If y=1/x which x,y can be represented by float numbers without rounding, and MIN_VALUE<=x,y,z,z*y<=MAX_VALUE, is z*y==z/x always true?

Question

I was viewing some code and found some parts use num/2, while others use num * 0.5. That inspired me a question : within the range of MIN_VALUE and MAX_VALUE, are num/2 and num*0.5 interchangeable?

I was not only interested in the case that num divided by 2, consider:

  0.5 = 1/2
  4 = 1/0.25
  0.125 = 1/8

which y=1/x and both x any y can be perfectly represented with float numbers. My question is, for other cases that x , y can be perfectly represented with float numbers without rounding, and Float.MIN_VALUE<=x,y,z,z*y<=Float.MAX_VALUE, is z * y == z/x always true?

I tested some case, which is true:

document.write(123.456*4 == 123.456/0.25);
document.write("<br/>");
document.write(24.68/16 == 24.68*0.0625);
document.write("<br/>");
document.write(98.765/2 == 98.765*0.5);

I also tried to verify some cases by for-loop:

for(let i=0;i<10000;i++){
  const r=Math.random()*10000;
  if(r*0.5!=r/2){
    document.write(r+"<br/>");
    break;
  }
}

for(let i=0;i<10000;i++){
  const r=Math.random()*10000;
  if(r*4!=r/0.25){
    document.write(r+"<br/>");
    break;
  }
}

for(let i=0;i<10000;i++){
  const r=Math.random()*10000;
  if(r*0.125!=r/8){
    document.write(r+"<br/>");
    break;
  }
}

tested it several times and no counter examples printed. But I don't know if it is true for all other cases.

Note : y doesn't includes numbers like 0.1 and 0.2 because 0.1 and 0.2 are not perfectly representable by float numbers, while 0.5 and 0.25 can be. Also y is not something like 1.25 because x=1/1.25=0.8, which 0.8 is also not representable perfectly with float numbers.

Answer 1

JavaScript is an implementation of ECMAScript, and ECMAScript specifies that the IEEE 754 floating-point standard is used. IEEE 754 defines general operations to produce a result as if the real-number arithmetic result were computed with infinite precision and then rounded according to the applicable rounding rule. Therefore, if the mathematical expressions z•y and z/x are equal, which they must be if y = 1/x with no rounding error, then the floating-point computations z*y and z/x produce the same result.

Specifically, IEEE 754-2019 4.3 says:

… Except where stated otherwise, every operation shall be performed as if it first produced an intermediate result correct to infinite precision and with unbounded range, and then rounded that result according to one of the attributes in this clause…

ECMAScript 2020 Language Specification does not make a general statement of conformity to IEEE 754 but specifies individually that operations conform. For example, 6.1.6.1.4 discusses multiplication and says:

… The result of a floating-point multiplication is governed by the rules of IEEE 754-2019 binary double-precision arithmetic:…

Answer 2

Floating point math is broken, therefore generally (x * 1/y) != (x / y):

> 0.2 / 10
< 0.02
> 0.2 * (1/10)
< 0.020000000000000004