Add autodiff example

examples/src/main/kotlin/space/kscience/kmath/expressions/autodiff.kt

@@ -0,0 +1,91 @@
+/*
+ * Copyright 2018-2023 KMath contributors.
+ * Use of this source code is governed by the Apache 2.0 license that can be found in the license/LICENSE.txt file.
+ */
+
+package space.kscience.kmath.expressions
+
+import space.kscience.kmath.UnstableKMathAPI
+// Only kmath-core is needed.
+
+// Let's declare some variables
+val x by symbol
+val y by symbol
+val z by symbol
+
+@OptIn(UnstableKMathAPI::class)
+fun main() {
+    // Let's define some random expression.
+    val someExpression = Double.autodiff.differentiate {
+        // We bind variables `x` and `y` to the builder scope,
+        val x = bindSymbol(x)
+        val y = bindSymbol(y)
+
+        // Then we use the bindings to define expression `xy + x + y - 1`
+        x * y + x + y - 1
+    }
+
+    // Then we can evaluate it at any point ((-1, -1) in the case):
+    println(someExpression(mapOf(x to -1.0, y to -1.0)))
+    // >>> -2.0
+
+    // We can also construct its partial derivatives:
+    val dxExpression = someExpression.derivative(x) // ∂/∂x. Must be `y+1`
+    val dyExpression = someExpression.derivative(y) // ∂/∂y. Must be `x+1`
+    val dxdxExpression = someExpression.derivative(x, x) // ∂^2/∂x^2. Must be `0`
+
+    // We can evaluate them as well
+    println(dxExpression(mapOf(x to 57.0, y to 6.0)))
+    // >>> 7.0
+    println(dyExpression(mapOf(x to -1.0, y to 179.0)))
+    // >>> 0.0
+    println(dxdxExpression(mapOf(x to 239.0, y to 30.0)))
+    // >>> 0.0
+
+    // You can also provide extra arguments that obviously won't affect the result:
+    println(dxExpression(mapOf(x to 57.0, y to 6.0, z to 42.0)))
+    // >>> 7.0
+    println(dyExpression(mapOf(x to -1.0, y to 179.0, z to 0.0)))
+    // >>> 0.0
+    println(dxdxExpression(mapOf(x to 239.0, y to 30.0, z to 100_000.0)))
+    // >>> 0.0
+
+    // But in case you forgot to specify bound symbol's value, exception is thrown:
+    println( runCatching { someExpression(mapOf(z to 4.0)) } )
+    // >>> Failure(java.lang.IllegalStateException: Symbol 'x' is not supported in ...)
+
+    // The reason is that the expression is evaluated lazily,
+    // and each `bindSymbol` operation actually substitutes the provided symbol with the corresponding value.
+
+    // For example, let there be an expression
+    val simpleExpression = Double.autodiff.differentiate {
+        val x = bindSymbol(x)
+        x pow 2
+    }
+    // When you evaluate it via
+    simpleExpression(mapOf(x to 1.0, y to 57.0, z to 179.0))
+    // lambda above has the context of map `{x: 1.0, y: 57.0, z: 179.0}`.
+    // When x is bound, you can think of it as substitution `x -> 1.0`.
+    // Other values are unused which does not make any problem to us.
+    // But in the case the corresponding value is not provided,
+    // we cannot bind the variable. Thus, exception is thrown.
+
+    // There is also a function `bindSymbolOrNull` that fixes the problem:
+    val fixedExpression = Double.autodiff.differentiate {
+        val x = bindSymbolOrNull(x) ?: const(8.0)
+        x pow -2
+    }
+    println(fixedExpression())
+    // >>> 0.015625
+    // It works!
+
+    // The expression provides a bunch of operations:
+    // 1. Constant bindings (via `const` and `number`).
+    // 2. Variable bindings (via `bindVariable`, `bindVariableOrNull`).
+    // 3. Arithmetic operations (via `+`, `-`, `*`, and `-`).
+    // 4. Exponentiation (via `pow` or `power`).
+    // 5. `exp` and `ln`.
+    // 6. Trigonometrical functions (`sin`, `cos`, `tan`, `cot`).
+    // 7. Inverse trigonometrical functions (`asin`, `acos`, `atan`, `acot`).
+    // 8. Hyperbolic functions and inverse hyperbolic functions.
+}