How to find first pi beginning with 3.14159 - c++

Question

For determining how many terms are required for the first time getting pi that begins with 3.14159 I wrote the following program that calculates terms as (pi = 4 - 4/3 + 4/5 - 4/7 + ...).
My problem is that I reached 146063 terms as the result but when I checked, there are many pis that begin similarly before that.
//piEstimation.cpp
//estima mathematical pi and detrmin when
//to get a value beganing with 3.14159
#include <iostream>
#include <string>
#include <iomanip>
using namespace std;
int main(){
//initialize vars
double denominator{1.0};
double pi{0};
string piString;
double desiredPi;
int terms;
int firstDesiredTerm;
//format output floating point numbers to show 10 digits after
// decimal poin
cout << setprecision (10) <<fixed;
for (terms = 1; ; terms++){
if(0 == terms % 2){ //if term is even
pi -= 4/denominator;
}
else{ //if term is odd
pi += 4/denominator;
}
// draw table
cout << terms << "\t" << pi << endl;
//determin first time the pi begains with 3.14159
piString = to_string(pi).substr(0,7);
if(piString == "3.14159"){
firstDesiredTerm = terms;
desiredPi = pi;
break;
}
denominator += 2;
}//end for
cout << "The first time that pi value begans with 3.14159 "
<< "the number of terms are " << firstDesiredTerm << " and pi value is "<< desiredPi <<endl;
}//end main

Answer 1

A number x begins with 3.14159 if x >= 3.14159 && x < 3.1416. There is no need to use strings and compare characters. to_string has to use some kind of round operation. Without the string the algorithm finds the result after 136121 steps
#include <iostream>
#include <iomanip>
int main(){
//initialize vars
double denominator{1.0};
double pi{0};
double desiredPi;
int terms;
int firstDesiredTerm;
//format output floating point numbers to show 10 digits after
// decimal poin
std::cout << std::setprecision (20) << std::fixed;
for (terms = 1; ; terms++){
if(0 == terms % 2){ //if term is even
pi -= 4/denominator;
}
else{ //if term is odd
pi += 4/denominator;
}
// draw table
std::cout << terms << "\t" << pi << std::endl;
if(pi >= 3.14159 && pi < 3.1416){
firstDesiredTerm = terms;
desiredPi = pi;
break;
}
denominator += 2;
}//end for
std::cout << "The first time that pi value begans with 3.14159 "
<< "the number of terms are " << firstDesiredTerm
<< " and pi value is "<< desiredPi << std::endl;
}
Output:
The first time that pi value begans with 3.14159 the number of terms are 136121 and pi value is 3.14159999999478589672
Here you can see how to_string rounds the result:
#include <iostream>
#include <iomanip>
#include <string>
int main(){
std::cout << std::setprecision (20) << std::fixed;
std::cout << std::to_string(3.14159999999478589672) << '\n';
}
Output:
3.141600
You can read on cppreference
std::string to_string( double value ); Converts a floating point value to a string with the same content as what std::sprintf(buf, "%f", value) would produce for sufficiently large buf.
You can read on cppreference
f F Precision specifies the exact number of digits to appear after the decimal point character. The default precision is 6
That means that std::to_string rounds after 6 digits.