Floating-Point Representation, Overflow, and Underflow

/**
 * Fintech Math Bootcamp · Lesson 111 of 120
 * Floating-Point Representation, Overflow, and Underflow
 * Module 12: Statistical Computing and Reproducibility
 *
 * Scenario: Why a familiar decimal can fail exact equality
 * Rule:     binary floating point is finite, rounded representation
 *
 * Try it:   Is Number.EPSILON a universal absolute tolerance for all values?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/statistical-computing-and-reproducibility/floating-point-representation-overflow-and-underflow/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

export function lesson111() {
  const sum=.1+.2;
  const overflow=Number.MAX_VALUE*2;
  const underflow=Number.MIN_VALUE/2;
  const result={sum,exactlyPointThree:sum===.3,
    overflow:String(overflow),underflow,epsilon:Number.EPSILON};
  return result;
}

export const checkedResult = {"sum":0.30000000000000004,"exactlyPointThree":false,"overflow":"Infinity","underflow":0,"epsilon":2.220446049250313e-16};

// Run this file directly: npx tsx lessons/12-statistical-computing-and-reproducibility/111-floating-point-representation-overflow-and-underflow.ts
if (process.argv[1] && import.meta.url.endsWith(process.argv[1].replace(/\\/g, "/").split("/").pop()!)) {
  console.log(JSON.stringify(lesson111(), null, 2));
}

# Fintech Math Bootcamp · Lesson 111 of 120
# Floating-Point Representation, Overflow, and Underflow
# Module 12: Statistical Computing and Reproducibility
#
# Scenario: Why a familiar decimal can fail exact equality
# Rule:     binary floating point is finite, rounded representation
#
# Try it:   Is Number.EPSILON a universal absolute tolerance for all values?
#
# Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/statistical-computing-and-reproducibility/floating-point-representation-overflow-and-underflow/
# Free course:    https://courses.thefintechbuilder.com
# Synthetic teaching example, not financial advice or a production library.
#
# Run: python main.py

import json
import math
import sys


def js_string(x):
    # JavaScript's String(Infinity) is "Infinity"; Python's str() would give "inf".
    if math.isinf(x):
        return "Infinity" if x > 0 else "-Infinity"
    return repr(x)


def lesson_111():
    total = 0.1 + 0.2
    overflow = sys.float_info.max * 2  # Number.MAX_VALUE
    # Number.MIN_VALUE is the smallest subnormal, math.ulp(0.0);
    # sys.float_info.min is the smallest *normal* double, a different value.
    underflow = math.ulp(0.0) / 2
    return {
        "sum": total,
        "exactlyPointThree": total == 0.3,
        "overflow": js_string(overflow),
        "underflow": underflow,
        "epsilon": sys.float_info.epsilon,  # Number.EPSILON: gap between 1 and the next double
    }


if __name__ == "__main__":
    print(json.dumps(lesson_111(), indent=2))

/*
 * Fintech Math Bootcamp - Lesson 111 of 120
 * Floating-Point Representation, Overflow, and Underflow
 * Module 12: Statistical Computing and Reproducibility
 *
 * Scenario: Why a familiar decimal can fail exact equality
 * Rule:     binary floating point is finite, rounded representation
 *
 * Try it:   Is Number.EPSILON a universal absolute tolerance for all values?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/statistical-computing-and-reproducibility/floating-point-representation-overflow-and-underflow/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 *
 * Run: javac Main.java && java Main
 */

import java.util.*;

public class Main {
    static Map<String, Object> lesson111() {
        double sum = .1 + .2;
        double overflow = Double.MAX_VALUE * 2;
        double underflow = Double.MIN_VALUE / 2; // Java's MIN_VALUE is also the smallest subnormal
        double epsilon = Math.ulp(1.0); // Number.EPSILON: gap between 1 and the next double
        // String.valueOf(Infinity) is "Infinity" in Java, exactly like JavaScript's String(Infinity).
        return obj("sum", sum, "exactlyPointThree", sum == .3, "overflow", String.valueOf(overflow),
                "underflow", underflow, "epsilon", epsilon);
    }

    public static void main(String[] args) {
        System.out.println(toJson(lesson111(), ""));
    }

    // Minimal JSON writer: insertion-ordered Map, List, Number, Boolean, String and null.
    static String toJson(Object value, String indent) {
        if (value == null) return "null";
        if (value instanceof String) return quote((String) value);
        if (value instanceof Boolean) return value.toString();
        if (value instanceof Double) return formatNumber((Double) value);
        if (value instanceof Number) return value.toString();
        String inner = indent + "  ";
        StringBuilder sb = new StringBuilder();
        if (value instanceof Map) {
            Map<?, ?> map = (Map<?, ?>) value;
            if (map.isEmpty()) return "{}";
            sb.append("{\n");
            int i = 0;
            for (Map.Entry<?, ?> e : map.entrySet()) {
                sb.append(inner).append(quote(e.getKey().toString())).append(": ").append(toJson(e.getValue(), inner));
                sb.append(++i < map.size() ? ",\n" : "\n");
            }
            return sb.append(indent).append("}").toString();
        }
        List<?> list = (List<?>) value;
        if (list.isEmpty()) return "[]";
        sb.append("[\n");
        for (int i = 0; i < list.size(); i++) {
            sb.append(inner).append(toJson(list.get(i), inner)).append(i + 1 < list.size() ? ",\n" : "\n");
        }
        return sb.append(indent).append("]").toString();
    }

    // Integral doubles print without ".0", as JavaScript does; others use Java's round-trip form.
    static String formatNumber(double x) {
        if (x == Math.rint(x) && Math.abs(x) < 1e15) return Long.toString((long) x);
        return Double.toString(x);
    }

    static String quote(String s) {
        StringBuilder sb = new StringBuilder("\"");
        for (char c : s.toCharArray()) {
            if (c == '"' || c == '\\') sb.append('\\').append(c);
            else if (c < 0x20) sb.append(String.format("\\u%04x", (int) c));
            else sb.append(c);
        }
        return sb.append('"').toString();
    }

    // Builds an insertion-ordered object from alternating keys and values.
    static Map<String, Object> obj(Object... keysAndValues) {
        Map<String, Object> map = new LinkedHashMap<String, Object>();
        for (int i = 0; i < keysAndValues.length; i += 2) map.put((String) keysAndValues[i], keysAndValues[i + 1]);
        return map;
    }

    static List<Object> list(double... values) {
        List<Object> out = new ArrayList<Object>();
        for (double v : values) out.add(v);
        return out;
    }
}

// Fintech Math Bootcamp · Lesson 111 of 120
// Floating-Point Representation, Overflow, and Underflow
// Module 12: Statistical Computing and Reproducibility
//
// Scenario: Why a familiar decimal can fail exact equality
// Rule:     binary floating point is finite, rounded representation
//
// Try it:   Is Number.EPSILON a universal absolute tolerance for all values?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/statistical-computing-and-reproducibility/floating-point-representation-overflow-and-underflow/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
//
// Run: go run main.go

package main

import (
	"encoding/json"
	"fmt"
	"math"
	"strconv"
)

type Result struct {
	Sum               float64 `json:"sum"`
	ExactlyPointThree bool    `json:"exactlyPointThree"`
	Overflow          string  `json:"overflow"`
	Underflow         float64 `json:"underflow"`
	Epsilon           float64 `json:"epsilon"`
}

// jsString mirrors JavaScript's String(x); Go would format infinity as "+Inf".
func jsString(x float64) string {
	if math.IsInf(x, 1) {
		return "Infinity"
	}
	if math.IsInf(x, -1) {
		return "-Infinity"
	}
	return strconv.FormatFloat(x, 'g', -1, 64)
}

func lesson111() Result {
	// Variables, not constants: Go evaluates constant expressions exactly,
	// so a constant 0.1 + 0.2 would be exactly 0.3 and hide the rounding.
	a, b := 0.1, 0.2
	sum := a + b
	maxValue := math.MaxFloat64 // Number.MAX_VALUE
	overflow := maxValue * 2
	minValue := math.SmallestNonzeroFloat64 // Number.MIN_VALUE, the smallest subnormal
	underflow := minValue / 2
	epsilon := math.Nextafter(1, 2) - 1 // Number.EPSILON: gap between 1 and the next double
	return Result{
		Sum:               sum,
		ExactlyPointThree: sum == 0.3,
		Overflow:          jsString(overflow),
		Underflow:         underflow,
		Epsilon:           epsilon,
	}
}

func main() {
	out, err := json.MarshalIndent(lesson111(), "", "  ")
	if err != nil {
		panic(err)
	}
	fmt.Println(string(out))
}

// Fintech Math Bootcamp · Lesson 111 of 120
// Floating-Point Representation, Overflow, and Underflow
// Module 12: Statistical Computing and Reproducibility
//
// Scenario: Why a familiar decimal can fail exact equality
// Rule:     binary floating point is finite, rounded representation
//
// Try it:   Is Number.EPSILON a universal absolute tolerance for all values?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/statistical-computing-and-reproducibility/floating-point-representation-overflow-and-underflow/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
//
// Run: g++ -std=c++17 -O1 -o lesson main.cpp && ./lesson

#include <algorithm>
#include <charconv>
#include <cmath>
#include <cstdint>
#include <cstdio>
#include <deque>
#include <iostream>
#include <limits>
#include <map>
#include <optional>
#include <stdexcept>
#include <string>
#include <utility>
#include <variant>
#include <vector>

// Minimal JSON value: objects keep insertion order; numbers print in shortest round-trip form.
struct Json {
    enum class Kind { Null, Bool, Number, String, Array, Object };
    Kind kind = Kind::Null;
    bool flag = false;
    double number = 0.0;
    std::string text;
    std::vector<std::string> keys;  // object keys, parallel to items
    std::vector<Json> items;        // array items, or object values
};

Json jnull() { return Json{}; }
Json jbool(bool value) { Json j; j.kind = Json::Kind::Bool; j.flag = value; return j; }
Json jnum(double value) { Json j; j.kind = Json::Kind::Number; j.number = value; return j; }
Json jstr(const std::string& value) { Json j; j.kind = Json::Kind::String; j.text = value; return j; }
Json jarr(std::vector<Json> items) { Json j; j.kind = Json::Kind::Array; j.items = std::move(items); return j; }

Json jobj(std::vector<std::pair<std::string, Json>> fields) {
    Json j;
    j.kind = Json::Kind::Object;
    for (auto& field : fields) {
        j.keys.push_back(field.first);
        j.items.push_back(std::move(field.second));
    }
    return j;
}

Json jnums(const std::vector<double>& values) {
    std::vector<Json> items;
    for (double v : values) items.push_back(jnum(v));
    return jarr(std::move(items));
}

Json jnums(const std::vector<std::optional<double>>& values) {
    std::vector<Json> items;
    for (const auto& v : values) items.push_back(v ? jnum(*v) : jnull());
    return jarr(std::move(items));
}

std::string formatNumber(double value) {
    char buffer[64];
    auto result = std::to_chars(buffer, buffer + sizeof buffer, value);  // shortest round-trip form
    return std::string(buffer, result.ptr);
}

std::string quote(const std::string& text) {
    std::string out = "\"";
    for (char c : text) {
        if (c == '"' || c == '\\') {
            out += '\\';
            out += c;
        } else if (static_cast<unsigned char>(c) < 0x20) {
            char buffer[8];
            std::snprintf(buffer, sizeof buffer, "\\u%04x", static_cast<unsigned>(c));
            out += buffer;
        } else {
            out += c;
        }
    }
    return out + "\"";
}

void writeJson(std::ostream& out, const Json& value, const std::string& indent = "") {
    const std::string inner = indent + "  ";
    switch (value.kind) {
        case Json::Kind::Null: out << "null"; return;
        case Json::Kind::Bool: out << (value.flag ? "true" : "false"); return;
        case Json::Kind::Number: out << formatNumber(value.number); return;
        case Json::Kind::String: out << quote(value.text); return;
        case Json::Kind::Array:
        case Json::Kind::Object: {
            const bool isObject = value.kind == Json::Kind::Object;
            if (value.items.empty()) {
                out << (isObject ? "{}" : "[]");
                return;
            }
            out << (isObject ? "{\n" : "[\n");
            for (std::size_t i = 0; i < value.items.size(); ++i) {
                out << inner;
                if (isObject) out << quote(value.keys[i]) << ": ";
                writeJson(out, value.items[i], inner);
                out << (i + 1 < value.items.size() ? ",\n" : "\n");
            }
            out << indent << (isObject ? "}" : "]");
        }
    }
}

// Mirrors JavaScript's String(x); C++ streams would print "inf".
std::string jsString(double x) {
    if (std::isinf(x)) return x > 0 ? "Infinity" : "-Infinity";
    return formatNumber(x);
}

Json lesson111() {
    double sum = .1 + .2;
    double overflow = std::numeric_limits<double>::max() * 2;         // Number.MAX_VALUE * 2
    double underflow = std::numeric_limits<double>::denorm_min() / 2;  // Number.MIN_VALUE is the smallest subnormal
    double epsilon = std::numeric_limits<double>::epsilon();           // Number.EPSILON
    return jobj({{"sum", jnum(sum)},
                 {"exactlyPointThree", jbool(sum == .3)},
                 {"overflow", jstr(jsString(overflow))},
                 {"underflow", jnum(underflow)},
                 {"epsilon", jnum(epsilon)}});
}

int main() {
    writeJson(std::cout, lesson111());
    std::cout << "\n";
    return 0;
}

// Fintech Math Bootcamp · Lesson 111 of 120
// Floating-Point Representation, Overflow, and Underflow
// Module 12: Statistical Computing and Reproducibility
//
// Scenario: Why a familiar decimal can fail exact equality
// Rule:     binary floating point is finite, rounded representation
//
// Try it:   Is Number.EPSILON a universal absolute tolerance for all values?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/statistical-computing-and-reproducibility/floating-point-representation-overflow-and-underflow/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
//
// Run: rustc -O main.rs && ./main

#![allow(dead_code)]

/// Minimal JSON value; objects keep insertion order.
enum Json {
    Null,
    Bool(bool),
    Num(f64),
    Str(String),
    Arr(Vec<Json>),
    Obj(Vec<(String, Json)>),
}

fn obj(fields: Vec<(&str, Json)>) -> Json {
    Json::Obj(fields.into_iter().map(|(k, v)| (k.to_string(), v)).collect())
}

fn nums(values: &[f64]) -> Json {
    Json::Arr(values.iter().map(|&v| Json::Num(v)).collect())
}

fn optional_nums(values: &[Option<f64>]) -> Json {
    Json::Arr(values.iter().map(|v| v.map_or(Json::Null, Json::Num)).collect())
}

fn quote(text: &str) -> String {
    let mut out = String::from("\"");
    for c in text.chars() {
        match c {
            '"' => out.push_str("\\\""),
            '\\' => out.push_str("\\\\"),
            c if (c as u32) < 0x20 => out.push_str(&format!("\\u{:04x}", c as u32)),
            c => out.push(c),
        }
    }
    out.push('"');
    out
}

// Display for f64 prints the shortest string that round-trips, as JavaScript does.
fn write_json(value: &Json, indent: &str, out: &mut String) {
    let inner = format!("{}  ", indent);
    match value {
        Json::Null => out.push_str("null"),
        Json::Bool(b) => out.push_str(if *b { "true" } else { "false" }),
        Json::Num(n) => out.push_str(&format!("{}", n)),
        Json::Str(s) => out.push_str(&quote(s)),
        Json::Arr(items) if items.is_empty() => out.push_str("[]"),
        Json::Obj(fields) if fields.is_empty() => out.push_str("{}"),
        Json::Arr(items) => {
            out.push_str("[\n");
            for (i, item) in items.iter().enumerate() {
                out.push_str(&inner);
                write_json(item, &inner, out);
                out.push_str(if i + 1 < items.len() { ",\n" } else { "\n" });
            }
            out.push_str(indent);
            out.push(']');
        }
        Json::Obj(fields) => {
            out.push_str("{\n");
            for (i, (key, item)) in fields.iter().enumerate() {
                out.push_str(&inner);
                out.push_str(&quote(key));
                out.push_str(": ");
                write_json(item, &inner, out);
                out.push_str(if i + 1 < fields.len() { ",\n" } else { "\n" });
            }
            out.push_str(indent);
            out.push('}');
        }
    }
}

/// Mirrors JavaScript's String(x); Rust's Display would print "inf".
fn js_string(x: f64) -> String {
    if x.is_infinite() {
        (if x > 0.0 { "Infinity" } else { "-Infinity" }).to_string()
    } else {
        format!("{}", x)
    }
}

fn lesson_111() -> Json {
    let sum = 0.1_f64 + 0.2;
    let overflow = f64::MAX * 2.0; // Number.MAX_VALUE * 2
    // Number.MIN_VALUE is the smallest subnormal, f64::from_bits(1);
    // f64::MIN_POSITIVE is the smallest *normal* double, a different value.
    let underflow = f64::from_bits(1) / 2.0;
    obj(vec![
        ("sum", Json::Num(sum)),
        ("exactlyPointThree", Json::Bool(sum == 0.3)),
        ("overflow", Json::Str(js_string(overflow))),
        ("underflow", Json::Num(underflow)),
        ("epsilon", Json::Num(f64::EPSILON)), // Number.EPSILON
    ])
}

fn main() {
    let mut out = String::new();
    write_json(&lesson_111(), "", &mut out);
    println!("{}", out);
}

// Fintech Math Bootcamp · Lesson 111 of 120
// Floating-Point Representation, Overflow, and Underflow
// Module 12: Statistical Computing and Reproducibility
//
// Scenario: Why a familiar decimal can fail exact equality
// Rule:     binary floating point is finite, rounded representation
//
// Try it:   Is Number.EPSILON a universal absolute tolerance for all values?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/statistical-computing-and-reproducibility/floating-point-representation-overflow-and-underflow/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
//
// Run: dotnet run

using System;
using System.Collections.Generic;
using System.Globalization;
using System.Linq;
using System.Text.Encodings.Web;
using System.Text.Json;

var options = new JsonSerializerOptions { WriteIndented = true, Encoder = JavaScriptEncoder.UnsafeRelaxedJsonEscaping };
Console.WriteLine(JsonSerializer.Serialize(Lesson111(), options));

static object Lesson111()
{
    double sum = .1 + .2;
    double overflow = double.MaxValue * 2;
    // C#'s double.Epsilon is the smallest subnormal, i.e. JavaScript's Number.MIN_VALUE.
    double underflow = double.Epsilon / 2;
    // JavaScript's Number.EPSILON is the gap between 1 and the next double.
    double epsilon = Math.BitIncrement(1.0) - 1.0;
    // .NET formats infinity as "∞"; JavaScript's String(Infinity) is "Infinity".
    string overflowText = double.IsPositiveInfinity(overflow) ? "Infinity" : overflow.ToString(CultureInfo.InvariantCulture);
    return new { sum, exactlyPointThree = sum == .3, overflow = overflowText, underflow, epsilon };
}