Z Microsystems 14 Uživatelský manuál

Sun Microsystems Unifying Tests for Square RootMichael ParksSun Microsyste

Sun Microsystems page 10Derivation, 2•Formulate boundary cases•For nearest

Sun Microsystems page 11Derivation, 3•For directed modes, seekx = z ±x =

Sun Microsystems page 12Derivation, 4•Recast the problem: Given k, find R

Sun Microsystems page 13Derivation, 5•Directed modes set up already•For ne

Sun Microsystems page 14Derivation, 6•Transformed the problem into simple

Sun Microsystems page 15Derivation, 7•Hensel's technique>Choose a

Sun Microsystems page 16Derivation, 8•Equivalently, solvez = 2j −1− z•Eac

Sun Microsystems page 17Derivation, 9•Hensel's Recurrencezj= zj − 1z3

Sun Microsystems page 18Test summary•A little algebra puts the test argume

Sun Microsystems page 19Example•Single precision n = 24 bits, about 7 digi

Sun Microsystems page 2Outline•Machine arithmetic•Rounding boundaries•Test

Sun Microsystems page 20Implementation in C language•User selects:• #defi

Sun Microsystems page 21Comparison•Existing UCBTEST implementation sqrtest

Sun Microsystems page 22Software•Test programs for >FMUL directed>FD

Sun Microsystems Core: p-adic arithmeticx =∑k = N∞xkpk∣x∣p=1pNdpx , y=∣x

Sun Microsystems Core: p-adic arithmetic∣x∣p=1pNℚpis a fieldℤp=rin g of p

Sun Microsystems Hensel Lifting in p-adic Arithmeticf z ' ≡0 2j 

Sun Microsystems Hensel Lifting in p-adic Arithmetic•Newton's method

Sun Microsystems page 27MottoSeek singularities!There lie all the errors.

Sun Microsystems page 28Results•Nothing found yet:>SPARC>AMD Opteron

Sun Microsystems page 29References•Companion to >W. Kahan, “A Test for

Sun Microsystems page 3Verification•Test the accuracy of FSQRT(x) subrouti

Sun Microsystems page 30Conclusions•Fold near and directed tests together

Sun Microsystems page 31Merci beaucoup!•Appreciatedieee754 @ yahoo.com

Sun Microsystems page 4IEEE 754 Arithmetic•Fixed precision n•Representatio

Sun Microsystems page 5Rounding modese =⌊log2s⌋minf =tru ncnear s=2e −

Sun Microsystems page 6Normalize•Scale2nx =4nx...~x16~x4~ x ~ 4 x ~16 x

Sun Microsystems page 7Table-Maker's Dilemma•Seeking cases for which

Sun Microsystems page 8Test strategy•Land close to the rounding boundaries

Sun Microsystems page 9Derivation, 1•Fundamental integer range•The test ar