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Diffstat (limited to 'std/Math/Real.bruijn')
| -rw-r--r-- | std/Math/Real.bruijn | 68 |
1 files changed, 53 insertions, 15 deletions
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 |