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380791
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2008-12-19T11:59:52.507
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2008-12-19T11:59:52.507
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As several others have poited out, fractions of 64 can be precicely represented by IEEE-floats. This means we can also convert to a fraction by moving and masking bits. This is not the place to explain all details of floating point representations, please refer to <a href="http://en.wikipedia.org/wiki/IEEE_754-1985" rel="nofollow noreferrer">wikipedia</a> for details. Briefly: a floating point number is stored as (sign)(exp)(frac) where sign is 1 bit, exp is 11 bits and frac is the fraction part (after 1.) and is 52 bits. This is enterpreted as the number: <pre><code>(sign == 1 ? -1 : 1) * 1.(frac) * 2^(exp-1023) </code></pre> Thus, we can get the 64th by moving the point accoring to the exponent and masking out the 6 bits after the point. In Java: <pre><code>private static final long MANTISSA_FRAC_BITMAP = 0xfffffffffffffl; private static final long MANTISSA_IMPLICIT_PREFIX = 0x10000000000000l; private static final long DENOM_BITMAP = 0x3f; // 1/64 private static final long DENOM_LEN = 6; private static final int FRAC_LEN = 52; public String floatAsFrac64(double d) { long bitmap = Double.doubleToLongBits(d); long mantissa = bitmap & MANTISSA_FRAC_BITMAP | MANTISSA_IMPLICIT_PREFIX; long exponent = ((bitmap >> FRAC_LEN) & 0x7ff) - 1023; boolean negative = (bitmap & (1l << 63)) > 0; // algorithm: // d is stored as SE(11)F(52), implicit "1." before F // move point to the right <exponent> bits to the right: if(exponent > FRAC_LEN) System.out.println("warning: loosing precision, too high exponent"); int pointPlace = FRAC_LEN-(int)exponent; // get the whole part as the number left of the point: long whole = mantissa >> pointPlace; // get the frac part as the 6 first bits right of the point: long frac = (mantissa >> (pointPlace-DENOM_LEN)) & DENOM_BITMAP; // if the last operation shifted 1s out to the right, we lost precision, check with // if any of these bits are set: if((mantissa & ((MANTISSA_FRAC_BITMAP | MANTISSA_IMPLICIT_PREFIX) >> (pointPlace - DENOM_LEN))) > 0) { System.out.println("warning: precision of input is smaller than 1/64"); } if(frac == 0) return String.format("%d", whole); int denom = 64; // test last bit, divide nom and demon by 1 if not 1 while((frac & 1) == 0) { frac = frac >> 1; denom = denom >> 1; } return String.format("%d %d/%d", whole, frac, denom); } </code></pre> (this code can probably be made shorter, but reading bit-flipping-code like this is hard enough as it is...)
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1. POFormat Double as Fraction
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1. USRolf Rander
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