Hexadecimal floating-point (HFP) is a format for encoding floating-point numbers first introduced on the IBM System/360 computers, and supported on subsequent machines based on that architecture, as well as machines which were intended to be application-compatible with System/360. In comparison to IEEE 754 floating point, the HFP format has a longer significand, and a shorter exponent. All HFP formats have 7 bits of exponent with a bias of 64. The normalized range of representable numbers is from 16−65 to 1663 (approx. 5.39761 × 10−79 to 7.237005 × 1075). The number is represented as the following formula: (−1)sign × 0.significand × 16exponent−64.
Single-precision 32-bit A single-precision HFP number (called "short" by IBM) is stored in a 32-bit word:
In this format the initial bit is not suppressed, and the radix (hexadecimal) point is set to the left of the significand (fraction in IBM documentation and the figures). Since the base is 16, the exponent in this form is about twice as large as the equivalent in IEEE 754, in order to have similar exponent range in binary, 9 exponent bits would be required.
Example Consider encoding the value −118.625 as an HFP single-precision floating-point value. The value is negative, so the sign bit is 1. The value 118.62510 in binary is 1110110.1012. This value is normalized by moving the radix point left four bits (one hexadecimal digit) at a time until the leftmost digit is zero, yielding 0.011101101012. The remaining rightmost digits are padded with zeros, yielding a 24-bit fraction of .0111 0110 1010 0000 0000 00002. The normalized value moved the radix point two hexadecimal digits to the left, yielding a multiplier and exponent of 16+2. A bias of +64 is added to the exponent (+2), yielding +66, which is 100 00102. Combining the sign, exponent plus bias, and normalized fraction produces this encoding:
In other words, the number represented is −0.76A00016 × 1666 − 64 = −0.4633789… × 16+2 = −118.625
Largest representable number
The number represented is +0.FFFFFF16 × 16127 − 64 = (1 − 16−6) × 1663 ≈ +7.2370051 × 1075
Smallest positive normalized number
The number represented is +0.116 × 160 − 64 = 16−1 × 16−64 ≈ +5.397605 × 10−79.
Zero
Zero (0.0) is represented in normalized form as all zero bits, which is arithmetically the value +0.016 × 160 − 64 = +0 × 16−64 ≈ +0.000000 × 10−79 = 0. Given a fraction of all-bits zero, any combination of positive or negative sign bit and a non-zero biased exponent will yield a value arithmetically equal to zero. However, the normalized form generated for zero by CPU hardware is all-bits zero. This is true for all three floating-point precision formats. Addition or subtraction with other exponent values can lose precision in the result.
Precision issues Since the base is 16, there can be up to three leading zero bits in the binary significand. That means when the number is converted into binary, there can be as few as 21 bits of precision. Because of the "wobbling precision" effect, this can cause some calculations to be very inaccurate. This has caused considerable criticism. A good example of the inaccuracy is representation of decimal value 0.1. It has no exact binary or hexadecimal representation. In hexadecimal format, it is represented as 0.19999999...16 or 0.0001 1001 1001 1001 1001 1001 1001...2, that is:
This has only 21 bits, whereas the binary version has 24 bits of precision. Six hexadecimal digits of precision is roughly equivalent to six decimal digits (i.e. (6 − 1) log10(16) ≈ 6.02). A conversion of single precision hexadecimal float to decimal string would require at least 9 significant digits (i.e. 6 log10(16) + 1 ≈ 8.22) in order to convert back to the same hexadecimal float value.
Double-precision 64-bit The double-precision HFP format (called "long" by IBM) is the same as the "short" format except that the fraction field is wider and the double-precision number is stored in a double word (8 bytes):
The exponent for this format covers only about a quarter of the range as the corresponding IEEE binary format. 14 hexadecimal digits of precision is roughly equivalent to 17 decimal digits. A conversion of double precision hexadecimal float to decimal string would require at least 18 significant digits in order to convert back to the same hexadecimal float value.
Extended-precision 128-bit Called extended-precision by IBM, a quadruple-precision HFP format was added to the System/370 series and was available on some S/360 models (S/360-85, -195, and others by special request or simulated by OS software). The extended-precision fraction field is wider, and the extended-precision number is stored as two double words (16 bytes):
28 hexadecimal digits of precision is roughly equivalent to 32 decimal digits. A conversion of extended precision HFP to decimal string would require at least 35 significant digits in order to convert back to the same HFP value. The stored exponent in the low-order part is 14 less than the high-order part, unless this would be less than zero.
Arithmetic operations Available arithmetic operations are add and subtract, both normalized and unnormalized, and compare. Prenormalization is done based on the exponent difference. Multiply and divide prenormalize unnormalized values, and truncate the result after one guard digit. There is a halve operation to simplify dividing by two. Starting in ESA/390, there is a square root operation. All operations have one hexadecimal guard digit to avoid precision loss. Most arithmetic operations truncate like simple pocket calculators. Therefore, 1 − 16−8 = 1. In this case, the result is rounded away from zero.
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