{"id":53,"date":"2022-05-16T20:41:17","date_gmt":"2022-05-16T19:41:17","guid":{"rendered":"https:\/\/blogs.glowscotland.org.uk\/glowblogs\/phshighercomputingscience\/?page_id=53"},"modified":"2022-05-16T20:48:14","modified_gmt":"2022-05-16T19:48:14","slug":"floating-point","status":"publish","type":"page","link":"https:\/\/blogs.glowscotland.org.uk\/glowblogs\/phshighercomputingscience\/floating-point\/","title":{"rendered":"Floating Point"},"content":{"rendered":"

In floating point notation, the real number is stored as 2 separate bits of data<\/span><\/p>\n

    \n
  1. A storage location called the\u00a0<\/span>mantissa<\/span><\/b>\u00a0holds the complete number without the point.<\/span><\/li>\n
  2. A storage location called the\u00a0<\/span>exponent<\/span><\/b>\u00a0holds the number of places that the point must be moved in the original number to place it at the left hand side.<\/span><\/li>\n<\/ol>\n

    To Work out the mantissa and exponent you need to\u00a0<\/strong><\/p>\n

      \n
    • Move the point all the way so the number is a fractional number<\/span><\/li>\n
    • The entire number without the point is the mantissa\u00a0<\/span><\/li>\n
    • The number of places the point was moved is the exponent<\/span><\/li>\n<\/ul>\n

      Example 1<\/span><\/span>\u00a0<\/span><\/h2>\n

      How would 1101.0011<\/strong> be represented in binary floating point representation using 16 bits<\/strong> for the mantissa<\/strong> (including the sign bit) and 8 bits<\/strong> for the exponent<\/strong>?<\/span><\/span>\u00a0<\/span><\/p>\n

      \"\"<\/p>\n

      The decimal place moved 4 places to the left<\/span><\/span><\/strong><\/p>\n

      Sign would be 0<\/strong><\/p>\n

      The mantissa is then 110100111 + 000000 <\/strong>to make it up to 15 bits<\/p>\n

      = 110100111000000\u00a0<\/strong><\/p>\n

      The exponent is then 4 = 00000100 in\u00a0 binary<\/span><\/span>\u00a0<\/span><\/strong><\/p>\n

      \"\"<\/p>\n

      Example 2 – Negative Exponent<\/span><\/span><\/h2>\n

      How would 0.0001001<\/strong> be represented in binary floating point representation using 16 bits<\/strong> for the mantissa<\/strong> (including the sign bit) and 8 bits<\/strong> for the exponent<\/strong>?<\/span><\/span>\u00a0<\/span><\/p>\n

      \"\"<\/p>\n

      The decimal place moved 3 places to the right\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/strong><\/p>\n

      Sign would be 0<\/strong><\/p>\n

      The mantissa is then 1001 + 00000000000 <\/strong>to make it up to 15 bits<\/p>\n

      = 100100000000000\u00a0<\/strong><\/p>\n

      The exponent is then -3\u00a0<\/span><\/span><\/strong><\/p>\n

      00000011\u00a0<\/strong>
      \n11111100\u00a0<\/strong>
      \n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 +1\u00a0<\/strong>
      \n11111101 would be the exponent<\/strong><\/p>\n

      \"\"\u00a0<\/strong><\/p>\n

      Example 3 – Negative Mantissa<\/span><\/span><\/h2>\n

      How would -101.00011<\/strong> be represented in binary floating point representation using 16 bits<\/strong> for the mantissa<\/strong> (including the sign bit) and 8 bits<\/strong> for the exponent?<\/span><\/span>\u00a0<\/span><\/p>\n

      \"\"<\/p>\n

      The decimal place moved 3 places to the left<\/strong><\/p>\n

      The exponent is then 3 \u00a0<\/strong><\/p>\n

      The Sign is 1 because the mantissa is negative\u00a0<\/strong><\/p>\n

      The mantissa is then 10100011 <\/span><\/span> + 0000000 <\/strong>to make it up to 15 bits<\/p>\n

      \"\"<\/p>\n

      Range and Accuracy<\/h2>\n

      It is possible to improve the\u00a0<\/span>accuracy<\/strong>\u00a0<\/strong>of a floating point number by\u00a0<\/span>increasing<\/strong>\u00a0<\/strong>the number of bits devoted to the\u00a0<\/span>mantissa<\/strong>. The\u00a0<\/span>range<\/strong>\u00a0<\/strong>of numbers held can be\u00a0<\/span>increased<\/strong>\u00a0<\/strong>if more bits are devoted to the storage of the\u00a0<\/span>exponent<\/strong>.<\/span><\/p>\n

      There will always be a\u00a0<\/span>trade-off between accuracy<\/strong> and range<\/span>\u00a0when using floating point notation, as there will always be a set number of bits allocated to storing real numbers, with the potential to increase or decrease the number of bits used for the mantissa against the number of bits used for the exponent.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

      In floating point notation, the real number is stored as 2 separate bits of data A storage location called the\u00a0mantissa\u00a0holds the complete number without the point. A storage location called the\u00a0exponent\u00a0holds the number of places that the point must be moved in the original number to place it at the left hand side. To Work […]<\/p>\n","protected":false},"author":79984,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-53","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/blogs.glowscotland.org.uk\/glowblogs\/phshighercomputingscience\/wp-json\/wp\/v2\/pages\/53","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.glowscotland.org.uk\/glowblogs\/phshighercomputingscience\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/blogs.glowscotland.org.uk\/glowblogs\/phshighercomputingscience\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.glowscotland.org.uk\/glowblogs\/phshighercomputingscience\/wp-json\/wp\/v2\/users\/79984"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.glowscotland.org.uk\/glowblogs\/phshighercomputingscience\/wp-json\/wp\/v2\/comments?post=53"}],"version-history":[{"count":4,"href":"https:\/\/blogs.glowscotland.org.uk\/glowblogs\/phshighercomputingscience\/wp-json\/wp\/v2\/pages\/53\/revisions"}],"predecessor-version":[{"id":64,"href":"https:\/\/blogs.glowscotland.org.uk\/glowblogs\/phshighercomputingscience\/wp-json\/wp\/v2\/pages\/53\/revisions\/64"}],"wp:attachment":[{"href":"https:\/\/blogs.glowscotland.org.uk\/glowblogs\/phshighercomputingscience\/wp-json\/wp\/v2\/media?parent=53"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}