How does computer convert floating point numbers to decimal string?

Question

When I run the following code, the output is accurately the number 2 ⁵⁰⁰ in decimal.

(g++ 5.3.1 on ubuntu)

#include<iostream>
#include<cmath>

using namespace std;

int main(){
    cout.precision(0);
    cout << fixed << pow(2.0,500.0);
    return 0;
}

I wonder how C++ converted this floating point number to its decimal string at such a high precision.

I know that 2 ⁵⁰⁰ can be accurately presented in IEEE 754 format. But I think mod 10 and divided by 10 can cause precision loss on floating point numbers. What algorithm is used when the conversion proceed?

Answer 1

Yes, there exists an exact double-precision floating-point representation for 2 ⁵⁰⁰ . You should not assume that pow(2.0,500.0) produces this value, though. There is no guarantee of accuracy for the function pow , and you may find SO questions that arose from pow(10.0, 2.0) not producing 100.0 , although the mathematical result was perfectly representable too.

But anyway, to answer your question, the conversion from the floating-point binary representation to decimal does not in general rely on floating-point operations, which indeed would be too inaccurate for the intended accuracy of the end result. In general, accurate conversion requires reliance on big integer arithmetics . In the case of 2 ⁵⁰⁰ , for instance, the naïve algorithm would be to repeatedly divide the big integer written in binary 1000…<500 zeroes in total>… by ten.

There are some cases where floating-point arithmetic can be used, for instance taking advantage of the fact that powers of 10 up to 10 ²³ are represented exactly in IEEE 754 double-precision. But correctly rounded conversion between binary floating-point and decimal floating-point always require big integer arithmetics in general, and this is particularly visible far away from 1.0.