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Sometimes a part of the type safety is implemented indirectly: e.g. the class BigDecimal represents a floating point number of arbitrary precision, but handles only numbers that can be expressed with a finite representation. The operation BigDecimal.divide() calculates a new object as the division of two numbers expressed as BigDecimal.
Ruby's standard library includes a BigDecimal class in the module bigdecimal. Java's standard library includes a java.math.BigDecimal class. In Objective-C, the Cocoa and GNUstep APIs provide an NSDecimalNumber class and an NSDecimal C data type for representing decimals whose mantissa is up to 38 digits long, and exponent is from −128 to 127.
Here we start with 0 in single precision (binary32) and repeatedly add 1 until the operation does not change the value. Since the significand for a single-precision number contains 24 bits, the first integer that is not exactly representable is 2 24 +1, and this value rounds to 2 24 in round to nearest, ties to even.
The IEEE 754 floating-point standard guarantees that add, subtract, multiply, divide, fused multiply–add, square root, and floating-point remainder will give the correctly rounded result of the infinite-precision operation. No such guarantee was given in the 1985 standard for more complex functions and they are typically only accurate to ...
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
IEEE 754-1985 [1] is a historic industry standard for representing floating-point numbers in computers, officially adopted in 1985 and superseded in 2008 by IEEE 754-2008, and then again in 2019 by minor revision IEEE 754-2019. [2]
divide-by-zero, set if the result is infinite given finite operands, returning an infinity, either +∞ or −∞. invalid , set if a finite or infinite result cannot be returned e.g. sqrt(−1) or 0/0, returning a quiet NaN.
Variable length arithmetic represents numbers as a string of digits of a variable's length limited only by the memory available. Variable-length arithmetic operations are considerably slower than fixed-length format floating-point instructions.