lecture 10 floating point arithmetic gpus in perspective
TRANSCRIPT
Announcements
• Interactive use on Forge • Trestles accounts? A4
©2012 Scott B. Baden /CSE 260/ Winter 2012 2
Today’s lecture
• Floating point arithmetic • GPU vs CPUs
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What is floating point? • A representation
u ±2.5732… × 1022 u NaN ∞ u Single, double, extended precision
• A set of operations u + = * / √ rem u Comparison < ≤ = ≠ ≥> u Conversions between different formats, binary to decimal u Exception handling
• IEEE Floating point standard P754 u Universally accepted u W. Kahan received the Turing Award in 1989 for design of IEEE
Floating Point Standard u Revision in 2008
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IEEE Floating point standard P754 • Normalized representation ±1.d…d× 2eps
u Macheps = Machine epsilon = ε = 2-#significand bits relative error in each operation
u OV = overflow threshold = largest number u UN = underflow threshold = smallest number
• ±Zero: ±significand and exponent = 0
©2012 Scott B. Baden /CSE 260/ Winter 2012
Format # bits #significand bits macheps #exponent bits exponent range ---------- -------- ---------------------- ------------ -------------------- ---------------------- Single 32 23+1 2-24 (~10-7) 8 2-126 - 2127 (~10+-38) Double 64 52+1 2-53 (~10-16) 11 2-1022 - 21023 (~10+-308) Double ≥80 ≥64 ≤2-64(~10-19) ≥15 2-16382 - 216383 (~10+-4932)
Jim Demmel 6
What happens in a floating point operation? • Round to the nearest representable floating point number
that corresponds to the exact value(correct rounding) • Round to nearest value with the lowest order bit =0
(rounding toward nearest even) • Others are possible • We don’t need the exact value to work this out! • Applies to + = * / √ rem
• Error formula: fl(a op b) = (a op b)*(1 + δ) where • op one of + , - , * , / • | δ | ≤ ε • assuming no overflow, underflow, or divide by zero
• Addition example u fl(∑ xi) = ∑i=1:n xi*(1+ei) u |ei| ∼< (n-1)ε
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Exception Handling • An exception occurs when the result of a floating point
operation is not representable as a normalized floating point number
u 1/0, √-1
• P754 standardizes how we handle exceptions u Overflow: - exact result > OV, too large to represent u Underflow: exact result nonzero and < UN, too small to represent u Divide-by-zero: nonzero/0 u Invalid: 0/0, √-1, log(0), etc. u Inexact: there was a rounding error (common)
• Two possible responses u Stop the program, given an error message u Tolerate the exeption
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An example • Graph the function
f(x) = sin(x) / x
• But we get a singularity @ x=0: 1/x = ∞ • This is an “accident” in how we represent the function
(W. Kahan) • f(0) = 1 • We catch the exception (divide by 0) • Substitute the value f(0) = 1
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Denormalized numbers • We compute if (a ≠ b) then x = a/(a-b) • We should never divide by 0, even if a-b is tiny • Underflow exception occurs when
exact result a-b < underflow threshold UN • We return a denormalized number for a-b
u ±0.d…d x 2min_exp
u sign bit, nonzero significand, minimum exponent value u Fill in the gap between 0 and UN
©2012 Scott B. Baden /CSE 260/ Winter 2012
Jim Demmel
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NaN (Not a Number) • Invalid exception
u Exact result is not a well-defined real number • We can have a quiet NaN or an sNan
u Quiet –does not raise an exception, but propagates a distinguished value
• E.g. missing data: max(3,NAN) = 3 u Signaling - generate an exception when accessed
• Detect uninitialized data
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Exception handling • Each of the 5 exceptions manipulates 2 flags • Sticky flag set by an exception, can be read and cleared by
the user • Exception flag: should a trap occur?
u If so, we can enter a trap handler u But requires precise interrupts, causes problems on a
parallel computer • We can use exception handling to build faster algorithms
u Try the faster but “riskier” algorithm u Rapidly test for accuracy
(possibly with the aid of exception handling) u Substitute slower more stable algorithm as needed
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When compiler optimizations alter precision • Let’s say we support 79+ bit extended format in registers • When we store values into memory, values a converted to
the lower precision format • Compilers can keep things in registers and we may lose
referential transparency • An example
float x, y, z; int j; …. x = y + z; if (x >= j) replace x by something smaller than j // x < j y=x;
• With optimization turned on, x is computed to extra precision; it is not an ordinary float
• If x is in a register, x≠y, no guarantee that the condition x < j will be preserved when x is stored in y, i.e. y >= j
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P754 on the GPU • Cuda Programming Guide (4.1)
“All compute devices follow the IEEE 754-2008 standard for binary floating-point arithmetic with the following deviations”
u There is no mechanism for detecting that a floating-point exception has occurred and all operations behave as if the exceptions are always masked… SNaN … are handled as quiet
• Cap. 2.x: FFMA … is an IEEE-754-2008 compliant fused multiply-add instruction … the full-width product … used in the addition & a single rounding occurs during generation of the final result
u rnd(A ☓ A + B) with FFMA (2.x) vs rnd(rnd(A ☓ A) + B) FMAD for 1.x
• FFMA can avoid loss of precision during subtractive cancellation when adding quantities of similar magnitude but opposite signs
• Also see Precision & Performance: Floating Point and IEEE 754 Compliance for NVIDIA GPUs,” by N. Whitehead and A. Fit-Florea
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Today’s lecture
• Floating point arithmetic • GPU vs CPUs
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Comparative results with Panfilov Method • Runs on Forge, single precision, 16 threads on the CPU • GPU: n= (1024, 1536); t=5.0
(84, 98) GF; Double prec: (49,54) • CPU: (62,65); Double precision: (35 , 17)
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Computing Platforms • NVIDIA GTX-280 – 200 series, like Lilliput • Intel core i7
u Superscalar, branch cache, out of order, 2-way hyperthreading u Each core: 32 KB I+D L1, 256KB L2 u Shared 8MB L3
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