2021-03-20 - still too much engineering

i started this year with a description how i started RE on device, only to learn that it's power cable was loose. and here we are – a quarter year later i did exactly the same. this time i've decided to memorize e (that i remember up to 2 decimal places) and pi (i remember 4 decimal places) up to 10 decimal places. what can i do with it? well – play with calc, of course. ;)

with e it was simple – ln(e)=1. so i started calc app on my phone: ln(2.7182818285)=1.00000000001506641592. sweet.

problem started with pi – sin(pi)=0. so i started calc app on my phone:

• sin(3)=0.1411…
• sin(3.14)=0.0015926…
• sin(3.1415)=0.054802050…
• sin(3.1415926535)=0.05480366…

crap! not only value is not equal to 0 on 2nd decimal place already, but it is RAISING with improved precision! problem seemed consistent on 3 different calculator apps – looks like hardware/API thing. what can i do? i've checked if CPU type – if it is 64-bit. turned out to be Cortex-A53, so 64-bit (i.e. not an issue of 32-bit and small floating-point precision). maybe some weirdo float vs. double in software? let's see on my PC:

#include <iostream>
#include <iomanip>
#include <cmath>
 
using namespace std;
 
int main()
{
  cout << "float:  sin(3.14000000) = " << std::fixed << std::setprecision(20) << sinf(3.14f) << endl;
  cout << "float:  sin(3.14159265) = " << std::fixed << std::setprecision(20) << sinf(3.14159265f) << endl;
  cout << endl;
 
  cout << "double: sin(3.14000000) = " << std::fixed << std::setprecision(20) << sin(3.14) << endl;
  cout << "double: sin(3.14159265) = " << std::fixed << std::setprecision(20) << sin(3.14159265) << endl;
}

and we have:

float:  sin(3.14000000) = 0.00159254798199981451
float:  sin(3.14159265) = -0.00000008742277657348

double: sin(3.14000000) = 0.00159265291648682823
double: sin(3.14159265) = 0.00000000358979302984

hmmm… sin(3.14) is very close to float/double results, but other results are more accurate. can't be that…

so i took another look at the calc app… and voila! trigonometric functions there use by default degrees – i.e. sin(180)=0. after switching to radians, sin(3.1415926535)=0.00000000008979323846. works. engineer – engineer never changes… ;)