit
error : bool
If True, raise exception if unit isn't found.
Returns
-------
PhysicalUnit
The actual unit object
"""
if isinstance(unit, str):
# Deal with 'as' for attoseconds
reg1 = re.compile(r'\bas\b')
unit = re.sub(reg1, 'as_', unit)
name = unit.strip()
try:
unit = _UNIT_CACHE[name]
except KeyError:
try:
unit = eval(name, {'__builtins__': None}, # nosec: scope limited
_UNIT_LIB.unit_table)
except Exception:
# This unit might include prefixed units that aren't in the
# unit_table. We must parse them ALL and add them to the
# unit_table.
# First character of a unit is always alphabet or $.
# Remaining characters may include numbers.
regex = re.compile('[A-Z,a-z]{1}[A-Z,a-z,0-9]*')
unit_table = _UNIT_LIB.unit_table
prefixes = _UNIT_LIB.prefixes
for item in regex.findall(name):
item = re.sub(reg1, 'as_', item)
# check if this was a compound unit, so each
# substring might be a unit
try:
eval(item, {'__builtins__': None}, unit_table) # nosec: scope limited
except Exception: # maybe is a prefixed unit then
base_unit = item[1:].rstrip('_')
# check for single letter prefix before unit
if (item[0] in prefixes and base_unit in unit_table):
add_unit(item, prefixes[item[0]] * unit_table[base_unit])
# check for double letter prefix before unit
elif (item[0:2] in prefixes and item[2:] in unit_table):
add_unit(item, prefixes[item[0:2]] * unit_table[item[2:]])
# no prefixes found, unknown unit
else:
if error:
raise ValueError(f"The units '{name}' are invalid.")
return None
unit = eval(name, {'__builtins__': None}, unit_table) # nosec: scope limited
_UNIT_CACHE[name] = unit
else:
name = unit
if not isinstance(unit, PhysicalUnit):
if error:
raise ValueError(f"The units '{name}' are invalid.")
return None
return unit
[docs]
def valid_units(unit):
"""
Return whether the given units are vaild.
Parameters
----------
unit : str
String representation of the units.
Returns
-------
bool
True for valid, False for invalid.
"""
return _find_unit(unit) is not None
[docs]
def conversion_to_base_units(units):
"""
Get the offset and scaler to convert from given units to base units.
Parameters
----------
units : str
String representation of the units.
Returns
-------
float
Offset to get to default unit: m (length), s(time), etc.
float
Mult. factor to get to default unit: m (length), s(time), etc.
"""
if not units: # dimensionless
return 0., 1.
unit = _find_unit(units, error=True)
return unit._offset, unit._factor
[docs]
def is_compatible(old_units, new_units):
"""
Check whether units are compatible in terms of base units.
e.g., m/s is compatible with ft/hr
Parameters
----------
old_units : str
Original units as a string.
new_units : str or None
New units to return the value in; if None, return in standard units.
Returns
-------
bool
Whether the units are compatible.
"""
if not old_units and not new_units: # dimensionless
return True
old_unit = _find_unit(old_units, error=True)
new_unit = _find_unit(new_units, error=True)
return old_unit.is_compatible(new_unit)
[docs]
def unit_conversion(old_units, new_units):
"""
Return conversion factor and offset between old and new units.
Parameters
----------
old_units : str
Original units as a string.
new_units : str
New units to return the value in.
Returns
-------
(float, float)
Conversion factor and offset.
"""
return _find_unit(old_units, error=True).conversion_tuple_to(_find_unit(new_units, error=True))
[docs]
def convert_units(val, old_units, new_units=None):
"""
Take a given quantity and return in different units.
Parameters
----------
val : float
Value in original units.
old_units : str or None
Original units as a string or None.
new_units : str or None
New units to return the value in or None.
Returns
-------
float
Value in new units.
"""
if not old_units or not new_units: # one side has no units
return val
old_unit = _find_unit(old_units, error=True)
if new_units:
new_unit = _find_unit(new_units, error=True)
else:
new_unit = old_unit.in_base_units()
(factor, offset) = old_unit.conversion_tuple_to(new_unit)
return (val + offset) * factor
def _has_val_mismatch(units1, val1, units2, val2):
"""
Return True if values differ after unit conversion or if values differ when units are None.
Parameters
----------
units1 : str or None
Units for first value.
val1 : float or ndarray
First value.
units2 : str or None
Units for second value.
val2 : float or ndarray
Second value.
"""
if units1 != units2:
if units1 is None or units2 is None:
return True
# convert units
try:
val1 = convert_units(val1, units1, new_units=units2)
except TypeError:
return True # units are not compatible
rtol = 1e-10
val1 = np.asarray(val1)
val2 = np.asarray(val2)
norm1 = np.linalg.norm(val1)
if norm1 == 0.:
return np.linalg.norm(val2) > rtol
else:
return np.linalg.norm(val2 - val1) / norm1 > rtol
[docs]
def simplify_unit(old_unit_str, msginfo=''):
"""
Simplify unit string using built-in naming method.
Unit string 'ft*s/s' becomes 'ft'.
Parameters
----------
old_unit_str : str
Unit string to simplify.
msginfo : str
A string prepended to the ValueError which is raised if the units are invalid.
Returns
-------
str
Simplified unit string.
"""
if old_unit_str is None:
return None
found_unit = _find_unit(old_unit_str)
if found_unit is None:
_msginfo = f'{msginfo}: ' if msginfo else ''
raise ValueError(f"{_msginfo}The units '{old_unit_str}' are invalid.")
new_str = found_unit.name()
if new_str == '1':
# Special Case. Unity always becomes None.
new_str = None
# Restore units 'as' (attoseconds).
if new_str:
reg1 = re.compile(r'\bas_\b')
new_str = reg1.sub('as', new_str)
return new_str
# Load in the default unit library
file_path = open(os.path.join(os.path.dirname(__file__), 'unit_library.ini'))
with file_path as default_lib:
import_library(default_lib)
if __name__ == '__main__':
for returned, expected in [
(conversion_to_base_units('cm'), (0., 1.0e-2)),
(conversion_to_base_units('km'), (0., 1.0e3)),
(convert_units(3.0, 'mm'), (3.0e-3)),
(convert_units(3.0, 'mm', 'cm'), (3.0e-1)),
(convert_units(100, 'degC', 'degF'), (212.))
]:
print(returned, 'should be', expected)
