The REPL and typechecker are available on the release page.
To install the libraries, you will can use To install the libraries, you can use ghcup to install cabal and GHC.
Then:
@@ -303,6 +318,29 @@The outer product ⊗ creates a table by applying some
+function.
It has type
+ > :ty \f.\x.\y. x f⊗ y
+(a → b → c) → Arr sh0 a → Arr sh1 b → Arr (sh0 ⧺ sh1) c > (frange 0 9 10) (*)⊗ (frange 0 9 10)
+[ [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
+, [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0]
+, [0.0, 2.0, 4.0, 6.0, 8.0, 10.0, 12.0, 14.0, 16.0, 18.0]
+, [0.0, 3.0, 6.0, 9.0, 12.0, 15.0, 18.0, 21.0, 24.0, 27.0]
+, [0.0, 4.0, 8.0, 12.0, 16.0, 20.0, 24.0, 28.0, 32.0, 36.0]
+, [0.0, 5.0, 10.0, 15.0, 20.0, 25.0, 30.0, 35.0, 40.0, 45.0]
+, [0.0, 6.0, 12.0, 18.0, 24.0, 30.0, 36.0, 42.0, 48.0, 54.0]
+, [0.0, 7.0, 14.0, 21.0, 28.0, 35.0, 42.0, 49.0, 56.0, 63.0]
+, [0.0, 8.0, 16.0, 24.0, 32.0, 40.0, 48.0, 56.0, 64.0, 72.0]
+, [0.0, 9.0, 18.0, 27.0, 36.0, 45.0, 54.0, 63.0, 72.0, 81.0] ] > (frange 0 4 5) [(x,y)]⊗ (frange 0 4 5)
+[ [(0.0*0.0), (0.0*1.0), (0.0*2.0), (0.0*3.0), (0.0*4.0)]
+, [(1.0*0.0), (1.0*1.0), (1.0*2.0), (1.0*3.0), (1.0*4.0)]
+, [(2.0*0.0), (2.0*1.0), (2.0*2.0), (2.0*3.0), (2.0*4.0)]
+, [(3.0*0.0), (3.0*1.0), (3.0*2.0), (3.0*3.0), (3.0*4.0)]
+, [(4.0*0.0), (4.0*1.0), (4.0*2.0), (4.0*3.0), (4.0*4.0)] ][(+)/ ((*)`((x::Vec n float)) y)]The rising factorial is defined as:
-x(n) = x(x + 1)⋯(x + n − 1)
+\[ x^{(n)} = x(x+1)\cdots +(x+n-1)\]
In apple this is
[(*)/ₒ 1 (⍳ x (x+y-1) 1)]/ₒ is a ternary operator, fold with seed.
[}.\`7 x]\p.\xs. (xs˙)'p⩪xsλx.λy.([{a⟜x->1;g⟜x->2;((a+g)%2,√(a*g))}]^:6 (x,y))->1Thence the complete +elliptic integral of the first kind:
+λk.
+{
+ agm ← λx.λy.([{a⟜x->1;g⟜x->2;((a+g)%2,√(a*g))}]^:6 (x,y))->1;
+ 𝜋%(2*agm 1 (√(1-k^2)))
+}Gauss’ approximation of the logarithm:
+λm.λx.
+{
+ amgm ← λx.λy.([{a⟜x->1;g⟜x->2;((a+g)%2,√(a*g))}]^:15 (x,y))->1;
+ -- m>2
+ 𝜋%(2*amgm 1 (0.5^(m-2)%x))-ℝ m*0.6931471805599453
+}\[ +{}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;x) = \sum_{n=0}^\infty +\frac{(a_1)_n\ldots (a_p)_n}{(b_1)_n\ldots +(b_q)_n}\frac{x^n}{n!}\]
+where \((x)_n\) is the rising factorial above.
+λa.λb.λz.
+{
+ rf ← [(*)/ₒ 1 (𝒻 x (x+y-1) (⌊y))]; fact ← rf 1;
+ Σ ← λN.λa. (+)/ₒ 0 (a'(⍳ 0 N 1)); Π ← [(*)/x];
+ Σ 30 (λn. {nn⟜ℝ n; (Π ((λa.rf a nn)'a)%Π((λb. rf b nn)'b))*(z^n%fact nn)})
+}We can use the REPL to inspect the type:
+ > :yank H math/hypergeometric.🍏
+ > :ty H
+Vec (i + 1) float → Vec (i + 1) float → float → float\[ x \mapsto \lambda - \lambda_0 \] +\[ y \mapsto \frac{1}{2} +\log{\left(\frac{1+\sin \phi}{1-\sin \phi}\right)} \]
+\𝜆₀.\𝜑.\𝜆.{a⟜sin.𝜑;(𝜆-𝜆₀,(_.((1+a)%(1-a)))%2)}λn.¬((∨)/ₒ #f ([(n|x)=0]'(⍳ 2 (⌊(√(ℝn))) 1)))Compute the radical of an integer n, ∏p|np
+class="math inline">\(n\), \(\displaystyle \prod_{p|n} p\)λn.
{ ni ⟜ ⌊(√(ℝn))
; isPrime ← λn.¬((∨)/ₒ #f ([(n|x)=0]'(⍳ 2 (⌊(√(ℝn))) 1)))
diff --git a/doc/apple-by-example.md b/doc/apple-by-example.md
index ce0617603..1bed2d455 100644
--- a/doc/apple-by-example.md
+++ b/doc/apple-by-example.md
@@ -12,7 +12,7 @@ The REPL and typechecker are available on the [release page](https://github.com/
### Libraries
-To install the libraries, you will can use [ghcup](https://www.haskell.org/ghcup/) to install cabal and GHC.
+To install the libraries, you can use [ghcup](https://www.haskell.org/ghcup/) to install cabal and GHC.
Then:
@@ -141,6 +141,40 @@ Reverse applied to a higher-dimensional array reverses elements (sub-arrays) alo
, [0, 1] ]
```
+## Outer Product
+
+The outer product `⊗` creates a table by applying some function.
+
+It has type
+
+```
+ > :ty \f.\x.\y. x f⊗ y
+(a → b → c) → Arr sh0 a → Arr sh1 b → Arr (sh0 ⧺ sh1) c
+```
+
+```
+ > (frange 0 9 10) (*)⊗ (frange 0 9 10)
+[ [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
+, [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0]
+, [0.0, 2.0, 4.0, 6.0, 8.0, 10.0, 12.0, 14.0, 16.0, 18.0]
+, [0.0, 3.0, 6.0, 9.0, 12.0, 15.0, 18.0, 21.0, 24.0, 27.0]
+, [0.0, 4.0, 8.0, 12.0, 16.0, 20.0, 24.0, 28.0, 32.0, 36.0]
+, [0.0, 5.0, 10.0, 15.0, 20.0, 25.0, 30.0, 35.0, 40.0, 45.0]
+, [0.0, 6.0, 12.0, 18.0, 24.0, 30.0, 36.0, 42.0, 48.0, 54.0]
+, [0.0, 7.0, 14.0, 21.0, 28.0, 35.0, 42.0, 49.0, 56.0, 63.0]
+, [0.0, 8.0, 16.0, 24.0, 32.0, 40.0, 48.0, 56.0, 64.0, 72.0]
+, [0.0, 9.0, 18.0, 27.0, 36.0, 45.0, 54.0, 63.0, 72.0, 81.0] ]
+```
+
+```
+ > (frange 0 4 5) [(x,y)]⊗ (frange 0 4 5)
+[ [(0.0*0.0), (0.0*1.0), (0.0*2.0), (0.0*3.0), (0.0*4.0)]
+, [(1.0*0.0), (1.0*1.0), (1.0*2.0), (1.0*3.0), (1.0*4.0)]
+, [(2.0*0.0), (2.0*1.0), (2.0*2.0), (2.0*3.0), (2.0*4.0)]
+, [(3.0*0.0), (3.0*1.0), (3.0*2.0), (3.0*3.0), (3.0*4.0)]
+, [(4.0*0.0), (4.0*1.0), (4.0*2.0), (4.0*3.0), (4.0*4.0)] ]
+```
+
# Examples
## Dot Product
@@ -233,6 +267,69 @@ To drop the first 6 elements:
\p.\xs. (xs˙)'p⩪xs
```
+## Numerics
+
+### Arithmetic-Geometric Mean
+
+```
+λx.λy.([{a⟜x->1;g⟜x->2;((a+g)%2,√(a*g))}]^:6 (x,y))->1
+```
+
+Thence the [complete elliptic integral of the first kind](https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind):
+
+```
+λk.
+{
+ agm ← λx.λy.([{a⟜x->1;g⟜x->2;((a+g)%2,√(a*g))}]^:6 (x,y))->1;
+ 𝜋%(2*agm 1 (√(1-k^2)))
+}
+```
+
+Gauss' approximation of the logarithm:
+
+```
+λm.λx.
+{
+ amgm ← λx.λy.([{a⟜x->1;g⟜x->2;((a+g)%2,√(a*g))}]^:15 (x,y))->1;
+ -- m>2
+ 𝜋%(2*amgm 1 (0.5^(m-2)%x))-ℝ m*0.6931471805599453
+}
+```
+
+### Hypergeometric Function
+
+$$ {}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;x) = \sum_{n=0}^\infty \frac{(a_1)_n\ldots (a_p)_n}{(b_1)_n\ldots (b_q)_n}\frac{x^n}{n!}$$
+
+where $(x)_n$ is the [rising factorial](#rising-factorial) above.
+
+```
+λa.λb.λz.
+{
+ rf ← [(*)/ₒ 1 (𝒻 x (x+y-1) (⌊y))]; fact ← rf 1;
+ Σ ← λN.λa. (+)/ₒ 0 (a'(⍳ 0 N 1)); Π ← [(*)/x];
+ Σ 30 (λn. {nn⟜ℝ n; (Π ((λa.rf a nn)'a)%Π((λb. rf b nn)'b))*(z^n%fact nn)})
+}
+```
+
+We can use the REPL to inspect the type:
+
+```
+ > :yank H math/hypergeometric.🍏
+ > :ty H
+Vec (i + 1) float → Vec (i + 1) float → float → float
+```
+
+## Geography
+
+### Mercator
+
+$$ x \mapsto \lambda - \lambda_0 $$
+$$ y \mapsto \frac{1}{2} \log{\left(\frac{1+\sin \phi}{1-\sin \phi}\right)} $$
+
+```
+\𝜆₀.\𝜑.\𝜆.{a⟜sin.𝜑;(𝜆-𝜆₀,(_.((1+a)%(1-a)))%2)}
+```
+
## Number Theory
### Primality Check
@@ -243,7 +340,7 @@ To drop the first 6 elements:
### Radical
-Compute the radical of an integer $n$, $\prod_{p|n} p$
+Compute the radical of an integer $n$, $\displaystyle \prod_{p|n} p$
```
λn.
diff --git a/docs/index.html b/docs/index.html
new file mode 120000
index 000000000..102919948
--- /dev/null
+++ b/docs/index.html
@@ -0,0 +1 @@
+../doc/apple-by-example.html
\ No newline at end of file