Double precision (64-bit) representation of numeric value in R (sign, exponent, significand)

Question

R FAQ states that:

The only numbers that can be represented exactly in R’s numeric type are integers and fractions whose denominator is a power of 2. All other numbers are internally rounded to (typically) 53 binary digits accuracy.

R uses IEEE 754 double-precision floating-point numbers which is

• 1 bit for sign
• 11 bits for exponent
• 52 bits for mantissa (or significand)

which sums up to 64-bits.

For the numeric number 0.1, R represents

sprintf("%.60f", 0.1)
[1] "0.100000000000000005551115123125782702118158340454101562500000"

Double (IEEE754 Double precision 64-bit) gives us this binary representation for 0.1 :

00111111 10111001 10011001 10011001 10011001 10011001 10011001 10011010

How we can get this representation in R and how does it relate to the output given by sprintf in our example?

Answer 1

The answer to the question raised by @chux in the comments is "yes"; R supports the %a format:

sprintf("%a", 0.1)
#> [1] "0x1.999999999999ap-4"

If you want to access the underlying bit pattern, you will have to reinterpret the double as a 64bit integer. For this task one can use C++ via Rcpp:

Rcpp::cppFunction('void print_hex(double x) {
    uint64_t y;
    static_assert(sizeof x == sizeof y, "Size does not match!");
    std::memcpy(&y, &x, sizeof y);
    Rcpp::Rcout << std::hex << y << std::endl;
}', plugins = "cpp11", includes = "#include <cstdint>")
print_hex(0.1)
#> 3fb999999999999a

This hexadecimal representation is identical to your binary representation. How does one get to the decimal representation?

• The first bit is zero, hence the sign is positive
• The exponent is 0x3fb, i.e. 1019 in decimal. Given the exponent bias this corresponds to an actual exponent of -4.
• The mantissa is 0x1999999999999a × 2^-52 including the implicit 1, i.e. 2^−52 × 7,205,759,403,792,794.

• In total this gives 2^−56 × 7,205,759,403,792,794:

  sprintf("%.60f", 2^-56 * 7205759403792794)
  #> [1] "0.100000000000000005551115123125782702118158340454101562500000"
  

Answer 2

Take for example 0.3 into account. Run in R console

> sprintf("%a", 0.3)
[1] "0x1.3333333333333p-2"

Mantissa or Significand

The hex representation 3333333333333 to binary would give us the mantissa (or significand) part. That is

0011001100110011001100110011001100110011001100110011

Exponent

The exponent part (11 bits) should be the offset from 2^(11-1) - 1 = 1023 so as the trailing 3 is p-2 (in the output given by sprintf) we have

-2 + 1023 = 1021

and its binary representation fixed in 11 bits is

01111111101

Sign

As for the sign bit, its 0 for positive and 1 otherwise

Double Precision Representation

So the complete representation is

0 | 01111111101 | 0011001100110011001100110011001100110011001100110011

---

Another example:

> sprintf("%a", -2.94)
[1] "-0x1.7851eb851eb85p+1"

# Mantissa or Significand
(7851eb851eb85) # base 16 
(0111100001010001111010111000010100011110101110000101) # base 2

# Exponent
1 + 1023 = 1024 # base 10
10000000000 # base 2

# So the complete representation is
1 | 10000000000 | 0111100001010001111010111000010100011110101110000101

Answer 3

From decimal to normalized double precion:

library(BMS)

from10toNdp <- function(my10baseNumber) {
out <- list()

# Handle special cases (0, Inf, -Inf)
if (my10baseNumber %in% c(0,Inf,-Inf)) {
if (my10baseNumber==0)    { out <- "0000000000000000000000000000000000000000000000000000000000000000" }
if (my10baseNumber==Inf)  { out <- "0111111111110000000000000000000000000000000000000000000000000000" }
if (my10baseNumber==-Inf) { out <- "1111111111110000000000000000000000000000000000000000000000000000" }
} else {

signBit <- 0 # assign initial value

from10to2 <- function(deciNumber) {
  binaryVector <- rep(0, 1 + floor(log(deciNumber, 2)))
  while (deciNumber >= 2) {
    theExpo <- floor(log(deciNumber, 2))
    binaryVector[1 + theExpo] <- 1
    deciNumber <- deciNumber - 2^theExpo  }
  binaryVector[1] <- deciNumber %% 2
  paste(rev(binaryVector), collapse = "")}

#Sign bit
if (my10baseNumber<0) { signBit <- 1 
} else { signBit <- 0 }

# Biased Exponent
BiasedExponent <- strsplit(from10to2(as.numeric(substr(sprintf("%a", my10baseNumber), which(strsplit( sprintf("%a", my10baseNumber), "")[[1]]=="p")+1, length( strsplit( sprintf("%a", my10baseNumber), "")[[1]]))) + 1023), "")[[1]] 
BiasedExponent <- paste(BiasedExponent, collapse='')
if (nchar(BiasedExponent)<11) {BiasedExponent <-  paste(c(  rep(0,11-nchar(BiasedExponent)), BiasedExponent),collapse='')    }

# Significand
significand <- BMS::hex2bin(substr( sprintf("%a", my10baseNumber) , which(strsplit( sprintf("%a", my10baseNumber), "")[[1]]=="x")+3, which(strsplit( sprintf("%a", my10baseNumber), "")[[1]]=="p")-1))

significand <- paste(significand, collapse='')
if (nchar(significand)<52) {significand <-  paste(c( significand,rep(0,52-nchar(significand))),collapse='')    }

out <- paste(c(signBit, BiasedExponent, significand), collapse='')
}

out
}

Hence,

from10toNdp(0.1)
# "0011111110111001100110011001100110011001100110011001100110011010"