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Applied Mathematical Sciences, Vol. 8, 2014, no. 87, 4323 - 4341

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2014.45338

Very Simply Explicitly Invertible Approximations of

Normal Cumulative and Normal Quantile Function

Alessandro Soranzo

Dipartimento di Matematica e Geoscienze

University of Trieste

Trieste – Italy

Emanuela Epure

European Commission

DG Joint Research Center

Ispra - Italy

Copyright c

2014 Alessandro Soranzo and Emanuela Epure. This is an open access

article distributed under the Creative Commons Attribution License, which permits unre-

stricted use, distribution, and reproduction in any medium, provided the original work is

properly cited.

Abstract

For the normal cumulative distribution function: Φ(x) we give the

new approximation 2**(-22**(1-41**(x/10))) for any x>0, which is very

simple (with only integer constants and operations - and / and power

elevation **) and is very simply explicitly invertible having 1 entry of x.

It has 3 decimals of precision having absolute error less than 0.00013. We

compute the inverse which approximates the normal quantile function,

or probit, and it has the relative precision of 1 percent (from 0.5) till

beyond 0.999. We give an open problem and a noticeable bibliography.

We report several other approximations.

Mathematics Subject Classiﬁcation: 33B20 , 33F05 , 65D20 , 97N50

Keywords: normal distribution function, normal cumulative, normal cdf,

Φ, normal quantile, probit, error function, erf, erfc, Q function, cPhi, inverse

erf, erf−1, approximation

4324 Alessandro Soranzo and Emanuela Epure

1 Introduction

This paper deals with the approximation of 2 special functions, Φ(x) and φα.

Let’s remember that Φ(x) and its inverse φα:= Φ−1(α) play a central role in

Statistics, essentially as a consequence of the Central Limit Theorem.

Papers [11] and recent [24] list several approximations of Φ(x), which were

published in literature directly as approximations, or bounds, for that func-

tion, or are immediately derived from approximations or bounds for related

functions (see Remark 8 below), and give new ones.

Remark 1. Though computers now allow to compute them with arbitrary

precision, such approximations are still valuable for several reasons, includ-

ing to catch the soul of the considered functions, allowing to understand at a

glance their behaviour. Furthermore, here we produce only explicitly invertible

(and, in fact, simply) approximations, which allow to keep coherence working

contemporarily with the considered functions and their respective inverses.

Let’s add, ﬁnally, that despite technologic progress, those functions – of wide

practical use – are not always available on pocket calculators.

Remark 2. The research about approximating Φ(x) ﬂoats among:

•exactness, but requiring limits, as series and continued fractions

•width of domains of approximation (usually x≥0 but not always)

•precision of approximations, but aﬀecting their simplicity

•simplicity of approximations, but aﬀecting their precision:

there are few and/or short decimal constants

if possible there are no decimal constants

•explicit invertibility by elementary functions.

Remark 3. The invertibility generates this categories:

(a) not explicitly invertible

(b) explicitly invertible solving a quartic equation

(c) explicitly invertible solving a generic cubic equation

(d) explicitly invertible solving a particular cubic equations x3+ax +b= 0

(e) simply explicitly invertible solving a quadratic (or biquadratic) equation

(f) very simply explicitly invertible, with only 1 entry of x.

Remark 4. Some special functions – among which those we consider in this

paper – are monotonic and then invertible, though not by elementary functions.

Remark 5. Of course the inverse of an approximation of an invertible func-

tion fis an approximation (how good, it has to be seen) of the inverse of f.

Very simply explicitly invertible approximations 4325

Remark 6. Usually the approximations of Φ(x) are not designed to be ex-

plicitly invertible by means of elementary functions, but sometimes they are,

solving cubic or quartic equations (after obvious substitutions) or rarely in

simpler manners.

Remark 7. As well known, it is possible to explicitly solve cubic and even

quartic equations, by complicate formulas, but it is not a standard procedure in

usual mathematical practice. (In literature, such explicit invertibility usually

is not even stated when presenting the approximations of Φ(x)).

2 Preliminary Notes

Remark 8. Similar things as in Remarks 1-7 may be said for the functions

erf(x), erfc(x) and Q(x) we are going to deﬁne.

Deﬁnition 1. (Most standard; unluckily there are ambiguities in literature).

Normal cumulative distribution function:

Φ(x) := Zx

−∞

1

√2πe−t2

2dt (1)

Error function:

erf (x) := Zx

0

2

√πe−t2dt (2)

Q-function:

Q(x) := Z+∞

x

1

√2πe

−t2

2dt. (3)

Complementary error function:

erf c(x) = 1

2+Z+∞

x

2

√πe−t2dt . (4)

Remark 8. Mutual relations, holding for any x∈IR:

Φ(x) = 1

2+1

2erf x

√2(5)

erf (x) = 2Φ(x√2) −1 (6)

Q(x) := 1 −Φ(x) (7)

erf c(x) := 1 −erf(x) (8)

Remark 9. We wrote := both in (3) and (7) because both are used as def-

initions in literature. We wrote := in (8) because that is usually used as

4326 Alessandro Soranzo and Emanuela Epure

deﬁnition, and not (4).

Remark 10. The approximation of Φ(x) for x≥0 and of its inverse for

0≤α≤1

2are suﬃcient because of symmetries:

Φ(−x) = 1 −Φ(x)∀x∈IR (9)

φ1−α=−φα∀α∈]0,1[.(10)

3 Our Results

3.1 New Approximation of Φ(x)

Denoting by |ε(x)|the absolute error and by εr(x) the relative error, we give

the following approximation:

(A) Φ(x)'2−221−41x/10

|ε(x)|<1.28 ·10−4

|εr(x)|<1.66 ·10−4∀x≥0

Let η(x) be the approximation of Φ(x) considered in Formula (A):

η(x) := 2−221−41x/10

.

The function 1.3·10−4−|Φ(x)−η(x)|is positive for 0 ≤x≤5 as may be seen

by plotting it (see Figures 1 and 2). All the graphs may be obtained by profes-

sional software Mathematica(R)or for free at the site www.wolframalpha.com:

for the considered(1 ) case, write Plot[

1.3 10ˆ(-4) - Abs[1/2+(1/2) Erf[x/Sqrt[2]] - 2ˆ(-22ˆ(1 - 41ˆ(x/10)))],{x,0,5}].

1All graphs may be obtained by these instructions, using as options (for example)

WorkingPrecision ->100, PlotStyle ->Black :

phi[x ] = 1/2+ (1/2) Erf[x/Sqrt[2]]

iphi[α] = Sqrt[2] InverseErf[2 α- 1]

PHI41[x ] = 2∧(−22∧(1 −41∧(x/10)))

iPHI41[α] = (10/Log[41]) Log[ 1 - (Log[(-Log[α])/Log[2]])/Log[22]]

Fig. 1 : Plot[{0, 128/10∧6 - Abs[PHI41[x] - phi[x]]},{x, 0, 5}, (options)]

Fig. 2 : Plot[{0, 128/10∧6 - Abs[PHI41[x] - phi[x]]},{x, 2.6, 2.8}...

Fig. 3 : Plot[{0, 166/10∧6 - Abs[(PHI41[x] - phi[x])/phi[x]]},{x, 0, 5}...

Fig. 4 : Plot[{0, 166/10∧6 - Abs[(PHI41[x] - phi[x])/phi[x]]},{x, 0.16, 0.18}...

Fig. 5 : Plot[{0, 5/1000 - Abs[iPHI41[x] - iphi[x]]},{x, 0.5, 0.9926}...

Fig. 6 : Plot[{0, 5/1000 - Abs[iPHI41[x] - iphi[x]]},{x, 0.9924, 0.9926}...

Fig. 7 : Plot[{0, 1/100 - Abs[(iPHI41[x] - iphi[x])/iphi[x]]},{x, 0.5, 0.99909}...

Fig. 8 : Plot[{0, 1/100 - Abs[(iPHI41[x] - iphi[x])/iphi[x]]},{x, 0.99907, 0.99909}...

Very simply explicitly invertible approximations 4327

For x > 5 let’s consider that it is Φ(5) = 0.9999997... and Φ(x)→1 and Φ is

increasing, then

(∀x > 5) |1−Φ(x)|<10−6.(11)

It is, for x > 5,

x > 5>3.6378... =10

log 41 log 1−1

log 22 log log(1 −10−4)

−log 2

10 log41(1 −log22(−log2(1 −10−4))) < x

log41(1 −log22(−log2(1 −10−4))) < x/10

1−log22(−log2(1 −10−4)) <41x/10

log22(−log2(1 −10−4)) >1−41x/10

log2(1 −10−4)<−221−41x/10

1−10−4<2−221−41x/10

0<1−2−221−41x/10

<10−4

that is to say

(∀x > 5) |1−η(x)|<10−4.(12)

By (11) and (12) it is

(∀x > 5) |Φ(x)−η(x)| ≤ |1−Φ(x)|+|1−η(x)|<10−6+ 10−4<1.3·10−4.

Then, for the relative error of Formula (A), for 0 ≤x≤5, see Fig. 3 and Fig.

4, and for x≥5 it is Φ(x)>0.9 (see above) and then

|2−221−41x/10 −Φ(x)|

|Φ(x)|<|2−221−41x/10 −Φ(x)|

0.9=|ε(x)|

0.9<1.3 10−4

0.9<1.7 10−4.

Fig. 1 Absolute error [PHI41] Fig. 2 Its zoom [PHI41Z]

1

2

3

4

5

0.00002

0.00004

0.00006

0.00008

0.00010

0.00012

2.65

2.70

2.75

2.80

1.¥10-6

2.¥10-6

3.¥10-6

4.¥10-6

Fig. 3 Relative error [PHI41R] Fig. 4 Its zoom [PHI41RZ]

1

2

3

4

5

0.00005

0.00010

0.00015

0.165

0.170

0.175

0.180

1.¥10-7

2.¥10-7

3.¥10-7

4.¥10-7

5.¥10-7

6.¥10-7

7.¥10-7

4328 Alessandro Soranzo and Emanuela Epure

3.2 Inversion: Approximation of φα.

Remembering Remark 5, inverting (A), and still denoting by |ε(x)|the absolute

error and by ε(x) the relative error, we give the following approximation of the

normal quantile function φα= Φ−1(α):

(a)φα'10

log41 log1−log((−log α)/log 2)

log 22

|ε(α)|<5·10−3∀α∈[0.5,9925]

|εr(α)|<1% ∀α∈[0.5,0.99908]

For the absolute error of (a) see Figures 5 and 6. For the relative error of (a)

see Figures 7 and 8.

Fig. 5 Absolute error [iPHI41] Fig. 6 Its zoom [iPHI41Z]

0.6

0.7

0.8

0.9

1.0

0.001

0.002

0.003

0.004

0.005

0.99245

0.99250

0.99255

0.99260

-0.00005

0.00005

0.0001

Fig. 7 Absolute error [iPHI41R] Fig. 8 Its zoom [iPHI41RZ]

0.6

0.7

0.8

0.9

1.0

0.002

0.004

0.006

0.008

0.010

0.999075

0.999080

0.999085

0.999090

-0.00004

-0.00002

0.00002

0.00004

4 Conclusions

In this paper for the normal cumulative distribution function Φ(x) and the

normal quantile function φαrespectively we gave these very simply explicitly

invertible (with 1 entry of x)corresponding approximations:

(A) Φ(x)'2−221−41x/10 ∀x≥0

(a)φα'10

log41 log1−log((−log α)/log 2)

log 22 0.5≤α < 1

Very simply explicitly invertible approximations 4329

As quantiﬁed more precisely in Sections 3.1 and 3.2, the approximation (A) of

Φ(x) grants abundantly 3 decimals of precision (having absolute error less than

0.00013), is very simple – with only 1 entry of x– and very simply explicitly

invertible, and the inverse (a) has essentiallly the same characteristics, giving

an approximation of the normal quantile function φαwhich maintains the 1%

precision (from 0.5) till 0.999???

In the end we remember that by the symmetry Formulas (9) and (10) the

approximations of Φ(x) for x≥0 and of φαfor 0.5≤α < 1 are suﬃcient.

Remark 11. Because of the mutual relations (see Remark 8) among the func-

tions Φ(x), erf(x), Q(x) and erfc(x), to approximate one of them is equivalent

to approximate the others.

We searched in a wide literature approximations published not only for Φ(x),

but also the approximations of Φ(x) implicitly contained in the approxima-

tions of the other 3 functions.

Remark 12. We will report other’s Author’s Formulas in a standard format.

This allows easy comparison.

We use xas independent variable. We write Φ(x)', and always consider both

absolute and relative errors, in absolute value, and write respectively |ε(x)|and

|εr(x)|. Authors not always report both. And they write them with diﬀerent

precisions. We found and wrote those errors with 2 digits after decimal point,

in the form a.bc ·10−n.

Of course any function may be written in several ways. We did our best in

reporting other Author’s formulas, sometimes changing the formal appearance.

In particular

1

2+1

2q1−ef(x)= 0.5+0.5(1 −exp f(x))0.5=1 + (1 −exp f(x)) 1

2

2

and we will write in the ﬁrst way whenever possible.

Remark 13. The most recent approximation of Φ(x) we have found in liter-

ature is in paper [24] (2014), which gives this new approximation

Φ(x)'1−1

√2π

e−x2

2

0.226 + 0.64x+ 0.33√x2+ 3 x > 0

for which we found |ε(x)|<1.93 ·10−4and |εr(x)|<3.86 ·10−4, not explicitly

invertible. The same paper lists 16 other approximations of Φ(x); the last is

Φ(x)'1

2+1

2s1−e

−x2

2

4

π+0.147 x2

2

1+0.147 x2

2(13)

4330 Alessandro Soranzo and Emanuela Epure

holding for x≥0, for which we found |ε(x)|<6.21 ·10−5and |εr(x)|<

6.30 ·10−5, originally published in [109] as

erf (x)'r1−e−x2

4

π+0.147x2

1+0.147x2∀x≥0.(14)

Both (13) and (14) are explicitly invertible, essentially by solving a biquadratic

equation, after obvious substitutions, just as the following improvements of

(13) which we already made available on the net in [95]

Φ(x)'1

2+1

2r1−e−x217+x2

26.694+2x2

|ε(x)|<4.00 ·10−5

|εr(x)|<4.53 ·10−5∀x≥0

and in [94]

Φ(x)'1

2+1

2r1−e

−1.2735457x2−0.0743968x4

2+0.1480931x2+0.0002580x4

|ε(x)|<1.14 ·10−5

|εr(x)|<1.78 ·10−5∀x≥0.

Both the above improvements reach 4 decimals of precision.

Remark 14. As far as we know, the most recent new approximations (all of

2013) of Q(x) or erf(x) or erfc(x) (from which one could immediately obtain

approximations of Φ(x)) are this double inequality

1

x+√4 + x2s2

πe−x2

2≤Q(x)≤1

qx2+x+8

πs2

πe−x2

2

in [19] (originally published for qπ

2e−x2

2Q(x), and notice that the lower bound

is of [10]), this bound

Q(x)≤1

√2π

1

√1 + x2e−x2/2

in [39] (year 2013, originally published for √2πQ(x)) and a family

Q(x)≤Σn

k=0

ak

xe−bkx2

of upper bounds in [41] (year 2013 too) and this family

Q(x)≥Σn

k=0 akx e−bkx2

of lower bounds in [42] (year 2013 too), and from those lower and upper bounds

one could obtain approximations of Φ(x) which are not explicitly invertible by

Very simply explicitly invertible approximations 4331

elementary functions. Those approximations are especiallly valuable not only

because bounds, but also for little relative errors for the function Q(x) for

great values of x. (Notice that Q(x)→0).

Remark 15. As far as we know, the most recent new approximation of Φ(x)

or Q(x) or erf(x) or erfc(x), having 1 entry of x, is this

Φ(x)'1−0.24015 e−0.5616x2

originally published as

erf c(√x)'ΣN

k=1ake−kbx N:= 1 a1= 0.4803; b= 1.1232

in [78] and [79] (both year 2012); (then the Authors give other approximations,

with 2 and 3 entries of x). Clearly that approximation is not intended to min-

imize the absolute error, which in 0 is about 0.52 for erfc(x) (and 0.26 for

the derived approximation of Φ(x)); and in fact its quality is the little relative

error for the function erf c(x) for great values of x. (Notice that erf c(x)→0).

Another recent (2009) approximation of Φ(x) (or Q(x) or erf(x) or erfc(x))

having 1 entry of xis this of [11]

Φ(x)'1

1 + e−1.702xx∈IR

for which we found |ε(x)|<9.49 ·10−3and |εr(x)|<1.35 ·10−2: it is simple

and very simply explicitly invertible, but not so precise; the same paper gives

also this approximation

Φ(x)'1

1 + e−0.07056x3−1.5976xx∈IR

for which we found |ε(x)|<1.42 ·10−4and |εr(x)|<2.08 ·10−4, which is

explicitly invertible solving a particular cubic equation.

Both the approximations have the quality of holding on the whole IR.

Remark 16. (Conclusions) As far as we know, before our Formula (A), the

most precise (with respect both to the absolute error and to the relative error)

approximation of Φ(x)

(α) published as approximations or bounds for Φ(x) or Q(x) or erf(x) or erfc(x)

(β) holding at least for x≥0 (and, then, Φ(−x)=1−Φ(x))

(γ) deﬁned by a single expression (or, not piecewise deﬁned)

(δ) very simply explicitly invertible, with 1 entry of x

4332 Alessandro Soranzo and Emanuela Epure

was this of [6]

Φ(x)'1

2+1

2q1−e−√π

8x2x≥0

for which we found |ε(x)|<1.98 ·10−3and |εr(x)|<2.04 ·10−3. The Author

provides also the inverse, approximating the normal quantile function φα.

Our Formula (A) approximating the normal cumulative distribution function

Φ(x), having |ε(x)|<1.28 ·10−4and |εr(x)|<1.66 ·10−4, appears really quite

noticeable for simplicity, precision and explicit invertibility.

That makes also quite valuable our Formula (a) for the approximation of the

normal quantile function φαinverse of Φ(x).

Remark 16. (Open problem). Modify constants to approximate erf(x)

applying erf(x) = 2Φ(x√2) −1 and our Formula (A), possibly avoiding √2.

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Received: May 11, 2014