Tests passing

examples/examples.jl

@@ -5,11 +5,11 @@ function sawtooth(X::IntervalBox)
 
     x, y = X
 
-    x = x ⊓ (-1..1)
-    y = y ⊓ (-2..2)
+    x = x ⊓ interval(-1, 1)
+    y = y ⊓ interval(-2, 2)
 
     y = y ⊓ (2x)
-    x = x ⊓ (y/2)
+    x = x ⊓ (y / 2)
 
     return IntervalBox(x, y)
 end
@@ -38,7 +38,7 @@ function cube0(X::IntervalBox)  # contractor for y=x^3, x>=0
 
     x, y = X
 
-    x = x ⊓ (0..∞)
+    x = x ⊓ (interval(0, Inf))
 
     y = y ⊓ (x ^ 3)
     x = x ⊓ interval(inf(y) ^ (1/3), sup(y)^(1/3))   # not rigorous!

src/IntervalContractors.jl

@@ -20,8 +20,6 @@ using IntervalBoxes
 
 const half_pi = ExactReal(0.5) * interval(pi)
 const two_pi = ExactReal(2.0) * interval(pi)
-
-const ∞ = Inf
 #
 # Base.:⊔(f::Function, g::Function) = X -> ( f(X) ⊔ g(X) )
 # Base.:⊓(f::Function, g::Function) = X -> ( f(X) ⊓ g(X) )  # or f(g(X)) for contractors

src/arithmetic.jl

@@ -148,8 +148,8 @@ function power_rev(a::Interval{T}, b::Interval{T}, n::Integer) where T  # a = b^
         b2 = b ⊓ (-root)
 
     elseif isodd(n)
-        pos_root = (a ⊓ (0..∞)) ^ (1//n)
-        neg_root = -( ( (-a) ⊓ (0..∞) ) ^ (1//n) )
+        pos_root = (a ⊓ interval(0, Inf)) ^ (1//n)
+        neg_root = -( ( (-a) ⊓ (interval(0, Inf)) ) ^ (1//n) )
 
         b1 = b ⊓ pos_root
         b2 = b ⊓ neg_root
@@ -207,7 +207,7 @@ sqrt_rev(a,b) = sqrt_rev(promote(a,b)...)
     sqr_rev(c::Interval[, x::Interval])
 
 Reverse square. Calculates the preimage of `a = x²`. If `x` is not provided, then
-byt default ``[-∞, ∞]`` is used. See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
+byt default ``[-Inf, Inf]`` is used. See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
 
 ### Output
 
@@ -230,7 +230,7 @@ end
     abs_rev(c::Interval[, x::Interval])
 
 Reverse absolute value. Calculates the preimage of `a = |x|`. If `x` is not provided, then
-byt default ``[-∞, ∞]`` is used. See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
+byt default ``[-Inf, Inf]`` is used. See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
 
 ### Output
 
@@ -241,7 +241,7 @@ The pair `(c, x_new)` where
 """
 function abs_rev(y, x = entireinterval(y))   # y = abs(x); refine x
 
-    y_new = y ⊓ (0..∞)
+    y_new = y ⊓ (interval(0, Inf))
 
     x1 = y_new ⊓ x
     x2 = -(y_new ⊓ (-x))
@@ -254,9 +254,9 @@ Reverse sign
 """
 function sign_rev(a::Interval, b::Interval)  # a = sqrt(b)
 
-    (a == 1.0) && b = b ⊓ (0..∞)
-    (a == 0.0) && b = b ⊓ (0.0..0.0)
-    (a == -1.0) && b = b ⊓ (-∞..0.0)
+    (a == 1.0) && b = b ⊓ (interval(0, Inf))
+    (a == 0.0) && b = b ⊓ (0.interval(0, 0).0)
+    (a == -1.0) && b = b ⊓ (-interval(Inf, 0).0)
 
     return a, b
 end
@@ -269,7 +269,7 @@ sign_rev(a,b) = sign_rev(promote(a,b)...)
     mul_rev_IEEE1788(b::Interval, c::Interval[, x::Interval])
 
 Reverse multiplication. Computes the preimage of ``c = x * b`` with respect to `x`.
-If `x` is not provided, then by default ``[-∞, ∞]`` is used.
+If `x` is not provided, then by default ``[-Inf, Inf]`` is used.
 See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
 
 ### Output
@@ -282,7 +282,7 @@ mul_rev_IEEE1788(b, c, x = entireinterval(b)) = mul_rev(c, x, b)[2]
     pow_rev1(b::Interval, c::Interval[, x::Interval])
 
 Reverse power 1. Computes the preimage of ``c=xᵇ`` with respect to `x`. If `x` is not provided,
-then byt default ``[-∞, ∞]`` is used.. See section 10.5.4 of the
+then byt default ``[-Inf, Inf]`` is used.. See section 10.5.4 of the
 IEEE 1788-2015 standard for interval arithmetic.
 
 ### Output
@@ -297,7 +297,7 @@ end
     pow_rev2(b::Interval, c::Interval[, x::Interval])
 
 Reverse power 2. Computes the preimage of ``c = aˣ`` with respect to `x`. If `x` is not provided, then
-byt default ``[-∞, ∞]`` is used. See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
+byt default ``[-Inf, Inf]`` is used. See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
 
 ### Output
 
@@ -319,11 +319,11 @@ See section 10.5.5 of the IEEE 1788-2015 standard for interval arithmetic.
 ### Example
 
 ```jldoctest
-julia> mul_rev_to_pair(-1..1, 1..2)
-([-∞, -1], [1, ∞])
+julia> mul_rev_to_pair(-interval(1, 1), interval(1, 2))
+([-Inf, -1], [1, Inf])
 
-julia> mul_rev_to_pair(1..2, 3..4)
-([1.5, 4], ∅)
+julia> mul_rev_to_pair(interval(1, 2), interval(3, 4))
+([1.5, 4], emptyinterval())
 
 """
 mul_rev_to_pair(b::Interval, c::Interval) = extended_div(c, b)

src/exponential.jl

@@ -11,7 +11,7 @@ end
 Reverse function for `exp`.
 """
 function exp_rev(y::Interval, x::Interval)
-    y_new = y ⊓ (0..∞)
+    y_new = y ⊓ (interval(0, Inf))
     x_new = x ⊓ log(y)
     return y_new, x_new
 end
@@ -20,7 +20,7 @@ end
 Reverse function for `exp2`.
 """
 function exp2_rev(y::Interval, x::Interval)
-    y_new = y ⊓ (0..∞)
+    y_new = y ⊓ (interval(0, Inf))
     x_new = x ⊓ log2(y)
 
     return y_new, x_new
@@ -30,7 +30,7 @@ end
 Reverse function for `exp10`.
 """
 function exp10_rev(y::Interval, x::Interval)
-    y_new = y ⊓ (0..∞)
+    y_new = y ⊓ (interval(0, Inf))
     x_new = x ⊓ log10(y)
 
     return y_new, x_new
@@ -40,7 +40,7 @@ end
 Reverse function for `expm1`.
 """
 function expm1_rev(y::Interval, x::Interval)
-    y_new = y ⊓ (-1.0..∞)
+    y_new = y ⊓ (interval(-1, Inf))
     x_new = x ⊓ log1p(y)
 
     return y_new, x_new

src/extrema.jl

@@ -12,10 +12,10 @@ function max_rev(a::Interval, b::Interval, c::Interval)  # a = max(b,c)
     (sup(b) < sup(c)) && (C_hi = min(sup(c),sup(a)))
 
     if isempty_interval(b)
-        isempty_interval(c) && (return a, ∅, ∅)
-        return a, ∅, interval(C_lo,C_hi)
+        isempty_interval(c) && (return a, emptyinterval(), emptyinterval())
+        return a, emptyinterval(), interval(C_lo,C_hi)
     else
-        isempty_interval(c) && (return a, interval(B_lo,B_hi), ∅)
+        isempty_interval(c) && (return a, interval(B_lo,B_hi), emptyinterval())
         return a, interval(B_lo,B_hi), interval(C_lo,C_hi)
     end
 end
@@ -35,10 +35,10 @@ function min_rev(a::Interval, b::Interval, c::Interval)
     (sup(b) < sup(c)) && (C_hi = min(sup(b),sup(a)))
 
     if isempty_interval(b)
-        isempty_interval(c) && (return a, ∅, ∅)
-        return a, ∅, interval(C_lo,C_hi)
+        isempty_interval(c) && (return a, emptyinterval(), emptyinterval())
+        return a, emptyinterval(), interval(C_lo,C_hi)
     else
-        isempty_interval(c) && (return a, interval(B_lo,B_hi), ∅)
+        isempty_interval(c) && (return a, interval(B_lo,B_hi), emptyinterval())
         return a, interval(B_lo,B_hi), interval(C_lo,C_hi)
     end
 end

src/hyperbolic.jl

@@ -2,7 +2,7 @@
     sinh_rev(c::Interval[, x::Interval])
 
 Reverse hyperbolic sine. Calculates the preimage of `a = sinh(x)`. If `x` is not provided, then
-byt default ``[-∞, ∞]`` is used. See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
+byt default ``[-Inf, Inf]`` is used. See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
 
 ### Output
 
@@ -21,7 +21,7 @@ end
     cosh_rev(c::Interval[, x::Interval])
 
 Reverse square root. Calculates the preimage of `a = cosh(x)`. If `x` is not provided, then
-byt default ``[-∞, ∞]`` is used. See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
+byt default ``[-Inf, Inf]`` is used. See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
 
 ### Output
 
@@ -31,7 +31,7 @@ The pair `(c, x_new)` where
 - `x_new` is the interval hull of the set ``{x ∈ b : cosh(x) ∈ a}``
 """
 function cosh_rev(y::Interval, x::Interval = entireinterval(y))
-    y_new = y ⊓ interval(1.,∞)
+    y_new = y ⊓ interval(1.,Inf)
     x = (x ⊓ acosh(y)) ⊔ (x ⊓ -acosh(y))
 
     return y_new, x
@@ -41,7 +41,7 @@ end
     tanh_rev(c::Interval[, x::Interval])
 
 Reverse square root. Calculates the preimage of `a = tanh(x)`. If `x` is not provided, then
-byt default ``[-∞, ∞]`` is used. See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
+byt default ``[-Inf, Inf]`` is used. See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
 
 ### Output
 

src/inverse_hyperbolic.jl

@@ -1,7 +1,7 @@
 """
 Reverse function for `asinh`.
 """
-function asinh_rev(y::Interval,x::Interval)
+function asinh_rev(y::Interval, x::Interval = entireinterval(y))
     x = x ⊓ sinh(y)
 
     return y, x
@@ -10,8 +10,8 @@ end
 """
 Reverse function for `acosh`.
 """
-function acosh_rev(y::Interval,x::Interval)
-    y_new = y ⊓ interval(0.0,∞)
+function acosh_rev(y::Interval, x::Interval = entireinterval(y))
+    y_new = y ⊓ interval(0.0,Inf)
     x = x ⊓ cosh(y)
 
     return y_new, x
@@ -20,7 +20,7 @@ end
 """
 Reverse function for `atanh`.
 """
-function atanh_rev(y::Interval,x::Interval)
+function atanh_rev(y::Interval, x::Interval = entireinterval(y))
     x = x ⊓ tanh(y)
 
     return y, x

src/inverse_trig.jl

@@ -25,7 +25,7 @@ end
 Reverse `acos`.
 """
 function acos_rev(y::Interval, x::Interval)
-        y_new = y ⊓ interval(0.0,sup(two_pi))
+        y_new = y ⊓ interval(0.0, sup(two_pi))
         x_new = x ⊓ cos(y_new)
 
         return y_new, x_new

src/powers.jl

@@ -11,7 +11,7 @@ function square_pos(X::IntervalBox)
 
     x, y = X
 
-    x = x ⊓ (0..∞)
+    x = x ⊓ (interval(0, Inf))
 
     y = y ⊓ (x^2)
     x = x ⊓ √y
@@ -27,7 +27,7 @@ function cube_pos(X::IntervalBox)  # contractor for y=x^3, x>=0
 
     x, y = X
 
-    x = x ⊓ (0..∞)
+    x = x ⊓ (interval(0, Inf))
 
     y = y ⊓ (x ^ 3)
     x = x ⊓ interval(inf(y) ^ (1/3), sup(y)^(1/3))   # not rigorous!

src/transformations.jl

@@ -68,15 +68,15 @@ function periodise(C, period)
     X -> begin
         x, y = X
 
-        x2 = -∞..∞
+        x2 = entireinterval()
         x2, y = C(IntervalBox(x2, y))
 
-        isempty(IntervalBox(x2, y)) && return(IntervalBox(∅, ∅))
+        isempty(IntervalBox(x2, y)) && return(IntervalBox(emptyinterval(), emptyinterval()))
 
         # periods where the periodization of x intersects with x2:
         periods = integer_contractor((x - x2) / period)
 
-        isempty_interval(periods) && return(IntervalBox(∅, ∅))
+        isempty_interval(periods) && return(IntervalBox(emptyinterval(), emptyinterval()))
 
         x3 = x2 + periods*period
         x = x ⊓ x3

src/trig.jl

@@ -9,7 +9,7 @@ function sin_main(X::IntervalBox)
     x, y = X
 
     x_range = interval(-sup(half_pi), sup(half_pi))
-    y_range = -1..1
+    y_range = interval(-1, 1)
 
     x = x ⊓ x_range
     y = y ⊓ y_range
@@ -40,7 +40,7 @@ sin!(X::IntervalBox) = periodise(sin_main, two_pi)(X) ⊔ periodise(sin_reverse,
     sin_rev(c::Interval[, x::Interval])
 
 Reverse sine. Calculates the preimage of `a = sin(x)`. If `x` is not provided, then
-byt default ``[-∞, ∞]`` is used. See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
+byt default ``[-Inf, Inf]`` is used. See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
 
 ### Output
 
@@ -68,7 +68,7 @@ function cos_main(X::IntervalBox)
     x, y = X
 
     x_range = interval(0, inf(interval(π)))
-    y_range = -1..1
+    y_range = interval(-1, 1)
 
     x = x ⊓ x_range
     y = y ⊓ y_range
@@ -100,7 +100,7 @@ cos!(X::IntervalBox) = periodise(cos_main, two_pi)(X) ⊔ periodise(cos_reverse,
     cos_rev(c::Interval[, x::Interval])
 
 Reverse cosine. Calculates the preimage of `a = cos(x)`. If `x` is not provided, then
-byt default ``[-∞, ∞]`` is used. See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
+byt default ``[-Inf, Inf]`` is used. See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
 
 ### Output
 
@@ -145,7 +145,7 @@ tan!(X::IntervalBox) = periodise(tan_main, interval(π))(X)
     tan_rev(c::Interval[, x::Interval])
 
 Reverse tangent. Calculates the preimage of `a = tan(x)`. If `x` is not provided, then
-byt default ``[-∞, ∞]`` is used. See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
+byt default ``[-Inf, Inf]`` is used. See section 10.5.4 of the IEEE 1788-2015 standard for interval arithmetic.
 
 ### Output