{"id":234296,"date":"2025-06-14T04:01:11","date_gmt":"2025-06-14T04:01:11","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=234296"},"modified":"2025-06-14T04:01:14","modified_gmt":"2025-06-14T04:01:14","slug":"answer-the-following-questions-related-to-ieee754-single-precision-floating-point-numbers","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/14\/answer-the-following-questions-related-to-ieee754-single-precision-floating-point-numbers\/","title":{"rendered":"Answer the following questions related to IEEE754 single-precision floating-point numbers"},"content":{"rendered":"\n

Answer the following questions related to IEEE754 single-precision floating-point numbers: a. Encode the following number: -7.4375 s = , e = <\/strong><\/em><\/strong>, f = _<\/strong> (base 2 for all fields) b. Suppose a floating point number (float x) contains a value of
. What would be the smallest value that can be added to x without a loss of precision?<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

ChatGPT said:<\/h6>\n\n\n\n

IEEE 754 Single-Precision Floating-Point Number Analysis<\/strong><\/h3>\n\n\n\n
\n\n\n\n

(a) Encode the number: -7.4375<\/strong><\/h4>\n\n\n\n

IEEE 754 single-precision format consists of 32 bits<\/strong>, divided as follows:<\/p>\n\n\n\n

    \n
  • Sign bit (s)<\/strong>: 1 bit<\/li>\n\n\n\n
  • Exponent (e)<\/strong>: 8 bits (with a bias of 127)<\/li>\n\n\n\n
  • Fraction\/Mantissa (f)<\/strong>: 23 bits<\/li>\n<\/ul>\n\n\n\n

    Let\u2019s break down the encoding of -7.4375<\/strong> step by step:<\/p>\n\n\n\n

    Step 1: Determine the sign bit (s)<\/strong><\/h5>\n\n\n\n

    Since the number is negative,
    s = 1<\/strong><\/p>\n\n\n\n

    Step 2: Convert the absolute value (7.4375) to binary<\/strong><\/h5>\n\n\n\n

    Split it into integer and fractional parts:<\/p>\n\n\n\n

      \n
    • Integer part:
      710=11127_{10} = 111_2710\u200b=1112\u200b<\/li>\n\n\n\n
    • Fractional part:
      0.4375100.4375_{10}0.437510\u200b:
      0.4375\u00d72=0.875\u219200.4375 \\times 2 = 0.875 \u2192 00.4375\u00d72=0.875\u21920
      0.875\u00d72=1.75\u219210.875 \\times 2 = 1.75 \u2192 10.875\u00d72=1.75\u21921
      0.75\u00d72=1.5\u219210.75 \\times 2 = 1.5 \u2192 10.75\u00d72=1.5\u21921
      0.5\u00d72=1.0\u219210.5 \\times 2 = 1.0 \u2192 10.5\u00d72=1.0\u21921
      \u2192 So, 0.4375=0.011120.4375 = 0.0111_20.4375=0.01112\u200b<\/li>\n<\/ul>\n\n\n\n

      Thus,
      7.4375 = 111.0111 in binary<\/strong><\/p>\n\n\n\n

      Step 3: Normalize the binary number<\/strong><\/h5>\n\n\n\n

      Convert 111.0111<\/strong> to scientific form:1.110111\u00d7221.110111 \\times 2^21.110111\u00d722<\/p>\n\n\n\n

        \n
      • Mantissa (f): Take the bits after the decimal point:
        f = 11011100000000000000000<\/strong> (23 bits)<\/li>\n\n\n\n
      • Exponent (e):
        E=2+127=129\u2192e=100000012E = 2 + 127 = 129 \u2192 e = 10000001_2E=2+127=129\u2192e=100000012\u200b<\/li>\n<\/ul>\n\n\n\n
        \n\n\n\n

        Final IEEE754 Encoding:<\/strong><\/p>\n\n\n\n

          \n
        • s = 1<\/strong><\/li>\n\n\n\n
        • e = 10000001<\/strong><\/li>\n\n\n\n
        • f = 11011100000000000000000<\/strong><\/li>\n<\/ul>\n\n\n\n
          \n\n\n\n

          (b) What is the smallest value that can be added to a float x without loss of precision?<\/strong><\/h4>\n\n\n\n

          This smallest increment is known as machine epsilon (\u03b5)<\/strong> and depends on the current value of x<\/strong>. In general, for IEEE 754 single precision:\u03f5(x)=2e\u221223\\epsilon(x) = 2^{e – 23}\u03f5(x)=2e\u221223<\/p>\n\n\n\n

          Where e<\/code> is the exponent of the normalized float. This value is the distance between x and the next representable float.<\/p>\n\n\n\n

          For example<\/strong>, if:<\/p>\n\n\n\n

            \n
          • x = 1.0<\/strong>, its binary is 1.0=1.00000000000000000000000\u00d7201.0 = 1.00000000000000000000000 \\times 2^01.0=1.00000000000000000000000\u00d720<\/li>\n\n\n\n
          • Then smallest addable value is: \u03f5=2\u221223\u22481.1920929\u00d710\u22127\\epsilon = 2^{-23} \u2248 1.1920929 \\times 10^{-7}\u03f5=2\u221223\u22481.1920929\u00d710\u22127<\/li>\n<\/ul>\n\n\n\n

            This is because IEEE754 can represent 23 fractional bits, and 2\u2212232^{-23}2\u221223 is the smallest change we can make at that magnitude without being rounded off.<\/p>\n\n\n\n

            \n\n\n\n

            Summary Answers<\/strong><\/h3>\n\n\n\n

            a. For -7.4375:<\/strong><\/p>\n\n\n\n

              \n
            • s = 1<\/strong><\/li>\n\n\n\n
            • e = 10000001<\/strong><\/li>\n\n\n\n
            • f = 11011100000000000000000<\/strong><\/li>\n<\/ul>\n\n\n\n

              b. Smallest value that can be added to x (e.g., x = 1.0):<\/strong><\/p>\n\n\n\n

                \n
              • \u03b5 = 2\u207b\u00b2\u00b3 \u2248 1.1920929 \u00d7 10\u207b\u2077<\/strong>
                This is the unit in the last place (ULP)<\/strong> and indicates floating-point precision.<\/li>\n<\/ul>\n\n\n\n
                \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

                Answer the following questions related to IEEE754 single-precision floating-point numbers: a. Encode the following number: -7.4375 s = , e = , f = _ (base 2 for all fields) b. Suppose a floating point number (float x) contains a value of. What would be the smallest value that can be added to x without […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-234296","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/234296","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=234296"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/234296\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=234296"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=234296"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=234296"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}