Water Vapor  Other Formulas
My programs implement a number of formulas which claim to compute saturated water pressure
at a given temperature.
However, from time to time I run across new formulas  and occasionally, flat out wrong formulas.
Since I don't want to be continuously updating my program,
I will simply add these "new" formulas here.
The Engineering ToolBox
The Engineering ToolBox
is a source of fairly reliable data.
As a result, I was shocked (and disappointed) to find that their
Saturation Pressure of Water Vapor
formula is totally different than all the other formulas I've collected.
p_{ws} = e^{(77.3450 + 0.0057 T  7235 / T)} / T^{8.2}
T in kelvins

In general, the results agree fairly well with GolfGratch up to about 0°C,
then it begins to strongly diverge. For instance,
(based on this table)
T (C) GolfGratch (mb) EngToolBox (mb)
15 1.91 1.9
 9 3.09 3.0
2 7.05 6.9
25 31.652 31.30
46 100.84 101.4

Only GISSAR4 (NASA) is worse.
Unfortunately, they do not reference their source for this equation 
it is merely presented as "a statement of fact".
A Google search for that equation
returned numerous hits.
Unfortunately, all the examples I checked are obviously just quoting The Engineering ToolBox
 either an explicit reference or the identical text and formatting.
In addition, on their
Dry Bulb, Wet Bulb and Dew Point Temperatures
page they say
The SI unit is Kelvin (K). Zero Kelvin equals to 273°C.

which is, of course, wrong!
It should be 273.15°C.
(Yes  it does make a difference! Particularly in a reference book.)
NASA Tech Note D8401
Equations for the Determination of Humidity from Dewpoint and Psychrometric Data

NASA Tech Note D8401,
Parish and Putnam, January 1977
 provides yet another set of equations with derivations.
They claim that the following is the Magnus formula.
log (e_{SW}) = 4.9283 Log T  2937.4/T + 22.5518

which has the general form
e_{SW} = 10^{(c + b/T)} T^{a} where e_{SW} is the saturation water vapor pressure

None of the other formulas I have seen to date have that form.
The MagnusTetens formula is
To be clear, several formulas contain T^{a},
but a is usually positive and there are always a lot more terms.
Their derivations of the constants  a, b, c  are provided in Appendix A.
They always use 273 to convert between K and C (should be 273.15)
and they use 2 vapor pressure callibration points  both are at 0°C (32°F).
Over water  Over Ice


6.11 mb  6.107 mb

Appendix A, Example 2 computes relative humidity from a dew point below freezing.
Their value is 17.3%, my program, using MurphyKoop, computes 19.2% over water and 17.1% over ice.
Example 3 uses 2 temperatures above freezing  they compute 84.6%, my program finds the same
for most of the provided algorithms.
In addition, their definition of relative humidity
(based on that giving in the Smithsonian Meteorological Tables, 1971)
is a bit strange.
It is the ratio of the actual partial pressure of water vapor
over ice or water (depending on whether the air temperture is above or below freezing)
divided by the saturation pressure over water.

I have not seen any other papers specify that the denominator always
be with respect to the saturation perssure over liquid water.
New Magnus Coefficients
Improved Magnus Form Approximation of Saturation Vapor Pressure,
Alduchov (Ru) and Eskridge (US), April 1996,
Journal of Applied Meteorology
This paper argues that the standard Magnus coefficients should be replaced
so that the Magnus equation can be used instead of the much more complex equations of
Wexler (1976),
Sonntag (1990), and
Goff and Gratch (1946).
As an aside, these dates do not match those provided by
Vömel

Wexler (1983), and
Sonntag (1994)

and the
MurphyKoop (2005)
formulation was published after this paper.
Part of the justification of this approach is that atmospheric temperature is
reported to only 0.1°C accuracy world wide and only 0.5°C in the US.
That, plus the general inaccuracies of the reported humidity proxies (such as dew point)
means that the true accuracy of any computation is low enough that
several formulations produce calculated saturation pressures
that are "the same" within some margin of error.
This is the general form of the Magnus formula.
P = C e^{AT/(B+T)} where T is in °C

Alduchov provides 6 different sets of Magnus coefficients by others, and one
"better formulation" suggested by himself.
The following table only shows 3  MagnusTetens from Vömel, one of the six "other" formulations,
and the new "suggested" Alduchov formulation.
Constant  MagnusTetens 1967  Sonntag 1990  Alduchov 1996


Over  Water  Ice  Water  Ice  Water  Ice


A  17.269388  21.8745584  17.62  22.46  17.625  22.587


B  237.3  265.5  243.12  272.62  243.04  273.86


C  6.1078  6.1078  6.112  6.112  6.1094  6.1121


MagnusTetens (1967) via
Vömel
 e_{w} =
6.1078 e^{17.269388 * }^{(T273.16)
/ (T  35.86)}
  Over water
  e_{i} =
6.1078 e^{21.8745584 * (T273.16)
/ (T  7.66)}
  Over ice
 
with T in [K] and e_{w} & e_{i} in [hPa] 

Full equations
The Alduchov paper discusses 4 "complex" equations by 3 authors 
its purpose was to replace these "difficult to compute" formulations
with a simpler Magnus form.
(With modern computers, this goal is questionable.)

The full Sonntag equation is identical in both sources
(and different from Sonntag's Magnus approximation used in the table above).

The Goff Gratch equation differs between Vömel and this paper in the fouth term.
Vömel  Goff Gratch 1946, 1984
Log10 ew = 7.90298 (373.16/T1)
+ 5.02808 Log10(373.16/T)
 1.3816 10^{7} (10^{11.344 (1T/373.16)} 1)
+ 8.1328 10^{3} (10^{3.49149 (373.16/T1)} 1)
+ Log10(1013.246)
Alduchov  Goff Gratch 1945
Log10 ew = 7.90298 (1  Ts/T)
+ 5.02808 Log10(Ts/T)
+ 1.3816 10^{7} (1  10^{11.344 (1T/Ts)})
 8.1328 10^{3} (1  10^{3.49149 (1Ts/T)})
+ Log10(1013.246)
Where Ts = 373.16 K the steam point at 1 atm

This is the problem in the 4th term.
10^{3.49149 (Ts/T  1)} vs 10^{3.49149 (1  Ts/T)}

There is no way to know if this is just a typo or if it invalidates the paper's conclusions.

The 1946 Goff Gratch equation is identical to the equation identified by Vömel
as WMO (Goff, 1957; WMO, 2012).
Note that the 1946 equation is based on the steam point (373.16 K), and that the 1957 equation
is based on the tripple point (273.16 K).

The Wexler (1976) equation is identical to the equation provided in
New Equations for Computing Vapor Pressure and Enhancement Factor,
Arden L.Buck, 1981
(see the next section) 
except that most for the coefficients were normalized with one digit left
of the decimal point.
Wexler
New Equations for Computing Vapor Pressure and Enhancement Factor,
Arden L.Buck, 1981.
This paper contains a good overview of the progress toward developing equations
to compute saturation vapor pressure.
It includes a modification to the Magnus equation by Bögel
and 2 equations by Wexler
that are not included in
Vömel.
Wexler
 e_{w} =
0.01 exp[2991.2729 T^{2}
 6017.0128 T^{1}
+ 18.87643854
 0.028354721 T
+ 0.17838301 10^{4} T^{2}
 0.84150417 10^{9} T^{3}
+ 0.44412543 10^{12} T^{4}
+ 2.858487 ln(T)]
  1976  Over water
  e_{i} =
0.01 exp[ 5865.3696 T^{1}
+ 22.241033
+ 0.013749042 T
 0.34031775 10^{4} T^{2}
+ 0.26967687 10^{7} T^{3}
+ 0.6918651 ln(T)]
  1977  Over ice
 
with T in [K] and e_{w} & e_{i} in [hPa] 

washington.edu skewt.js
While developing a skewT charting application, I found
washington.edu skewt.js.
This is their saturation vapor pressure subroutine (no source given)
function ESAT(t) {
// Computes the saturation vapour pressure over water at temperature t.
// ESAT in mb, t in K. Log to base 10 is needed for this function.
// Dim a0 As Double, a1 As Double, a2 As Double
a0 = 23.832241  5.02808 * Log10(t);
a1 = 0.00000013816 * 10 ^ (11.344  0.0303998 * t);
a2 = 0.0081328 * 10 ^ (3.49149  1302.8844 / t);
ESAT = 10 ^ (a0  a1 + a2  2949.076 / t);
return ESAT;
}

This is almost identical to the Goff Gratch equation (only one constant is different),
but with the terms multiplied out
(which presumably makes the computation faster)
which, unfortunately, obscures the use of the triple point temperature.
Goff Gratch equation
(Smithsonian Tables, 1984, after Goff and Gratch, 1946):
Log10 ew = 7.90298 (373.16/T1)
+ 5.02808 Log10(373.16/T)
 1.3816 10^7 (10^[11.344 (1T/373.16)] 1)
+ 8.1328 10^3 (10^[3.49149 (373.16/T1)] 1)
+ Log10(1013.246)
with T in [K] and ew in [hPa]
voemel

The following computations were performed using a spreadsheet.
7.90298 * 373.16 = 2949.0760168
5.02808 * Log10(373.16) = 12.931694236
11.344 / 373.16 = 0.0303998285
3.49149 * 373.16 = 1302.8844084
Log10(1013.246) = 3.0057148979
7.90298 + 12.931694236 + 1.3816*10^7  8.1328*10^3 + 3.0057148979 = 23.8322564721
This is the only significant difference  23.832241 vs 23.832256

A few test computations show the same results as Goff Gratch to 2 decimal places 
4 significant figures.
Formula
 20 C  0 C  15 C  30 C


Goff Gratch
 1.25292  6.10336  17.03281  42.40599


washington.edu
 1.252880448  6.1031443925  17.0322055233  42.4044797454


MurphyKoop
 1.25504  6.11213  17.0588  42.46814


Author:
Robert Clemenzi