From b1a500ee7f29d2216b3d748d9a30538b4ac06a4d Mon Sep 17 00:00:00 2001
From: Alexey Stukalov <astukalov@gmail.com>
Date: Fri, 19 Nov 2021 15:58:24 +0100
Subject: [PATCH 01/11] add exp(-1) constant

useful in many situations, also -exp(-1) is the branching point of
Lambert's W_0(t) and W_{-1}(t)
---
 src/IrrationalConstants.jl | 3 ++-
 src/stats.jl               | 2 ++
 test/runtests.jl           | 3 +++
 3 files changed, 7 insertions(+), 1 deletion(-)

diff --git a/src/IrrationalConstants.jl b/src/IrrationalConstants.jl
index e9c2b27..54263e0 100644
--- a/src/IrrationalConstants.jl
+++ b/src/IrrationalConstants.jl
@@ -26,7 +26,8 @@ export
     logten,     # log(10)
     logπ,       # log(π)
     log2π,      # log(2π)
-    log4π       # log(4π)
+    log4π,      # log(4π)
+    invℯ        # 1 / ℯ
 
 include("stats.jl")
 
diff --git a/src/stats.jl b/src/stats.jl
index c221961..389dc50 100644
--- a/src/stats.jl
+++ b/src/stats.jl
@@ -28,3 +28,5 @@
 @irrational logπ    1.1447298858494001741 log(big(π))
 @irrational log2π   1.8378770664093454836 log(2 * big(π))
 @irrational log4π   2.5310242469692907930 log(4 * big(π))
+
+@irrational invℯ    0.367879441171442321595 inv(big(ℯ))
diff --git a/test/runtests.jl b/test/runtests.jl
index 0bab199..554ec86 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -40,3 +40,6 @@ end
   @test isapprox(log(4pi), log4π)
 end
 
+@testset "1/e" begin
+  @test isapprox(invℯ, exp(-1))
+end

From 50056d3538428f7a19d61ce383fae9412aaa4dfd Mon Sep 17 00:00:00 2001
From: Alexey Stukalov <astukalov@gmail.com>
Date: Fri, 19 Nov 2021 16:01:20 +0100
Subject: [PATCH 02/11] add Lambert's Omega constant

---
 src/IrrationalConstants.jl |  5 ++++-
 src/lambertw_omega.jl      | 32 ++++++++++++++++++++++++++++++++
 test/runtests.jl           |  8 ++++++++
 3 files changed, 44 insertions(+), 1 deletion(-)
 create mode 100644 src/lambertw_omega.jl

diff --git a/src/IrrationalConstants.jl b/src/IrrationalConstants.jl
index 54263e0..e9085a8 100644
--- a/src/IrrationalConstants.jl
+++ b/src/IrrationalConstants.jl
@@ -27,8 +27,11 @@ export
     logπ,       # log(π)
     log2π,      # log(2π)
     log4π,      # log(4π)
-    invℯ        # 1 / ℯ
+    invℯ,       # 1 / ℯ
+
+    LambertW_Ω  # Ω exp(Ω) = 1
 
 include("stats.jl")
+include("lambertw_omega.jl")
 
 end # module
diff --git a/src/lambertw_omega.jl b/src/lambertw_omega.jl
new file mode 100644
index 0000000..31a52e4
--- /dev/null
+++ b/src/lambertw_omega.jl
@@ -0,0 +1,32 @@
+# lazy-initialized LambertW Omega at 256-bit precision
+const LambertW_Omega_BigFloat256 = Ref{BigFloat}()
+
+# compute BigFloat Omega constant at arbitrary precision
+function compute_LambertW_Omega()
+    # initialize Omega_BigFloat256
+    isassigned(LambertW_Omega_BigFloat256) ||
+        (LambertW_Omega_BigFloat256[] = BigFloat("0.5671432904097838729999686622103555497538157871865125081351310792230457930866845666932194"))
+    o = LambertW_Omega_BigFloat256[] # initial value
+    precision(BigFloat) <= 256 && return o
+    # iteratively improve the precision of the constant
+    myeps = eps(BigFloat)
+    for _ in 1:100
+        o_ = (1 + o) / (1 + exp(o))
+        abs(o - o_) <= myeps && break
+        o = o_
+    end
+    return o
+end
+
+@irrational LambertW_Ω 0.567143290409783872999968662210355 compute_LambertW_Omega()
+
+"""
+Lambert's Omega (Ω) constant, such that Ω exp(Ω) = 1.
+
+*W(Ω) = 1*, where *W(t) = t exp(t)* is the *Lambert's W function*.
+
+# See
+https://en.wikipedia.org/wiki/Omega_constant
+"""
+LambertW_Ω
+
diff --git a/test/runtests.jl b/test/runtests.jl
index 554ec86..1c064e4 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -43,3 +43,11 @@ end
 @testset "1/e" begin
   @test isapprox(invℯ, exp(-1))
 end
+
+@testset "LambertW_Omega" begin
+  @test isapprox(LambertW_Ω * exp(LambertW_Ω), 1)
+  setprecision(BigFloat, 2048) do
+    o = big(LambertW_Ω)
+    @test isapprox(o * exp(o), 1, atol=eps(BigFloat))
+  end
+end

From 2552035b7d402b458635935243abb5374142b928 Mon Sep 17 00:00:00 2001
From: Alexey Stukalov <astukalov@gmail.com>
Date: Thu, 30 Dec 2021 22:39:34 +0100
Subject: [PATCH 03/11] add ASCII inve alias

---
 src/IrrationalConstants.jl | 2 +-
 src/stats.jl               | 1 +
 test/runtests.jl           | 1 +
 3 files changed, 3 insertions(+), 1 deletion(-)

diff --git a/src/IrrationalConstants.jl b/src/IrrationalConstants.jl
index e9085a8..2f8e8c8 100644
--- a/src/IrrationalConstants.jl
+++ b/src/IrrationalConstants.jl
@@ -27,7 +27,7 @@ export
     logπ,       # log(π)
     log2π,      # log(2π)
     log4π,      # log(4π)
-    invℯ,       # 1 / ℯ
+    invℯ, inve, # 1 / ℯ
 
     LambertW_Ω  # Ω exp(Ω) = 1
 
diff --git a/src/stats.jl b/src/stats.jl
index 389dc50..ce1dacc 100644
--- a/src/stats.jl
+++ b/src/stats.jl
@@ -30,3 +30,4 @@
 @irrational log4π   2.5310242469692907930 log(4 * big(π))
 
 @irrational invℯ    0.367879441171442321595 inv(big(ℯ))
+const inve = invℯ # ASCII alias
diff --git a/test/runtests.jl b/test/runtests.jl
index 1c064e4..f1fb4ce 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -42,6 +42,7 @@ end
 
 @testset "1/e" begin
   @test isapprox(invℯ, exp(-1))
+  @test isapprox(inve, exp(-1))
 end
 
 @testset "LambertW_Omega" begin

From e22a444630a268bf55f8fbd10c1ec14cb809133c Mon Sep 17 00:00:00 2001
From: Alexey Stukalov <astukalov@gmail.com>
Date: Thu, 30 Dec 2021 22:41:08 +0100
Subject: [PATCH 04/11] LambertW => lambertw, add lambertw_Omega alias

---
 src/IrrationalConstants.jl |  2 +-
 src/lambertw_omega.jl      | 18 +++++++++---------
 test/runtests.jl           | 11 ++++++++---
 3 files changed, 18 insertions(+), 13 deletions(-)

diff --git a/src/IrrationalConstants.jl b/src/IrrationalConstants.jl
index 2f8e8c8..89e316c 100644
--- a/src/IrrationalConstants.jl
+++ b/src/IrrationalConstants.jl
@@ -29,7 +29,7 @@ export
     log4π,      # log(4π)
     invℯ, inve, # 1 / ℯ
 
-    LambertW_Ω  # Ω exp(Ω) = 1
+    lambertw_Ω, lambertw_Omega  # Ω exp(Ω) = 1
 
 include("stats.jl")
 include("lambertw_omega.jl")
diff --git a/src/lambertw_omega.jl b/src/lambertw_omega.jl
index 31a52e4..1fa43ae 100644
--- a/src/lambertw_omega.jl
+++ b/src/lambertw_omega.jl
@@ -1,12 +1,12 @@
 # lazy-initialized LambertW Omega at 256-bit precision
-const LambertW_Omega_BigFloat256 = Ref{BigFloat}()
+const lambertw_Omega_BigFloat256 = Ref{BigFloat}()
 
 # compute BigFloat Omega constant at arbitrary precision
-function compute_LambertW_Omega()
-    # initialize Omega_BigFloat256
-    isassigned(LambertW_Omega_BigFloat256) ||
-        (LambertW_Omega_BigFloat256[] = BigFloat("0.5671432904097838729999686622103555497538157871865125081351310792230457930866845666932194"))
-    o = LambertW_Omega_BigFloat256[] # initial value
+function compute_lambertw_Omega()
+    # initialize lambertw_Omega_BigFloat256
+    isassigned(lambertw_Omega_BigFloat256) ||
+        (lambertw_Omega_BigFloat256[] = BigFloat("0.5671432904097838729999686622103555497538157871865125081351310792230457930866845666932194"))
+    o = lambertw_Omega_BigFloat256[] # initial value
     precision(BigFloat) <= 256 && return o
     # iteratively improve the precision of the constant
     myeps = eps(BigFloat)
@@ -18,7 +18,8 @@ function compute_LambertW_Omega()
     return o
 end
 
-@irrational LambertW_Ω 0.567143290409783872999968662210355 compute_LambertW_Omega()
+@irrational lambertw_Ω 0.567143290409783872999968662210355 compute_lambertw_Omega()
+const lambertw_Omega = lambertw_Ω # ASCII alias
 
 """
 Lambert's Omega (Ω) constant, such that Ω exp(Ω) = 1.
@@ -28,5 +29,4 @@ Lambert's Omega (Ω) constant, such that Ω exp(Ω) = 1.
 # See
 https://en.wikipedia.org/wiki/Omega_constant
 """
-LambertW_Ω
-
+lambertw_Ω
\ No newline at end of file
diff --git a/test/runtests.jl b/test/runtests.jl
index f1fb4ce..097d4a0 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -45,10 +45,15 @@ end
   @test isapprox(inve, exp(-1))
 end
 
-@testset "LambertW_Omega" begin
-  @test isapprox(LambertW_Ω * exp(LambertW_Ω), 1)
+@testset "lambertw_Omega" begin
+  @test isapprox(lambertw_Ω * exp(lambertw_Ω), 1)
+  @test lambertw_Omega == lambertw_Ω
+
   setprecision(BigFloat, 2048) do
-    o = big(LambertW_Ω)
+    o = big(lambertw_Ω)
     @test isapprox(o * exp(o), 1, atol=eps(BigFloat))
+
+    oalias = big(lambertw_Omega)
+    @test o == oalias
   end
 end

From ba45b7b60c08280383fdb5671c86980c6026e81f Mon Sep 17 00:00:00 2001
From: Alexey Stukalov <astukalov@gmail.com>
Date: Thu, 30 Dec 2021 22:41:35 +0100
Subject: [PATCH 05/11] improve lambertw_Omega docstring

---
 src/lambertw_omega.jl | 11 +++++++----
 1 file changed, 7 insertions(+), 4 deletions(-)

diff --git a/src/lambertw_omega.jl b/src/lambertw_omega.jl
index 1fa43ae..852d648 100644
--- a/src/lambertw_omega.jl
+++ b/src/lambertw_omega.jl
@@ -22,11 +22,14 @@ end
 const lambertw_Omega = lambertw_Ω # ASCII alias
 
 """
-Lambert's Omega (Ω) constant, such that Ω exp(Ω) = 1.
+Lambert's Omega (*Ω*) constant.
 
-*W(Ω) = 1*, where *W(t) = t exp(t)* is the *Lambert's W function*.
+Lambert's *Ω* is the solution to *W(Ω) = 1* equation,
+where *W(t) = t exp(t)* is the
+[Lambert's *W* function](https://en.wikipedia.org/wiki/Lambert_W_function).
 
-# See
-https://en.wikipedia.org/wiki/Omega_constant
+# See also
+  * https://en.wikipedia.org/wiki/Omega_constant
+  * [`lambertw()`][@ref SpecialFunctions.lambertw]
 """
 lambertw_Ω
\ No newline at end of file

From a72e8350022eacd8dc18c49d52346fbfb0ec1aad Mon Sep 17 00:00:00 2001
From: Alexey Stukalov <astukalov@gmail.com>
Date: Thu, 30 Dec 2021 22:42:10 +0100
Subject: [PATCH 06/11] lambertw_Omega: better handle BigFloat precision

---
 src/lambertw_omega.jl |  6 +++---
 test/runtests.jl      | 12 ++++++++++++
 2 files changed, 15 insertions(+), 3 deletions(-)

diff --git a/src/lambertw_omega.jl b/src/lambertw_omega.jl
index 852d648..f78fc13 100644
--- a/src/lambertw_omega.jl
+++ b/src/lambertw_omega.jl
@@ -5,9 +5,9 @@ const lambertw_Omega_BigFloat256 = Ref{BigFloat}()
 function compute_lambertw_Omega()
     # initialize lambertw_Omega_BigFloat256
     isassigned(lambertw_Omega_BigFloat256) ||
-        (lambertw_Omega_BigFloat256[] = BigFloat("0.5671432904097838729999686622103555497538157871865125081351310792230457930866845666932194"))
-    o = lambertw_Omega_BigFloat256[] # initial value
-    precision(BigFloat) <= 256 && return o
+        (lambertw_Omega_BigFloat256[] = BigFloat("0.5671432904097838729999686622103555497538157871865125081351310792230457930866845666932194", 256))
+    o = BigFloat(lambertw_Omega_BigFloat256[]) # initial value with current precision
+    precision(o) <= 256 && return o
     # iteratively improve the precision of the constant
     myeps = eps(BigFloat)
     for _ in 1:100
diff --git a/test/runtests.jl b/test/runtests.jl
index 097d4a0..2190d9c 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -49,8 +49,20 @@ end
   @test isapprox(lambertw_Ω * exp(lambertw_Ω), 1)
   @test lambertw_Omega == lambertw_Ω
 
+  # lower than default precision
+  setprecision(BigFloat, 196) do
+    o = big(lambertw_Ω)
+    @test precision(o) == 196
+    @test isapprox(o * exp(o), 1, atol=eps(BigFloat))
+
+    oalias = big(lambertw_Omega)
+    @test o == oalias
+  end
+
+  # higher than default precision
   setprecision(BigFloat, 2048) do
     o = big(lambertw_Ω)
+    @test precision(o) == 2048
     @test isapprox(o * exp(o), 1, atol=eps(BigFloat))
 
     oalias = big(lambertw_Omega)

From b4b290883a876595d40ae116b1728a1dd177bb12 Mon Sep 17 00:00:00 2001
From: Alexey Stukalov <astukalov@gmail.com>
Date: Thu, 30 Dec 2021 22:57:39 +0100
Subject: [PATCH 07/11] try to fix Omega precision for Julia 1.0

---
 src/lambertw_omega.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/lambertw_omega.jl b/src/lambertw_omega.jl
index f78fc13..3b79c9f 100644
--- a/src/lambertw_omega.jl
+++ b/src/lambertw_omega.jl
@@ -6,7 +6,7 @@ function compute_lambertw_Omega()
     # initialize lambertw_Omega_BigFloat256
     isassigned(lambertw_Omega_BigFloat256) ||
         (lambertw_Omega_BigFloat256[] = BigFloat("0.5671432904097838729999686622103555497538157871865125081351310792230457930866845666932194", 256))
-    o = BigFloat(lambertw_Omega_BigFloat256[]) # initial value with current precision
+    o = BigFloat(lambertw_Omega_BigFloat256[], precision(BigFloat)) # initial value with current precision
     precision(o) <= 256 && return o
     # iteratively improve the precision of the constant
     myeps = eps(BigFloat)

From ee9180e14fd88f945081f62ce3d369fecf6612c9 Mon Sep 17 00:00:00 2001
From: Alexey Stukalov <astukalov@gmail.com>
Date: Thu, 30 Dec 2021 23:11:43 +0100
Subject: [PATCH 08/11] Omega: remove BigFloat lazy-init to fix Julia 1.0

---
 src/lambertw_omega.jl | 8 +-------
 1 file changed, 1 insertion(+), 7 deletions(-)

diff --git a/src/lambertw_omega.jl b/src/lambertw_omega.jl
index 3b79c9f..818d38f 100644
--- a/src/lambertw_omega.jl
+++ b/src/lambertw_omega.jl
@@ -1,12 +1,6 @@
-# lazy-initialized LambertW Omega at 256-bit precision
-const lambertw_Omega_BigFloat256 = Ref{BigFloat}()
-
 # compute BigFloat Omega constant at arbitrary precision
 function compute_lambertw_Omega()
-    # initialize lambertw_Omega_BigFloat256
-    isassigned(lambertw_Omega_BigFloat256) ||
-        (lambertw_Omega_BigFloat256[] = BigFloat("0.5671432904097838729999686622103555497538157871865125081351310792230457930866845666932194", 256))
-    o = BigFloat(lambertw_Omega_BigFloat256[], precision(BigFloat)) # initial value with current precision
+    o = BigFloat("0.5671432904097838729999686622103555497538157871865125081351310792230457930866845666932194")
     precision(o) <= 256 && return o
     # iteratively improve the precision of the constant
     myeps = eps(BigFloat)

From d5aae009fa913743526bb2622c815d861ce690d5 Mon Sep 17 00:00:00 2001
From: Alexey Stukalov <astukalov@gmail.com>
Date: Thu, 30 Dec 2021 23:11:58 +0100
Subject: [PATCH 09/11] Omega: ref. for computation method

---
 src/lambertw_omega.jl | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/src/lambertw_omega.jl b/src/lambertw_omega.jl
index 818d38f..253bf6c 100644
--- a/src/lambertw_omega.jl
+++ b/src/lambertw_omega.jl
@@ -2,7 +2,8 @@
 function compute_lambertw_Omega()
     o = BigFloat("0.5671432904097838729999686622103555497538157871865125081351310792230457930866845666932194")
     precision(o) <= 256 && return o
-    # iteratively improve the precision of the constant
+    # iteratively improve the precision
+    # see https://en.wikipedia.org/wiki/Omega_constant#Computation
     myeps = eps(BigFloat)
     for _ in 1:100
         o_ = (1 + o) / (1 + exp(o))

From 3fc698095fc2e205d82d74d4049c58a89b8f4896 Mon Sep 17 00:00:00 2001
From: Alexey Stukalov <astukalov@gmail.com>
Date: Fri, 31 Dec 2021 00:11:23 +0100
Subject: [PATCH 10/11] Omega: allow 1000 iters, warn if prec not reached

---
 src/lambertw_omega.jl | 5 +++--
 1 file changed, 3 insertions(+), 2 deletions(-)

diff --git a/src/lambertw_omega.jl b/src/lambertw_omega.jl
index 253bf6c..a77252f 100644
--- a/src/lambertw_omega.jl
+++ b/src/lambertw_omega.jl
@@ -5,11 +5,12 @@ function compute_lambertw_Omega()
     # iteratively improve the precision
     # see https://en.wikipedia.org/wiki/Omega_constant#Computation
     myeps = eps(BigFloat)
-    for _ in 1:100
+    for _ in 1:1000
         o_ = (1 + o) / (1 + exp(o))
-        abs(o - o_) <= myeps && break
+        abs(o - o_) <= myeps && return o
         o = o_
     end
+    @warn "lambertw_Omega precision is less than current BigFloat precision ($(precision(BigFloat)))"
     return o
 end
 

From 1ee42a5b7b020b46f52085ae5b550e476eb26d0f Mon Sep 17 00:00:00 2001
From: Alexey Stukalov <astukalov@gmail.com>
Date: Sat, 1 Jan 2022 11:15:39 +0100
Subject: [PATCH 11/11] rename to LambertW.Omega

---
 src/IrrationalConstants.jl |  2 +-
 src/lambertw_omega.jl      | 12 ++++++++----
 test/runtests.jl           | 14 +++++++-------
 3 files changed, 16 insertions(+), 12 deletions(-)

diff --git a/src/IrrationalConstants.jl b/src/IrrationalConstants.jl
index 89e316c..99ccd5d 100644
--- a/src/IrrationalConstants.jl
+++ b/src/IrrationalConstants.jl
@@ -29,7 +29,7 @@ export
     log4π,      # log(4π)
     invℯ, inve, # 1 / ℯ
 
-    lambertw_Ω, lambertw_Omega  # Ω exp(Ω) = 1
+    LambertW    # module that defines Lambert'W Ω: Ω*exp(Ω) = 1
 
 include("stats.jl")
 include("lambertw_omega.jl")
diff --git a/src/lambertw_omega.jl b/src/lambertw_omega.jl
index a77252f..71bd088 100644
--- a/src/lambertw_omega.jl
+++ b/src/lambertw_omega.jl
@@ -10,12 +10,13 @@ function compute_lambertw_Omega()
         abs(o - o_) <= myeps && return o
         o = o_
     end
-    @warn "lambertw_Omega precision is less than current BigFloat precision ($(precision(BigFloat)))"
+    @warn "Omega precision is less than current BigFloat precision ($(precision(BigFloat)))"
     return o
 end
 
-@irrational lambertw_Ω 0.567143290409783872999968662210355 compute_lambertw_Omega()
-const lambertw_Omega = lambertw_Ω # ASCII alias
+@irrational lambertw_Omega 0.567143290409783872999968662210355 compute_lambertw_Omega()
+
+module LambertW
 
 """
 Lambert's Omega (*Ω*) constant.
@@ -28,4 +29,7 @@ where *W(t) = t exp(t)* is the
   * https://en.wikipedia.org/wiki/Omega_constant
   * [`lambertw()`][@ref SpecialFunctions.lambertw]
 """
-lambertw_Ω
\ No newline at end of file
+const Ω = parentmodule(@__MODULE__).lambertw_Omega
+const Omega = parentmodule(@__MODULE__).lambertw_Omega # ASCII alias
+
+end
diff --git a/test/runtests.jl b/test/runtests.jl
index 2190d9c..684f66d 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -45,27 +45,27 @@ end
   @test isapprox(inve, exp(-1))
 end
 
-@testset "lambertw_Omega" begin
-  @test isapprox(lambertw_Ω * exp(lambertw_Ω), 1)
-  @test lambertw_Omega == lambertw_Ω
+@testset "LambertW.Omega" begin
+  @test isapprox(LambertW.Ω * exp(LambertW.Ω), 1)
+  @test LambertW.Omega === LambertW.Ω
 
   # lower than default precision
   setprecision(BigFloat, 196) do
-    o = big(lambertw_Ω)
+    o = big(LambertW.Ω)
     @test precision(o) == 196
     @test isapprox(o * exp(o), 1, atol=eps(BigFloat))
 
-    oalias = big(lambertw_Omega)
+    oalias = big(LambertW.Omega)
     @test o == oalias
   end
 
   # higher than default precision
   setprecision(BigFloat, 2048) do
-    o = big(lambertw_Ω)
+    o = big(LambertW.Ω)
     @test precision(o) == 2048
     @test isapprox(o * exp(o), 1, atol=eps(BigFloat))
 
-    oalias = big(lambertw_Omega)
+    oalias = big(LambertW.Omega)
     @test o == oalias
   end
 end