Add an iterative rational implementation for exact results

Adds an implementation in golang using rational number types to
get exact results, and an iterative algorithm for performance
improvement compared to recursive approaches.

Outputs to CSV for easy data integration with tooling.

.gitignore

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+# Build go binary
+
+frog

README.md

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 # frog_problem
 Here is the frog problem code I wrote in an Edinburgh pub.
 
-Feel free to make better version. This one has not changed since I wrote it. V1 was the same code but saved before I changed it to work for different distances so I have not bothered upload it as well (it was still very unifinished).
+Feel free to make better version. This one has not changed since I wrote it. V1
+was the same code but saved before I changed it to work for different distances
+so I have not bothered upload it as well (it was still very unifinished).
 
 Video is here: https://youtu.be/ZLTyX4zL2Fc
+
+## Iterative Rational Implmentation
+
+Written in Golang by Taylor Wrobel
+
+Makes use of big rationals in golang to get exact expectation for all number of
+pads, up to and including the specified target number of pads (including target
+bank).  Makes use of an interative approach, which performs better than its
+recursive equivalent (calculating all results up to 1,000 pads takes ~12.5
+seconds for me).
+
+Outputs to STDOUT in CSV format with the following columns (output for `n=10`
+displayed in table form):
+
+n  | numerator | denominator | float64 representation
+-- | --------- | ----------- | ----------------------
+ 1 |         1 |           1 |               1.000000
+ 2 |         3 |           2 |               1.500000
+ 3 |        11 |           6 |               1.833333
+ 4 |        25 |          12 |               2.083333
+ 5 |       137 |          60 |               2.283333
+ 6 |        49 |          20 |               2.450000
+ 7 |       363 |         140 |               2.592857
+ 8 |       761 |         280 |               2.717857
+ 9 |      7129 |        2520 |               2.828968
+10 |      7381 |        2520 |               2.928968
+
+
+Building: `go build`
+Running: `./frog [n]` (`n` defaults to 10 if not provided)
+Outputting to file (on a UNIX system): `./frog 100 > output.csv`

go.mod

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+module wrobel.dev/frog
+
+go 1.12

main.go

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+package main
+
+import (
+	"encoding/csv"
+	"fmt"
+	"math/big"
+	"os"
+	"strconv"
+	"strings"
+)
+
+func main() {
+	// Parse input value from command line args.  Do simple validation.
+	n := 10
+	if len(os.Args) > 1 {
+		parsed, err := strconv.ParseInt(os.Args[1], 10, 64)
+		if err != nil || parsed < 1 {
+			panic("Input target must be a positive integer")
+		}
+		n = int(parsed)
+	}
+
+	// Likelihoods is a slice that's built up iteratively, as the expected value
+	// on lillypad n is dependent on the expected value for all lillypads from
+	// 1 to n-1.
+	likelihoods := make([]*big.Rat, n+1)
+	likelihoods[0] = big.NewRat(0, 1)
+
+	w := csv.NewWriter(os.Stdout)
+
+	for i := 1; i <= n; i++ {
+		expected := big.NewRat(0, 1)
+		for j := 0; j < i; j++ {
+			value := big.NewRat(0, 1)
+			value.Add(big.NewRat(1, 1), likelihoods[j])
+			value.Mul(value, big.NewRat(1, int64(i)))
+
+			expected.Add(expected, value)
+		}
+		likelihoods[i] = expected
+
+		numDenom := strings.Split(expected.String(), "/")
+		f, _ := expected.Float64()
+		w.Write([]string{fmt.Sprintf("%d", i), numDenom[0], numDenom[1], fmt.Sprintf("%f", f)})
+	}
+
+	w.Flush()
+}