Arrow

e4fc31fa-5a73-45ad-a088-59a7489fe0a6 159 9 12 decimal

&[DATE] &[TIME] - &[FILENAME]

x f 1 x 2^- :

y Inverse y x f ≡ x solve maple : 1 y - sqrt 1 y - sqrt - 21 mat

Inversing a function is solving for 'x', given the Cartesian 'y <= f(x)'. A most advanced maths work that Smath/Maple solves. Given 'y' we recover the two "x's"

y > 1 pure imaginary Y 0.625 :

target 0 Y X 1 el Y X 1 el 032 mat :

X 1 Y - sqrt 1 Y - sqrt - 21 mat : 0.61237 0.61237 - 21 mat

X f vectorize 0.625 0.625 21 mat

Arrow

xo 0 :

size 0.125 :

sub 0.0125 :

arrow1 xo - 1.5 size * sub i * 1.5 size * sub i * - 15 mat transpose : arrow2 xo - size 2 sub * i * size 2 sub * i * - 15 mat transpose : 21 sys

α rot α cos α sin α sin - α cos 22 mat :

Numeric A1 arrow1 Re 1.25 size * - arrow1 Im X 1 el + augment : A2 arrow2 Re size - X 1 el + 0 + arrow2 Im Y + augment : 21 sys

Arrow1 A1 90 ° * - rot * : Arrow2 A2 0 ° * - rot * : 21 sys

x f Arrow2 Arrow1 target 5 1 sys 1 x - sqrt 1 x - sqrt - 2 1 sys

1 x - sqrt

Y 0.625

X 1 el 0.6124

sanity <= X 1 el f 0.625

1 x - sqrt -

Understanding InverseFunction . . .

Example above looks naive, no application, just demo of the

reverse Engineering the maths.

NASA(h) calculates kPa from physical altitude [m].

Behind the pilot seat, that is useless.

However, pressure @ altitude is from physical barometer.

Then, from the Inversed NASA(h) now you can appreciate

your flying altitude [m] as fnct of measured kPa.

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NASA Atmospheric Pressure vs altitude kPa(m)... as published h NASA 101.29 288.14 0.00649 h * - 288.08 / (5.256^* 1000 - h ≤ (h 11000 ≤ (& 22.65 1.73 0.000157 h * - exp * 11000 h ≤ (h 25000 ≤ (& 2.488 141.89 0.00299 h * + 216.6 / (11.388 -^* cases :

0 NASA 101.4009

NASA Altitude vs Pressure m(kPa)... from Smath/Maple companions. kPa alt 20001440714404100 kPa * 10129 / ln 5.256 / exp * - (* 649 / 101.4009 kPa ≥ (kPa 22.7074 ≥ (& 1000017310020 kPa * 453 / ln * - (* 157 / 22.7074 kPa ≥ (kPa 2.5223 ≥ (& 100014189 - 21660125 kPa * 311 / ln 11.388 / - exp * + (* 299 / cases :

101.4009 alt 0

A380 cokpit elevation ~[m] 101.325 alt 6.3

collect/plot NASA regions x alt x c11 * x alt x c25 * x alt x c60 * 22.7 11000 + 20 black 15 mat 2.52 25000 + 20 black 15 mat 51 sys

x c11 1 101.4009 x ≥ (x 22.7074 ≥ (& cases : x c25 1 22.7074 x ≥ (x 2.5223 ≥ (& cases : x c60 1 2.5223 x ≥ (x 0.0279 ≥ (& cases : 31 sys

x alt x c11 * x alt x c25 * x alt x c60 * 22.7 11000 + 20 black 1 5 mat 2.52 25000 + 20 black 1 5 mat 5 1 sys

NASA atmosphere . . . [3 regions . . . m(kPa)]

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Another application . . .

NASA example is another form of OIAC, which defines altimetric pressure from physics.

NASA being expressed otherwise from equivalent OIAC physics.

So, OIAC is pure physics, useless as is. To be exploitable for purpose, we reverse Engineer .

On a similar principle, many thermodynamic temperature points are primary standard.

From primary lab, we measure the resistance of the Pt thermometer. Now, for evaluating

temperature from measured resistance, again we reverse Engineer the physics.

Not exemplified here, lot more involved as final installation. The Pt 100 is in fact the

secondary temperature standard in a large range.

Otherwise Inverse Function . . .

No maths may inverse some functions with too many parameters, what's the solution ?

AFAIK, any function can be inversed via discrete algorithms, most interesting then what ?

At this point of having in hand a discrete InverseFunction, you may attempt:

1. a model function [global/piecewise] ? ? ? 2. simply spline/interpolate

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Maple RootOf Inverse Function . . .

Functions that Maple does not inverse explicitely are returned in terms of ... maple RootOf

The system can be recast as Dual Solver, as simplified for Pt100 . . . offered below

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Pt100: Wr [Ω] vs temperature [t °C] t Wr Z K.p 1 α β α - (/ t 259.3467 + (- α K.p * / exp * + β β α - (/ t 259.3467 + (- β K.p * / exp * - (* + :

Wr_Pt100

Maple reflexive solver. The Maple root argument will be passed  to Smath Sol(w,t).
Given  [°C]  from lab thermal bath, will solve for 'w', providing the thermal bath 
is itself at least equivalent to a secondary standard of ITS 90.  This direction of
experiment will enable checking the °C reading in the control room . w sol t Wr w - t root :

w sol Z β α - (* K.p β α 1 - 259346710000 t * + 10000 α * K.p * / - exp + (* + β 259346710000 t * + 10000 β * K.p * / - exp * - (* + w β α - (* - β α - / t root

The XFR coefficients 
were obtained from MCD
via a specialy designed
..."osculatice method"...
NOT revealed [Jean Giraud] Z 0.000999622531208 : K.p 12.4435127532766 : α 1.72760216537988 : β 228.3303465992546 : 41 sys

User unit system: copy/paste argument
<= in the empty place holder of result. °C 1 : Ω 1 : 21 sys

The reflexive solvers

w t argument β α - (Z w - (* K.p β 1259346710000 t * + 10000 β * K.p * / - exp - (* α 1 - 259346710000 t * + 10000 α * K.p * / - exp + (* + (* + β α - / :

Solving for (t), given (w)... Roots(range) => 0,t,-260,970
For Wr 0.00099963 ... => 0,t,-260,-250 w t Sol w t argument 0 ≡ t 260 - 970 solve :

0 t 260 - 250 - 259.36 - 259.34 - 21 mat 15 mat

0.0110234 t Sol °C 248.59 -

Solving for (w), given (t)...Roots(range 0,w,-1,5 w t Sol w t argument 0 ≡ w 1 - 5 solve :

w 962 Sol Ω 4.2870822

EIT90 259.3467 - 0.00119007 248.5939 - 0.00844974 218.7916 - 0.09171804 189.3442 - 0.21585975 38.8344 - 0.84414211 0.01 1.0000000 29.7646 1.11813889 156.5985 1.60980185 231.928 1.89279768 419.527 2.56891730 660.323 3.37600860 961.78 4.28642053 122 mat :

°C_wr 259.3467 - 0.00119007 0.0009996 0.0001911 248.5939 - 0.00844974 0.0110234 0.0025737 - 218.7916 - 0.09171804 0.0982121 0.006494 - 189.3442 - 0.21585975 0.2149336 0.0009261 38.8344 - 0.84414211 0.8424528 0.0016893 0.01 1 0.9999903 0.0000097 29.7646 1.11813889 1.1192165 0.0010776 - 156.5985 1.60980185 1.6136646 0.0038627 - 231.928 1.89279768 1.8970493 0.0042516 - 419.527 2.56891730 2.5709767 0.0020594 - 660.323 3.37600860 3.3732862 0.0027224 961.78 4.28642053 4.2864505 0.00003 - 124 mat :

t EIT90 1 col :

t xfr Z K.p 1 α β α - (/ t 259.3467 + (- α K.p * / exp * + β β α - (/ t 259.3467 + (- β K.p * / exp * - (* + :

t xfr vectorize

w Sol w t argument 0 ≡ t 260 - 970 solve :

x y axes x 1 - y / - :

0.0110234 Sol °C 248.59 -

t0 1 time :

Design a discrete function inverser  in the L, H span , N points f L H N Inv dx H L - N / : i 1 N range ω i el L dx - (dx i * + : for j 1 ω rows range sol j el u xfr ω j el - 0 ≡ u 260 - 970 solve : for ω sol augment 51 line :

eit90 0.00119007 259.3467 - 0.00844974 248.5939 - 0.09171804 218.7916 - 0.21585975 189.3442 - 0.84414211 38.8344 - 1 0.01 1.11813889 29.7646 1.60980185 156.5985 1.89279768 231.928 2.5689173 419.527 3.3760086 660.323 4.28642053 961.78 122 mat :

Pt100 f 0.0110234 4.2864505 100 Inv :

Pt100 axes eit90 3 1 sys

Since EIT90: H2O [3] 0.01°C

1 time t0 - 64 s *

Piecewise Pt100

imported for convenience

from 3 advanced models

]]> Pt100 [-273.15 ... 1084.5 °C]

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Smath al_nleqsolve λ 4.582374490 0.689736236 - 4.153686384 - 50.944725014 - 41 mat :

Mathcad  Minerr LM μ 1015.59658242 0.05304418 1.32997985 - 2.19590058 - 41 mat :

Mathcad Minerr LM ζ 795.75522816 0.05179366 0.39683701 1.58191684 - 41 mat :

231.928 419.524 660.323 961.78 41 mat

... exact eit90 inverse in the subCryo  [-259.3467...-189.3442] x λ subCryo λ 1 el x λ 3 el - (λ 2 el x *^(* 1 x ln + x λ 3 el - (- exp + λ 4 el + (* :

... exact eit90 inverse mid range[-189.3442  ... 156.5985 x μ Φ μ 1 el x μ 3 el - (μ 2 el x *^(* 1 x 2 + ln + x μ 3 el - (- exp + μ 4 el + (* :

1.89279768 μ Φ 231.928 - 0.02

...exact eit90 inverse upper range [231.928 ... 961.78] x ζ Extrapol ζ 1 el x ζ 3 el - (ζ 2 el x *^(* 1 x ln + x ζ 3 el - (- exp + ζ 4 el + (* :

Ω °C Ω λ subCryo 0.00006230541 Ω ≤ (Ω 0.21585975 ≤ (& Ω μ Φ 0.21585975 Ω ≤ (Ω 1.60980185 ≤ (& Ω ζ Extrapol 1.60980185 Ω ≤ (Ω 4.62715523 ≤ (& 4.62715523 ζ Extrapol cases :

abs 0.00006230541 :

°C 1 :

Ω 1 :

Extrapolated eit90

eit90 273.15 - abs 268.935 - 0.000155487 259.3467 - 0.00119007 248.5939 - 0.00844974 218.7916 - 0.09171804 189.3442 - 0.21585975 38.8344 - 0.84414211 0.01 1 29.7646 1.11813889 156.5985 1.60980185 231.928 1.89279768 419.527 2.5689173 660.323 3.3760086 961.78 4.28642053 1064.43 4.57255961 1084.5 4.62715523 162 mat :

Deviation °C(Ω) @ eit90  fixed
Exceptional recovery depending
upon the accuracy of measured Ω Δ°C X eit90 2 col : Y eit90 1 col : n eit90 rows : i 1 eit90 rows range u i el X i el °C eval : for Y Y u - augment 51 line :

Δ°C 273.1500 - 0.0000 268.9350 - 0.0000 259.3467 - 0.0000 248.5939 - 0.0000 218.7916 - 0.0000 189.3442 - 0.0000 38.8344 - 0.0000 0.0100 0.0000 29.7646 0.0000 156.5985 0.0000 231.9280 0.0000 419.5270 0.0000 660.3230 0.0000 961.7800 0.0000 1064.4300 0.0000 1084.5000 0.0000 162 mat

ε 0.02 < 11 sys

X 1.89279768 :

balistic region
..mid segment.. 156.5985 231.9280 21 sys

X μ Φ 231.928 - 0.02

data eit90 2 col eit90 1 col augment : x sub 1 0.00006230541 x ≤ (x 0.21585975 ≤ (& cases : x mid 1 0.21585975 x ≤ (x 1.60980185 ≤ (& cases : x up 1 1.60980185 x ≤ (x 5 ≤ (& cases : fixed data o 7 black plot : 51 sys

x λ subCryo x sub * fixed abs 273.15 - + 40 red 1 5 mat 3 1 sys

Generalized discrete inverser

Solving NASA(h)

altitude[m] from measured kPa

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h NASA 101.29 288.14 0.00649 h * - 288.08 / (5.256^* 1000 - h ≤ (h 11000 ≤ (& 22.65 1.73 0.000157 h * - exp * 11000 h ≤ (h 25000 ≤ (& 2.488 141.89 0.00299 h * + 216.6 / (11.388 -^* cases :

0 NASA 101.4009

L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

L 60000 NASA : H 101.4009 : N 101 : 13 mat

u L H N xd :

sol i 1 u rows range m_altitude i el h NASA u i el - 0 ≡ h 0100000 solve : for u 4 round vectorize m_altitude 0 round vectorize augment 21 line :

t0 1 time :

NASA(h) solved [m_altitude] from measured kPa

sol 101.325 10000 101.325 10000 - 2 2 mat 2 1 sys

1 time t0 - min 0.4

Maple RootOf . . .

applied for solving PDF in statistics..

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α 1.727602165

β 228.330346599

Fréchet distribution
α > 0 ... β > 0 α β x Fr α β / β x / (α 1 +^* β x / (α^- exp * :

w αα ββ x Fr ≡ x solve maple w ββ * αα ββ_Z / (1 αα +^* ββ_Z / (αα^- exp * - RootOf

recast RootOf

Fr 51 x Fr 11 x Fr 21 x Fr 0.5 1 x Fr 41 sys :

plot Fr 1 - Y 1.5 Y 22 mat X 1 el Y . 14 black 15 mat X 2 el Y . 14 black 15 mat 41 sys :

α 5 : β 1 : Y 1 : 13 mat w x ipdf w β * α β x / (1 α +^* β x / (α^- exp * - : w x Sol w x ipdf 0 ≡ x 1 - 10 solve : X Y x Sol (: 41 sys

Fréchet distribution ... α >0 ... β > 0

Fr 1 - Y 1.5 Y 2 2 mat X 1 el Y . 14 black 1 5 mat X 2 el Y . 14 black 1 5 mat 4 1 sys

X α β X Fr vectorize 21 mat 0.809 1.234 21 mat 1121 mat 21 mat

Bar & Stem

Spectrum stem plot for QuickPlot unicolor data Stem stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - range V V stem j el stack : for V V data data rows row stack : 71 line :

Traditionally, Gumbel was the model "rainfall". It is now recognised that EV_II better represents "rainfall" from 154 years collection world wide. Here, we propose "Fréchet" Type II.

US 5.123 4 5.36 6 5.597 14 5.834 9 6.071 7 6.308 24 6.545 10 6.782 4 7.019 9 7.256 5 7.494 11 7.731 7 7.968 4 8.205 2 8.442 1 8.679 2 8.916 3 9.153 0 9.39 1 9.627 1 9.864 2 10.101 1 10.338 0 10.575 2242 mat :

Fréchet: extreme value-II K α β x f K α β / * β x / (α 1 +^* β x / (α^- exp * :

X US 1 col : Bar US 0.025 BarGraph : 21 line

XY i 1 X rows range Y i el K α β X i el f : for X Y augment 21 line :

K 31.664 : α 6.774 : β 6.212 : 31 line

plot K α β x f US Stem XY 31 sys :

plot

Lambert W(x) principal branch x W w w exp * x - 0 ≡ w 1 e / - roots :

Lambert [W(x)] is a solving function. Solves many DE's, algebraic expressions

x W 1 e / - 1 1 e / - 1 - 2 2 mat 2 1 sys

e W 1.000000000000000

ω expr tt ω e ω^* - eval :

ω expr ω solve maple tt LambertW

ω ω^z - 0 ≡ ω solve maple z ln z ln LambertW /

ω expr ω 3^e ω^- :

ω expr ω solve maple 313 / - LambertW * - 31 - 13 / - LambertW * - 31 i 3 sqrt * - 6 / LambertW * - 31 i 3 sqrt * + 6 / LambertW * - 41 mat

x ln x ln W /

x ln x ln W /

Gumbel PDF μ σ x Gp 1 σ / x μ - x μ - (- σ / exp σ * + (- σ / exp * :

yy m σ x Gp - x solve maple m yy σ * - LambertW - ln σ * -

μ 1 : σ 0.625 : 12 mat

solve Gumbel for two X's Ω μ σ x Gp 0.375 ≡ x 02 solve : 0.4873 1.7042 21 mat

Y μ σ 2 Gp : 0.264

X μ Y σ * - W - ln σ * - : 2

μ σ Ω Gp vectorize 0.375 0.375 21 mat

μ x σ * - W - ln σ * - μ σ x Gp X Y X 0 2 2 mat 2 1 sys

========== else solve ==========

Smath solve at(1)

Solve/evaluate contour

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L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

Universal ... most preferred discretizer

εlectron 816 π * sqrt * : 351.67021

isolate εlectron 351.67021 :

L 1.785686 : H 14.245529 : N 100 : 13 mat x L H N xd : 21 line

x y Φ 1 - 3 / x 2^y 2^+ sqrt * exp 2 y 2^* - x 2^+ εlectron / * - 0.005 + :

n 1 x length range Sols n el y x n el y Φ vectorize y solve at : for

# # at

plot branch_1 x Sols augment : branch_2 x Sols - augment : branch_3 x - Sols augment : branch_4 x - Sols - augment : 41 sys :

round limits:  more decimals Smath complains 1.78568577383018 14.2455293828432 21 mat 6 round vectorize 1.785686 14.245529 21 mat

plot

revisit Lambert W(x)

Solving a DE

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L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

Lambert two principal branches x W0 w w exp * x - 0 ≡ w 1 e / - roots : x W1 w w exp * x - 0 ≡ w 2 sqrt - roots : 21 line

L 0 : H 10 : N 200 : 13 mat U L H N xd : 21 line

Lambert complementary branches

x W0 x W1 1 e / - 1 1 e / - 5 - 2 2 mat 3 1 sys

Example: solving a DE

t C t diff Vm t C * - Km t C + / ≡

C0 1 :

Vm 2 :

Km 3 :

solution borrowed from Mathematica "ProductLog" DE i 1 N range p i el U i el Vm Km / * (- C0 C0 Km / exp * ln Km / + exp eval : d i el Km p i el W0 * : 21 line for d 21 line :

DEsol U DE augment :

Title 1.5 1.85 DE solution in Pharmacology 12 red 15 mat :

DEsol Title 2 1 sys

t C t diff Vm t C * - Km t C + / ≡

range roots wisely
NormalPDF [μ=0, σ=1] x G 12 π * sqrt / x 2^- 2 / exp * :

relevant Lambert calculus

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x Inverse xx G x - 0 ≡ xx 1 - roots : x inverse xx G x - 0 ≡ xx 1 roots : 21 sys

x W x diff x W 1 x W + (x * / ≡

x W x int x - x x W / + x W x * + ≡

sys(Scalar(x),-e,2,1)

x Inverse x inverse π 10 x * G * 3 1 sys x W 1 x W + ( x * / x - x x W / + x W x * + 1 e / - 10 1 e / - 2 - 2 2 mat 3 1 sys

algebraic Lambert

the most accurate solver

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Lambert identity x W y W + x y * 1 x W / 1 y W / + (* W ≡

Maple LambertW two branches ... x W0 w w exp * x - 0 ≡ w 1 e / - roots : x W1 w w exp * x - 0 ≡ w 2 sqrt - roots : x W w w exp * x - 0 ≡ w 1 e / - roots : 31 sys

# LambertW 1 - # LambertW x W # ProductLog ≡ 31 sys

X 0.25 - :

Lambert complementary branches

x W0 x W1 1 e / - 1 1 e / - 5 - 2 2 mat X 1 X 3 - 2 2 mat 4 1 sys

e W 1.000000000000000

X W0 X W1 21 mat 0.357403 - 2.153292 - 21 mat

ω expr ω 3^e ω^- :

ω expr ω solve maple 313 / - LambertW * - 31 - 13 / - LambertW * - 31 i 3 sqrt * - 6 / LambertW * - 31 i 3 sqrt * + 6 / LambertW * - 41 mat

... wrt Maple 31 - branch -1 ≡ 13 / - W * -

313 / - W * - (3^313 / - W * - exp - 0

Solving the xpresiion "completely"

313 / - W * - 313 / - W * - (3^313 / - W * - exp - 313 / - W1 * - 313 / - W1 * - (3^313 / - W1 * - exp - 61 mat 1.85718386020784 6.40567207898112 6.40567207898109 - 4.53640365497352 93.3544608350032 93.3544608350029 - 61 mat

31 i 3 sqrt * - 6 / abs W * - 31 i 3 sqrt * - 6 / abs W * - (3^31 i 3 sqrt * - 6 / abs W * - exp - 31 mat 0.772882959 - 0.461680143 - 0.461680143 - 31 mat

junction ... and linedup pairs

x 3 ^ x exp - 3 1 3 / - W * - 3 1 3 / - W * - ( 3 ^ 3 1 3 / - W * - 3 1 3 / - W * - exp - 2 2 mat 3 1 i 3 sqrt * - 6 / abs W * - 3 1 i 3 sqrt * - 6 / abs W * - exp - + 40 black 1 5 mat 4 1 sys

1rst pairs 1.85718386020784 2nd pairs 4.53640365497352

x y x y 2^x y 3^- ≡ x diff

t y 1144 t - exp * W + / ≡

0 y 0.2 ≡

t y a t 1 - y * ≡ t diff

for 'a = real' t y a W t * exp ≡

1 1 4 4 x - exp * W + / 0.25 W x * exp

more algebraic Lambert

collapsed in there

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more algebraic u 2^e u^- 0 ≡ a b v * + c 10 v -^* ≡ a b v * + c e v -^* ≡ 3 x^x 2^≡ α α exp * 1 e / - ≡ a x^b x * + c + 0 ≡ 2 t^5 t * - 0 ≡ x t^z - 0 ≡ z z b log * a - 0 ≡ 91 sys

y 1 υ e υ^* + ≡ υ solve maple y υ υ e υ^* + υ υ e υ^* - / ≡ υ solve maple y 1 υ e υ^* + 1 υ e υ^* - / ≡ υ solve maple 31 mat 1 - y + LambertW 1 - y + 1 y + / ln 1 - y + 1 y + / LambertW 31 mat

υ 2^e υ^- 0 ≡ υ solve maple 212 / - LambertW * - 212 / LambertW * - 21 mat

a b v * + c e v -^* ≡ v solve maple c a b / exp * b / LambertW b * - a + b / -

a b v * + c 10 v -^* ≡ v solve maple 10 ln a * b 10 ln c * 10 ln a * b / exp * b / LambertW * - 10 ln b * / -

a xx^b xx * + c + 0 ≡ xx solve maple a ln c a ln * b / - exp * b / LambertW b * c a ln * + a ln b * / -

3 xx^xx 2^≡ xx solve maple 23 ln 2 / - LambertW * 3 ln / - 23 ln 2 / LambertW * 3 ln / - 21 mat

a a exp * 1 e / - ≡ a solve maple 1 -

2 tt^5 tt * - 0 ≡ tt solve maple 2 ln 5 / - LambertW 2 ln / - 1 - 2 ln 5 / - LambertW 2 ln / - 21 mat

xx tt^z - 0 ≡ tt solve maple z ln xx ln /

xx tt^z - 0 ≡ xx solve maple z ln tt / exp

xx tt^z - 0 ≡ z solve maple xx tt^

z z log * a - 0 ≡ z solve maple a a LambertW /

MCD 11 a 10 ln a 10 ln * W / *

a 3 :

a a W / a 10 ln a 10 ln * W / * 21 mat 2.857390784 4.555535705 21 mat

LambertW principal branch

x ProductLog 15 decimals u x e > x ln x ln ln - x 1 e / + sqrt 1 e / sqrt - if 11 line : w 2 u * : u w - abs u abs / 1012 -^≥ w u : u u 2^x e u -^* + 1 u + / eval : 21 line while u 51 line :

x W w w exp * x - 0 ≡ w 1 e / - roots :

x ProductLog x W 2 1 sys

Except for the very low range => they are both equally accurate, rated 15 decimals

13 / - W 13 / - ProductLog 21 mat 0.619061286735947 - 0.619061286735950 - 21 mat

1 e / - 1010 -^+ W 1 e / - 1010 -^+ ProductLog 21 mat 0.999976683865137 - 0.999976683771404 - 21 mat

Lambert all branches MCD can't reproduce in Smath

revisit Lambert . . . vs solve(, , , ,)

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y xx e xx^* ≡ xx solve maple xx e xx^* aa - xx solve maple 21 mat y LambertW aa LambertW 21 mat

Lambert complementary branches

x W0 x W1 1 e / - 1 1 e / - 5 - 2 2 mat 3 1 sys x e x ^ * x W0 x W1 3 1 sys

2 ζζ^5 ζζ * - 0 ≡ ζζ solve maple 2 ln 5 / - LambertW 2 ln / - 1 - 2 ln 5 / - LambertW 2 ln / - 21 mat

2 ln 5 / - W0 2 ln / - 2 ln 5 / - W1 2 ln / - 21 mat 0.235455710071006 4.48800113601203 21 mat

L 2 x^5 x * - 0 ≡ x 05 solve : 0.235455710070995 4.48800113601193 21 mat

2 x ^ 5 x * - L 1 el 0 + 40 red 1 5 mat L 2 el 0 + 40 red 1 5 mat 3 1 sys

L H N Xd xd 0 : k 0 N range xd k 1 + el L H L - N / (k * + : for xd 31 line :

Maple LambertW two branches ... x W0 0 x LambertW maple : x W1 1 x LambertW maple : x W_1 1 - x LambertW maple : 31 sys

U 1 - 1199 Xd :

U W_1 vectorize #

x x cos * x sin / -