![]() ![]() The C languages offer a Quadruple-precision floating-point type, _int128, boasting up to 36 digits of precision.To compute an approximation with arbitrary (dynamic) precision, look into Java’s BigDecimal class. In other words, 3.141592653589793 is the probably the best approxmimation you can make out-of-the-box. Most programming languages offer a 32-bit and 64-bit (Double) floating-point type, allowing for 7 and 16 digits of precision respectively. Following this trend, the 4 occurs after the summation of 200 terms, then the 1 after 2000 terms, then the 5 after 20,000 terms, and so on.Īs exemplified by this trend, the number of operations needed to generate digits of pi grows exponentially with every digit each digit requires 10 times more computation than the previous. The second digit, 1, is set in stone after 20 = 2 * 10¹ iterations because 3.0916 is within 0.05 of Pi’s exact value. The first digit, 3, is finalized after 2 = 2 * 10⁰ iterations, since the approximation of 2.67 is within 0.5 of Pi’s exact value. ![]() When analyzing how many iterations are required various digit of precision, I accidentally discovered that every new digit of precision occurs after the summation of 2 times a power of 10 terms (amazing coincidence, right?)! Though I won’t walk through the formal derivation, the Taylor Series for 4 * arctan(1) is Rearranging this equation allows us to solve directly for Pi. Let’s start with a basic trig expression involving Pi. Our goal is to find the right Taylor Series such that the infinite summation converges to Pi itself. In calculus class, you probably studied the Taylor Series: an infinitely long summation that can model any function by matching all derivatives up to the nth degree. There is one method - one that I discovered on my own in high school - that isn’t as fast, but is most elegant in its simple mathematic groundwork. These methods are extraordinarily efficient, but somewhat complex. A number of algorithms are commonly used to approximate Pi, including the Gauss–Legendre, Borwein’s, and Salamin–Brent algorithms. Though the value of Pi is essential to nature, Pi does not come naturally to computers. Today, over 50 trillion digits have been found. ![]() I’ve always been amazed by how precisely computers can approximate Pi: the famous irrational constant 3.141592… plus infinitely more digits. ![]()
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