math/big: Add small complete example of big.Float usage

Updates #11241

Change-Id: I573be85d0cfcf410f6125ecd2be8a3d292c40bbb
Reviewed-on: https://go-review.googlesource.com/11245
Reviewed-by: Robert Griesemer <gri@golang.org>

src/math/big/example_test.go

@@ -7,6 +7,7 @@ package big_test
 import (
 	"fmt"
 	"log"
+	"math"
 	"math/big"
 )
 
@@ -83,3 +84,49 @@ func Example_fibonacci() {
 	// 1344719667586153181419716641724567886890850696275767987106294472017884974410332069524504824747437757
 	// false
 }
+
+// This example shows how to use big.Float to compute the square root of 2 with
+// a precision of 200 bits, and how to print the result as a decimal number.
+func Example_sqrt2() {
+	// We'll do computations with 200 bits of precision in the mantissa.
+	const prec = 200
+
+	// Compute the square root of 2 using Newton's Method. We start with
+	// an initial estimate for sqrt(2), and then iterate:
+	//     x_{n+1} = 1/2 * ( x_n + (2.0 / x_n) )
+
+	// Since Newton's Method doubles the number of correct digits at each
+	// iteration, we need at least log_2(prec) steps.
+	steps := int(math.Log2(prec))
+
+	// Initialize values we need for the computation.
+	two := new(big.Float).SetPrec(prec).SetInt64(2)
+	half := new(big.Float).SetPrec(prec).SetFloat64(0.5)
+
+	// Use 1 as the initial estimate.
+	x := new(big.Float).SetPrec(prec).SetInt64(1)
+
+	// We use t as a temporary variable. There's no need to set its precision
+	// since big.Float values with unset (== 0) precision automatically assume
+	// the largest precision of the arguments when used as the result (receiver)
+	// of a big.Float operation.
+	t := new(big.Float)
+
+	// Iterate.
+	for i := 0; i <= steps; i++ {
+		t.Quo(two, x)  // t = 2.0 / x_n
+		t.Add(x, t)    // t = x_n + (2.0 / x_n)
+		x.Mul(half, t) // x_{n+1} = 0.5 * t
+	}
+
+	// We can use the usual fmt.Printf verbs since big.Float implements fmt.Formatter
+	fmt.Printf("sqrt(2) = %.50f\n", x)
+
+	// Print the error between 2 and x*x.
+	t.Mul(x, x) // t = x*x
+	fmt.Printf("error = %e\n", t.Sub(two, t))
+
+	// Output:
+	// sqrt(2) = 1.41421356237309504880168872420969807856967187537695
+	// error = 0.000000e+00
+}