{"id":"floating-number-range-resolution","name":"Floating-point range, resolution and rounding","shortName":"Represent a number","role":"Explains which numerical values a finite bit pattern can represent before those values enter a learned computation.","description":"Floating-point formats distribute representable numbers across a range of magnitudes. An exponent controls scale while fraction bits provide resolution within that scale. A sign bit, exponent field and fraction field must be interpreted under an explicit encoding convention. The workbench defines its own seven-bit format: one sign bit, three exponent bits, three fraction bits and exponent bias 3. It reserves the all-ones exponent for special values, supports subnormals near zero and rounds ties toward an even low-order significand bit. This is a fully enumerated teaching format, not an assertion about a vendor’s FP8 implementation. Seeing the exact stored bits makes it possible to separate representability error from changes in an algorithm.","mechanism":["Separate the sign and magnitude, then identify the normal exponent interval or subnormal range.","Find adjacent representable magnitudes and round to the nearest, resolving exact midpoints with the declared even-bit rule.","Inspect the stored sign/exponent/fraction fields and the absolute difference from the input.","Probe the boundaries where spacing changes, a small value rounds to zero or a large value overflows."],"learningObjective":"Encode a value in a declared miniature format and diagnose rounding, gradual underflow and overflow.","engineeringExample":"In this format, 1.0625 lies halfway between 1 and 1.125 and rounds to 1. The midpoint 1.1875 rounds to 1.25. The smallest positive subnormal is 1/32=0.03125; 1/64 is a tie that rounds to zero. The largest finite value is 15, and 15.5 reaches the nearest-rounding overflow threshold.","evidenceNotes":"The workbench executes the declared small mechanism. Numeric examples are authored and independently checked, not measurements from the cited research. This authored seven-bit format uses bias 3, gradual underflow and nearest ties-to-even. Exponent 7 is special and excluded from the finite expression.","specs":[{"label":"Teaching specimen","value":"Represent a number","context":"In this format, 1.0625 lies halfway between 1 and 1.125 and rounds to 1. The midpoint 1.1875 rounds to 1.25. The smallest positive subnormal is 1/32=0.03125; 1/64 is a tie that rounds to zero. The largest finite value is 15, and 15.5 reaches the nearest-rounding overflow threshold."},{"label":"Evidence boundary","value":"Explicit computed example","context":"This authored seven-bit format uses bias 3, gradual underflow and nearest ties-to-even. Exponent 7 is special and excluded from the finite expression."}],"interfaces":[{"name":"Inputs and convention","direction":"in","signal":"Separate the sign and magnitude, then identify the normal exponent interval or subnormal range."},{"name":"Computed result and scope","direction":"out","signal":"Encode a value in a declared miniature format and diagnose rounding, gradual underflow and overflow."}],"tradeoff":"More exponent bits extend dynamic range; more fraction bits refine spacing. Neither change alone establishes model accuracy, latency or energy savings.","whyAI":"Training and inference depend on representable activations, weights, gradients and accumulations. Low precision needs explicit scale, accumulation and error contracts.","misconception":"A datatype name or bit count does not specify all rounding, saturation, subnormal and accumulation behavior.","check":{"question":"What primarily changes when fraction bits are increased at fixed exponent width?","options":["The spacing between representable values within a magnitude interval","The meaning of the target label","The physical number of training examples"],"answer":0,"explanation":"The spacing between representable values within a magnitude interval. This authored seven-bit format uses bias 3, gradual underflow and nearest ties-to-even. Exponent 7 is special and excluded from the finite expression."},"prerequisites":[],"related":["quantization","floating-point-reduction-order","tensor-core"],"sources":[{"title":"Goodfellow, Bengio and Courville · Numerical computation","url":"https://www.deeplearningbook.org/contents/numerical.html"}],"level":"model","scale":"Mathematical and data space; dimensions are not physical distance","kind":"model","branch":"mathematical-foundations","subbranch":"Represent a number","workstream":"W11","visualFamily":"mathematical","labId":"library","labScope":"Open Mathematical foundations for the bounded executable mechanism and its record. General numerical labs are an additional reference library.","science":[{"title":"Floating-point range, resolution and rounding","equation":"Subnormals use exponent minus two without the implicit leading one; normal finite values use biased exponent e minus three and an implicit leading one.","explanation":"In this format, 1.0625 lies halfway between 1 and 1.125 and rounds to 1. The midpoint 1.1875 rounds to 1.25. The smallest positive subnormal is 1/32=0.03125; 1/64 is a tie that rounds to zero. The largest finite value is 15, and 15.5 reaches the nearest-rounding overflow threshold.","assumptions":"This authored seven-bit format uses bias 3, gradual underflow and nearest ties-to-even. Exponent 7 is special and excluded from the finite expression."}]}