diff options
Diffstat (limited to 'std/Math/Real.bruijn')
| -rw-r--r-- | std/Math/Real.bruijn | 63 |
1 files changed, 47 insertions, 16 deletions
diff --git a/std/Math/Real.bruijn b/std/Math/Real.bruijn index 9a4b1bd..d334117 100644 --- a/std/Math/Real.bruijn +++ b/std/Math/Real.bruijn @@ -7,7 +7,7 @@ :import std/Math/Rational Q :import std/Math N -# a Real is just a Unary → Rational! +# a Real is just a Number → Rational! # converts a balanced ternary number to a real number number→real [[Q.number→rational 1]] ⧗ Number → Real @@ -18,7 +18,27 @@ number→real [[Q.number→rational 1]] ⧗ Number → Real approx-eq? [[[Q.eq? (1 2) (0 2)]]] ⧗ Number → Real → Real → Boolean # TODO: bigger value?? -…≈?… approx-eq? (+100u) +…≈?… approx-eq? (+100) + +# returns true if a real number is greater than another +gt? [[[Q.gt? (1 2) (0 2)]]] ⧗ Number → Real → Real → Boolean + +…>?… gt? + +# returns true if a real number is less than another +lt? [[[Q.lt? (1 2) (0 2)]]] ⧗ Number → Real → Real → Boolean + +…<?… lt? + +# returns true if a real number is positive +positive? [\(gt? 0) (+0.0r)] ⧗ Number → Real → Boolean + +>?‣ positive? + +# returns true if a real number is negative +negative? [\(lt? 0) (+0.0r)] ⧗ Number → Real → Boolean + +<?‣ negative? # adds two real numbers add φ Q.add ⧗ Real → Real → Real @@ -93,7 +113,7 @@ dec \sub (+1.0r) ⧗ Real → Real # 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 +unary-exp [[0 &[[[[[[[2 1 0 (Q.add 4 (1 : N.--0)) N.++3]] pow fac]]]]] start [[[[1]]]]]] ⧗ Unary → Real pow N.mul 6 4 fac N.mul 1 3 start [0 (+1) (+1) (+1.0q) (+1)] @@ -114,7 +134,7 @@ lim-exp [[pow-n [(N.add 2 1) : N.--1] 0]] ⧗ Number → Real # 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} +# 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] @@ -123,7 +143,8 @@ ln [[[L.nth-iterate &[[[op]]] start 1] (--1 / ++1 0) [[[1]]]]] ⧗ Real → Real :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) +derive [[[[((3 (2 + 0)) - (3 2)) / 0] ((+1.0r) / conv) 0]]] ⧗ (Real → Real) → (Real → Real) + conv number→real (N.pow (+2) (N.mul (+2) 0)) # power function: real^real pow [[exp (0 ⋅ (ln 1))]] ⧗ Real → Real → Real @@ -151,9 +172,9 @@ 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) + op [0 N.++5 (Q.add 4 ((N.mul 3 2) : N.--1)) num-pow num-fac denom] + num-pow N.mul 3 (+2) + num-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 @@ -177,21 +198,31 @@ hypot [[sqrt ((0 ⋅ 0) + (1 ⋅ 1))]] ⧗ Real → Real → Real ψ -(~φ) # golden ratio from fibonacci convergence -φ* [(L.index (L.iterate &[[0 : (N.add 1 0)]] ((+0) : (+1))) 0) [[1 : N.--0]]] +φ* ++[(L.nth-iterate &[[0 : (N.add 1 0)]] ((+0) : (+1)) 0) [[1 : N.--0]]] # real fibonacci fib [((pow φ 0) - (pow ψ 0)) / (sqrt (+5.0r))] # 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 +atan* [[[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) + op [0 ((3 Q.add Q.sub) 4 (Q.div 2 1)) \3 num denom] + num 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 +atan [[[fallback] (1 0)]] ⧗ Real → Real + normal Q.sub π/2-pm conj-atan + π/2-pm Q.mul (Q.>?0 (+1.0q) (-1.0q)) (π/2 1) + conj-atan atan* [Q.div (+1.0q) 1] 1 + fallback Q.lt? Q.|0 (+1.0) (atan* [1] 1) normal + +# 2-argument arctan +atan2 [[[[[go]] (2 0) (1 0)]]] ⧗ Real → Real → Real + go Q.add a (Q.mul b (Q.add c d)) + z (+0.0q) + a Q.=?0 z (atan [Q.div 2 1] 2) + b Q.sub (+1.0q) (Q.<?1 (+2.0q) z) + c Q.<?0 (π 2) z + d Q.=?0 (π/2 2) z |