3 files changed, 92 insertions, 18 deletions
diff --git a/std/Math/Complex.bruijn b/std/Math/Complex.bruijn
index a20717f..d3b5a7d 100644
--- a/std/Math/Complex.bruijn
+++ b/std/Math/Complex.bruijn
@@ -5,9 +5,29 @@
 :import std/Pair .
 :import std/Math/Real R
 
-ι (+0.0r) : (+1.0r)
+ι (+0.0+1.0i)
 
-# returns true of two complex numbers are equal approximately
+# converts a balanced ternary number to a complex number
+number→complex [[0 (R.number→real 1) (+0.0r)]] ⧗ Number → Complex
+
+:test (number→complex (+5)) ((+5.0+0.0i))
+
+# returns real part of a complex number
+real fst ⧗ Complex → Real
+
+:test (real (+5.0+2.0i)) ((+5.0r))
+
+# returns imaginary part of a complex number
+imag snd ⧗ Complex → Real
+
+:test (imag (+5.0+2.0i)) ((+2.0r))
+
+# approximates complex number by turning it into a pair of rationals
+approx &[[[(2 0) : (1 0)]]] ⧗ Complex → Number → (Pair Rational Rational)
+
+:test (approx (+5.0+2.0i) (+2)) ((+5.0q) : (+2.0q))
+
+# returns true if two complex numbers are equal approximately
 approx-eq? [[[R.approx-eq? 2 (1 2) (0 2)]]] ⧗ Number → Complex → Complex → Boolean
 
 # TODO: bigger value (higher performance first!)
@@ -48,3 +68,14 @@ invert &[[p : q]] ⧗ Complex → Complex
 	q R.div 0 (R.add (R.mul 1 1) (R.mul 0 0))
 
 ~‣ invert
+
+# ---
+
+:import std/List L
+
+# power function: complex^number
+pow-n [L.nth-iterate (mul 0) (+1.0+0.0i)] ⧗ Complex → Number → Complex
+
+ln [p : q] ⧗ Complex → Complex
+	p R.ln (&R.hypot 0)
+	q &R.atan2 0
diff --git a/std/Math/Rational.bruijn b/std/Math/Rational.bruijn
index 54c3a88..f1509b0 100644
--- a/std/Math/Rational.bruijn
+++ b/std/Math/Rational.bruijn
@@ -3,6 +3,7 @@
 
 # (p : q) ⇔ (1 / (q + 1))
 
+:import std/Logic .
 :import std/Combinator .
 :import std/Logic .
 :import std/Pair .
@@ -91,6 +92,10 @@ div [[1 ⋅ ~0]] ⧗ Rational → Rational → Rational
 :test ((+8.0) / (+4.0) =? (+2.0)) (true)
 :test ((+18.0) / (+12.0) =? (+1.5)) (true)
 
+gt? &[[&[[(N.gt? 1 3) ⋀? (N.gt? 0 2)]]]]
+
+lt? &[[&[[(N.lt? 1 3) ⋀? (N.lt? 0 2)]]]]
+
 # increments a rational number
 inc add (+1.0) ⧗ Rational → Rational
 
@@ -106,4 +111,4 @@ dec \sub (+1.0) ⧗ Rational → Rational
 :import std/List L
 
 # power function: rational^number
-pow-n [L.…!!… (L.iterate (mul 0) (+1.0))] ⧗ Rational → Number → Rational
+pow-n [L.nth-iterate (mul 0) (+1.0)] ⧗ Rational → Number → Rational
diff --git a/std/Math/Real.bruijn b/std/Math/Real.bruijn
index 49f4a92..28bc512 100644
--- a/std/Math/Real.bruijn
+++ b/std/Math/Real.bruijn
@@ -91,41 +91,79 @@ dec \sub (+1.0r) ⧗ Real → Real
 
 :import std/List L
 
-# ∑(1/n^2)
-# converges to 1.6449...
-unary-1/n² [0 &[[(Q.add 1 op) : N.++0]] start [[0]]] ⧗ Real
-	op (+1) : N.--(N.mul 0 0)
-	start (+0.0f) : (+1)
-
-# e^x using Taylor expansion: ∑(x^n/n!) = e^x
+# e^x using Taylor expansion
+# tex: \sum_{n=0}^\infty \frac{x^n}{n!}
 unary-exp [[0 &[[[[[[[2 1 0 (Q.add 4 (1 : N.--0)) N.++3]] pow fac]]]]] start [[[[1]]]]]] ⧗ Number → Real
 	pow N.mul 6 4
 	fac N.mul 1 3
-	start [0 (+1) (+1) (+1.0f) (+1)]
+	start [0 (+1) (+1) (+1.0q) (+1)]
 
 # equivalent to unary-exp but with ternary index using infinite list iteration
-exp [[L.index 0 (L.iterate &[[[[op]]]] start) [[[[1]]]] 0]] ⧗ Real → Real
+exp [[L.nth-iterate &[[[[op]]]] start 0 [[[[1]]]] 0]] ⧗ Real → Real
 	start [0 (+1.0r) (+1.0r) (+1.0r) (+1)]
 	op [[[2 1 0 (4 + (1 / 0)) N.++3]] pow fac]
 		pow 6 ⋅ 4
 		fac (number→real 1) ⋅ 3
 
 # power function: real^number
-pow-n [L.…!!… (L.iterate (mul 0) (+1.0r))] ⧗ Real → Number → Real
+pow-n [L.nth-iterate (mul 0) (+1.0r)] ⧗ Real → Number → Real
 
-# e^x using infinite limit: lim (1+x/n)^n, n to ∞
+# e^x using infinite limit
+# tex: \lim_{n\to\infty}(1+x/n)^n
 lim-exp [[pow-n [(N.add 2 1) : N.--1] 0]] ⧗ Number → Real
 
-# log_e using Taylor expansion
-ln [[[L.index 1 (L.iterate &[[[op]]] start)] (--1 / ++1 0) [[[1]]]]] ⧗ Real → Real
-	start [0 1 (+0.0f) (+0)]
+# natural logarithm using Taylor expansion
+# tex: \sum_{n=0}^\infty\frac{2}{2n+1}(\frac{x-1}{x+1})^{2n+1}
+# error: O((x-1)/2)^{2n+1}
+ln [[[L.nth-iterate &[[[op]]] start 1] (--1 / ++1 0) [[[1]]]]] ⧗ Real → Real
+	start [0 1 (+0.0q) (+0)]
 	op [0 pow (Q.add 2 go) N.++1]
-		pow Q.mul 3 (Q.pow-n 4 (+2))
+		pow Q.mul 3 (Q.mul 4 4)
 		go Q.mul ((+2) : (N.mul (+2) 1)) 3
 
+:test (Q.eq? (ln (+2.0r) (+2)) ((+168) : (+242))) (true)
+
 derive [[[[((3 (0 + 1)) - (3 0)) / 1]] ((+1.0r) / 0) 0]] ⧗ (Real → Real) → (Real → Real)
 
 # power function: real^real
 pow [[exp (0 ⋅ (ln 1))]] ⧗ Real → Real → Real
 
 …**… pow
+
+# square root by x^{0.5}
+sqrt* \pow (+0.5r) ⧗ Real → Real
+
+# Newton's/Heron's method, quadratic convergence
+# tex: x_{n+1}=\frac{x_n+a/x_n}{2}
+sqrt [[y [[[N.=?0 guess go]]] (1 0) 0]] ⧗ Real → Real
+	guess (+1.0q)
+	go [Q.div (Q.add 0 (Q.div 2 0)) (+2.0q)] (2 1 N.--0)
+
+# hypotenuse
+hypot [[sqrt ((0 ⋅ 0) + (1 ⋅ 1))]] ⧗ Real → Real → Real
+
+# tex: \sum_{n=0}^\infty\frac{2^n n!^2}{(2n+1)!}
+π/2 [L.nth-iterate &[[[[[op]]]]] start 0 [[[[[3]]]]]] ⧗ Real
+	start [0 (+1) (+0.0q) (+1) (+1) (+1)]
+	op [0 N.++5 (Q.add 4 ((N.mul 3 2) : N.--1)) enum-pow enum-fac denom]
+		enum-pow N.mul 3 (+2)
+		enum-fac N.mul 2 (N.mul 5 5)
+		denom [N.mul 2 (N.mul 0 N.++0)] (N.mul (+2) 5)
+
+# ratio of circle's circumference to its diameter
+# TODO: Gauss-Legendre
+π π/2 ⋅ (+2.0r) ⧗ Real
+
+# arctan by Taylor expansion, only for |x|<=1
+# tex: \sum_{n=0}^\infty(-1)^n \frac{x^{2n+1}}{2n+1}
+arctan* [[[L.nth-iterate &[[[[op]]]] start 1] (1 0) [[[[3]]]]]] ⧗ Real → Real
+	start [0 1 [[0]] (Q.pow-n 1 (+3)) (+3.0q)]
+	op [0 ((3 Q.add Q.sub) 4 (Q.div 2 1)) \3 enum denom]
+		enum Q.mul 2 (Q.mul 5 5)
+		denom Q.add 1 (+2.0q)
+
+# actual arctan for arbitrary x
+arctan [[Q.sub (π/2 0) (arctan* [Q.div (+1.0q) (2 1)] 0)]] ⧗ Real → Real
+
+# TODO: atan2
+atan2 [0] ⧗ Real → Real