Pi and e In Binary

Some people are curious about the binary representations of the mathematical constants pi and e. Mathematically, they’re like every other irrational number — infinite strings of 0s and 1s (with no discernible pattern). In a computer, they’re finite, making them only approximations to their true values. I will show you what their approximations look like in five different levels of binary floating-point precision.

The first 43 bits of pi and e

Binary Floating-Point Formats

In binary floating-point, infinitely precise values are rounded to finite precision. Here’s how rounding works in five different levels of precision:

• In half-precision, values are rounded to 11 significant bits.
• In single-precision, values are rounded to 24 significant bits.
• In double-precision, values are rounded to 53 significant bits.
• In extended-precision, values are rounded to 64 significant bits.
• In quadruple-precision, values are rounded to 113 significant bits.

The rounding rule used most often in practice is round-to-nearest, round-half-to-even; that’s the rule I will use. For pi and e, there are no “half to even” cases, since their binary expansions are infinite. This makes the rounding rule simple: if the rounding bit is 0, round down; if the rounding bit is 1, round up.

I will show the correctly rounded approximations of pi and e in these five formats.

Pi (π)

Here are the first 50 decimal digits of pi:

3.1415926535897932384626433832795028841971693993751…

Here are the first 128 bits of pi:

11.00100100001111110110101010001000100001011010001100001000110100
1100010011000110011000101000101110000000110111000001110011010001...

Here are the 128 bits again, with the rounding bit for each level of precision highlighted (bits 12, 25, 54, 65, and 114):

11.00100100001111110110101010001000100001011010001100001000110100
1100010011000110011000101000101110000000110111000001110011010001...

Here are the correctly rounded values of pi in each of the five levels of precision, shown in normalized binary scientific notation and as hexadecimal floating-point constants:

Half-Precision

pi = 1.1001001 x 21
   = 0x1.92p+1

This equals 3.140625 in decimal (all binary floating-point numbers have exact decimal representations), which approximates pi accurately to about 4 decimal digits.

(You can verify this conversion by hand, by adding the powers of two corresponding to the positions of the 1 bits: 11.001001 = 2¹ + 2⁰ + 2^-3 + 2^-6 = 3.140625.)

Single-Precision

pi = 1.10010010000111111011011 x 21
   = 0x1.921fb6p+1

This equals 3.1415927410125732421875, which approximates pi accurately to about 8 decimal digits.

Double-Precision

pi = 1.1001001000011111101101010100010001000010110100011 x 21
   = 0x1.921fb54442d18p+1

This equals 3.141592653589793115997963468544185161590576171875, which approximates pi accurately to about 16 decimal digits.

Extended-Precision

pi = 1.100100100001111110110101010001000100001011010001100001000110
     101 x 21
   = 0x1.921fb54442d1846ap+1

This equals

3.14159265358979323851280895940618620443274267017841339111328125

which approximates pi accurately to about 20 decimal digits.

Quadruple-Precision

pi = 1.100100100001111110110101010001000100001011010001100001000110
     1001100010011000110011000101000101110000000110111 x 21
   = 0x1.921fb54442d18469898cc51701b8p+1

This equals

3.141592653589793238462643383279502797479068098137295573004504331
874296718662975536062731407582759857177734375

which approximates pi accurately to about 35 decimal digits.

Euler’s Number (e)

Here are the first 50 decimal digits of e:

2.7182818284590452353602874713526624977572470936999…

Here are the first 128 bits of e:

10.10110111111000010101000101100010100010101110110100101010011010
1010111111011100010101100010000000100111001111010011110011110001...

Here are the 128 bits again, with the rounding bits for each level of precision highlighted:

10.10110111111000010101000101100010100010101110110100101010011010
1010111111011100010101100010000000100111001111010011110011110001...

Here are the correctly rounded values of e in each of the five levels of precision:

Half-Precision

e = 1.010111 x 21
  = 0x1.5cp+1

This equals 2.71875, which approximates e accurately to about 4 decimal digits.

Single-Precision

e = 1.010110111111000010101 x 21 
  = 0x1.5bf0a8p+1

This equals 2.71828174591064453125, which approximates e accurately to about 8 decimal digits.

Double-Precision

e = 1.0101101111110000101010001011000101000101011101101001 x 21
  = 0x1.5bf0a8b145769p+1

This equals 2.718281828459045090795598298427648842334747314453125, which approximates e accurately to about 16 decimal digits.

Extended-Precision

e = 1.010110111111000010101000101100010100010101110110100101010011
    011 x 21
  = 0x1.5bf0a8b145769536p+1

This equals

2.71828182845904523542816810799394033892895095050334930419921875

which approximates e accurately to about 20 decimal digits.

Quadruple-Precision

e = 1.010110111111000010101000101100010100010101110110100101010011
    010101011111101110001010110001000000010011100111101 x 21
  = 0x1.5bf0a8b1457695355fb8ac404e7ap+1

This equals

2.718281828459045235360287471352662314358421867193548862669230860
32766716801933881697550532408058643341064453125

which approximates e accurately to about 34 decimal digits.