Common Lisp library for unit conversions and unit verified formulas
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This is a library for unit conversions and defining formulas with automated unit consistency checking and conversions. This is similar to [], but written with ease of implementation over optimization. In other words, I understand how mine works better. Most of the actual unit data is taken from that program, which is why this is under GPLv2.

This is not yet well tested. Any comments are welcome.


  • iterate
  • alexandria


Unit definition language

Units are described by s-expression. Units are identified by symbols, but interpreted by symbol name, so the package is irrelevant. There is significant number of units already defined in unit-data.lisp, check there or keys of units-formula::*units* hashtable for a list. Units can be constructed from those by use of * / expt sqrt & formula operators. In a list with a unit name as first element * is implied. Numbers might be included in unit definition, they will be combined into contant factor.


Function reduce-unit unit-spec turns the list in above format to an unit object, which can be used in any place where unit definition can be used, to avoid repeated reduction of unit definition.

Function convert-unit unit-from unit-to takes two unit objects and returns a conversion factor between them, or :incorrect-conversion if the units do not match.


CL-USER> (unit-formulas:convert-unit '(/ parsec fortnight) '(/ km second)) 2.5487764e7

numbers can be included, as mentioned above, to convert between values, rather than to obtain conversion factor:

CL-USER> (unit-formulas:convert-unit '(5 kg) 'pound) 11.023113

Function same-unit-p unit1 unit2 takes two units and returns true if they are compatible. If key argument :factor is true, then equality between constant factors is also checked.

Function dimensionless-p unit returns t if the unit is dimensionless (ie. only the constant factor is relevant). It's value can be retrieved with (convert-unit dimensionless-value nil)

Macro define-operators list-of-operators kind-keyword allows definition of operators allowed in formulas. Right now only :agree and :dimensionless kinds are present, which require all arguments to be the same unit or dimensionless respectively.

Macro defformula name (&rest in-spec) formula-expression defines a formula. This creates a function named name, which takes a &rest argument forming an association list of form (name value unit) or (name unit-with-value). Argument in-spec is a list of form (name unit) or (name unit value). The second form creates a named constant which will be folded into the formula. Note that this has to literal number because this is folded at macroexpansion stage. Units in in-spec would in most cases be base units, which have synonym symbols with the name of what it is an unit of.

Formula-expression consists of operators defined in units-formula::*operators* hash table, which must have directly corresponding functions defined. Other allowed expressions are: symbol, naming first a binding defined in in-spec, which will be replaced either by function argument or constant value, if provided, a literal constant, either a number, an unit name, or (number unit-definition).

Created function returns an unit object, which can be converted to value in desired units with convert-unit, or queried directly with query-unit.


CL-USER> (unit-formulas:defformula K-np 
                                   ((effective-mass mass)
                                    (delta-e energy)
                                    (h-bar (/ (m m kg) s) #.(/ 6.62d-34 (* 2 pi)))
                                   (f electric-field))
               (/ (/ (* 4 (sqrt (* 2 effective-mass (expt delta-e 3))))
                     (abs f))
		      (* 3 elementary-charge h-bar)))
CL-USER> (k-np '(effective-mass 0.2 electron-mass) '(delta-e 0.8 eV) '(f 0.09 (/ V (nano m))))
#<UNIT-FORMULAS::UNIT 24.309902549224955d0 >

Function query-unit unit returns a property list with unit value and exponents of base SI units forming an unit.

Function identify-unit unit tries to find a quantity with the same units, and if found returns a keyword naming it.

Macro defformula* defines a formula using positional arguments, with much less error checking. If the wrong units are passed it will still fail, because symbols will be checked by the formula.

Macro defformulae* operates like defformula*, but formula-expression allows nesting of formulas defined using defformulae*. This is useful if you have formulas based on other formulas.

Macro define-units accepts unit-definitions as a list of ((unit-names) unit-definition) where unit-names is a list of synonyms for the unit and unit-definition defines a relationship to a base unit. The relationship definition can be a base-unit, a multiple of a base unit (ie. * / expt sqrt), or a list in the form of (formula :convert-to [formula-symbol-name] :convert-from [formula-symbol-name]). The latter defines the unit in terms of a formula defined by defformulae*, which allows for more complex unit definitions.


CL-USER> (unit-formulas:defformulae* celsius-to-kelvin ((c unity))
   (* (+ c 273.15) kelvin))
CL-USER> (unit-formulas:defformulae* kelvin-to-celsius ((k kelvin))
   (- k (273.15 kelvin)))

CL-USER> (unit-formulas:define-units ((celsius centigrade)
   (unit-formulas:formula :convert-to celsius-to-kelvin
	                  :convert-from kelvin-to-celsius)))

CL-USER> (unit-formulas:convert-unit '(100 celsius) 'kelvin)
CL-USER> (unit-formulas:convert-unit '(0 kelvin) 'centigrade)

In the above example two formulae were defined. The first allows a dimensionless unit to be converted into kelvin. The second formula converts from kelvin back to celsius. In the second case we are not concerned with units since convert-unit calls this formula and will be returning a unitless float.

The third form defines two synonyms celsius and centigrade that are units that use our formulae to convert-to and from a base unit instead of the defaults, which are * and /.

Function transform-units defines a generic-function that accepts an input-unit, an output-unit, and a unit-bag list of available units to use in performing dimensional analysis, converting the input-unit to a value in the output unit. This is a basic implementation and is far from optimized, but should work for all cases.


UNIT-FORMULAS> 	(defvar *molar-mass-of-water*
		  (reduce-unit '(18.01528 (/ grams mol))))
		(defvar *density-of-water*
  		  (reduce-unit '(1 (/ gram mL))))

		(transform-units '(100 gallons)
	   			 (list *density-of-water*