Faster multiprecision integer division William Hart June 22, 2015 - PowerPoint PPT Presentation

Outline Multiprecision integer division Faster multiprecision integer division William Hart June 22, 2015 William Hart Faster multiprecision integer division

Outline Multiprecision integer division Multiprecision integer division William Hart Faster multiprecision integer division

Outline Multiprecision integer division Multiprecision integer division • Given: • A = < a m − 1 , a m − 2 , a m − 3 , . . . , a 1 , a 0 > • D = < d n − 1 , . . . , d 1 , d 0 > William Hart Faster multiprecision integer division

Outline Multiprecision integer division Multiprecision integer division • Given: • A = < a m − 1 , a m − 2 , a m − 3 , . . . , a 1 , a 0 > • D = < d n − 1 , . . . , d 1 , d 0 > • Compute: • Approximate quotient Q ⋆ William Hart Faster multiprecision integer division

Outline Multiprecision integer division Multiprecision integer division • Given: • A = < a m − 1 , a m − 2 , a m − 3 , . . . , a 1 , a 0 > • D = < d n − 1 , . . . , d 1 , d 0 > • Compute: • Approximate quotient Q ⋆ • Exact quotient Q William Hart Faster multiprecision integer division

Outline Multiprecision integer division Multiprecision integer division • Given: • A = < a m − 1 , a m − 2 , a m − 3 , . . . , a 1 , a 0 > • D = < d n − 1 , . . . , d 1 , d 0 > • Compute: • Approximate quotient Q ⋆ • Exact quotient Q • Quotient Q and remainder R William Hart Faster multiprecision integer division

Outline Multiprecision integer division Machine primitive • Machine does 2 × 1 division William Hart Faster multiprecision integer division

Outline Multiprecision integer division Machine primitive • Machine does 2 × 1 division • < u 1 , u 0 > by < v 0 > , quotient and remainder both 1 limb William Hart Faster multiprecision integer division

Outline Multiprecision integer division Machine primitive • Machine does 2 × 1 division • < u 1 , u 0 > by < v 0 > , quotient and remainder both 1 limb • Require v 0 normalised William Hart Faster multiprecision integer division

Outline Multiprecision integer division Machine primitive • Machine does 2 × 1 division • < u 1 , u 0 > by < v 0 > , quotient and remainder both 1 limb • Require v 0 normalised • If B = 2 32 or 2 64 then B < v 0 ≤ B / 2 William Hart Faster multiprecision integer division

Outline Multiprecision integer division Machine primitive • Machine does 2 × 1 division • < u 1 , u 0 > by < v 0 > , quotient and remainder both 1 limb • Require v 0 normalised • If B = 2 32 or 2 64 then B < v 0 ≤ B / 2 • Also require u 1 < v 0 William Hart Faster multiprecision integer division

Outline Multiprecision integer division Schoolbook algorithm • Shift A and D left so that D is normalised William Hart Faster multiprecision integer division

Outline Multiprecision integer division Schoolbook algorithm • Shift A and D left so that D is normalised • Ensure top n limbs of A are less than D William Hart Faster multiprecision integer division

Outline Multiprecision integer division Schoolbook algorithm • Shift A and D left so that D is normalised • Ensure top n limbs of A are less than D • i ← m − n + 1 • while i ≥ 0 • q i ← quotient of < a n + i , a n + i − 1 > by < d n − 1 > • A ← A − q i DB i • i ← i − 1 William Hart Faster multiprecision integer division

Outline Multiprecision integer division Schoolbook algorithm • Shift A and D left so that D is normalised • Ensure top n limbs of A are less than D • i ← m − n + 1 • while i ≥ 0 • q i ← quotient of < a n + i , a n + i − 1 > by < d n − 1 > • A ← A − q i DB i • i ← i − 1 • Shift remainder right William Hart Faster multiprecision integer division

Outline Multiprecision integer division Problems • May have < a n + i , a n + i − 1 > ≥ d n − 1 B William Hart Faster multiprecision integer division

Outline Multiprecision integer division Problems • May have < a n + i , a n + i − 1 > ≥ d n − 1 B • q i D may be too large William Hart Faster multiprecision integer division

Outline Multiprecision integer division Problems • May have < a n + i , a n + i − 1 > ≥ d n − 1 B • q i D may be too large • Need tests and adjustments for both cases William Hart Faster multiprecision integer division

Outline Multiprecision integer division Standard improvement No. 1 • Multiply by precomputed inverse ν of d n − 1 instead of division William Hart Faster multiprecision integer division

Outline Multiprecision integer division Standard improvement No. 1 • Multiply by precomputed inverse ν of d n − 1 instead of division • 1986 Barret used B 2 n / d William Hart Faster multiprecision integer division

Outline Multiprecision integer division Standard improvement No. 1 • Multiply by precomputed inverse ν of d n − 1 instead of division • 1986 Barret used B 2 n / d • 1986 Beame, Cook, Hoover suggested using an under approximation of the inverse William Hart Faster multiprecision integer division

Outline Multiprecision integer division Standard improvement No. 1 • Multiply by precomputed inverse ν of d n − 1 instead of division • 1986 Barret used B 2 n / d • 1986 Beame, Cook, Hoover suggested using an under approximation of the inverse • 1951 Wallace, diode transistor circuits William Hart Faster multiprecision integer division

Outline Multiprecision integer division Standard improvement No. 1 • Multiply by precomputed inverse ν of d n − 1 instead of division • 1986 Barret used B 2 n / d • 1986 Beame, Cook, Hoover suggested using an under approximation of the inverse • 1951 Wallace, diode transistor circuits • 2000 BC, Babylonian “Clay tablet” PC’s William Hart Faster multiprecision integer division

Outline Multiprecision integer division Precomputed inverses • 1994, Granlund, Montgomery suggested ν = ⌊ ( B 2 − 1 ) / d n − 1 ⌋ − B William Hart Faster multiprecision integer division

Outline Multiprecision integer division Precomputed inverses • 1994, Granlund, Montgomery suggested ν = ⌊ ( B 2 − 1 ) / d n − 1 ⌋ − B • One mulhigh, one mul William Hart Faster multiprecision integer division

Outline Multiprecision integer division Precomputed inverses • 1994, Granlund, Montgomery suggested ν = ⌊ ( B 2 − 1 ) / d n − 1 ⌋ − B • One mulhigh, one mul • At most 3 correction steps for 2 × 1 division William Hart Faster multiprecision integer division

Outline Multiprecision integer division Precomputed inverses • 1994, Granlund, Montgomery suggested ν = ⌊ ( B 2 − 1 ) / d n − 1 ⌋ − B • One mulhigh, one mul • At most 3 correction steps for 2 × 1 division Additional problem: • Quotient limb may now be too small or too large!! William Hart Faster multiprecision integer division

Outline Multiprecision integer division M¨ oller-Granlund (2011) • Same precomputed inverse William Hart Faster multiprecision integer division

Outline Multiprecision integer division M¨ oller-Granlund (2011) • Same precomputed inverse • Algorithm improved to one mul, one mullow William Hart Faster multiprecision integer division

Outline Multiprecision integer division M¨ oller-Granlund (2011) • Same precomputed inverse • Algorithm improved to one mul, one mullow • One correction with unpredicted branch William Hart Faster multiprecision integer division

Outline Multiprecision integer division M¨ oller-Granlund (2011) • Same precomputed inverse • Algorithm improved to one mul, one mullow • One correction with unpredicted branch • One very unlikely correction William Hart Faster multiprecision integer division

Outline Multiprecision integer division Improvement No. 2 Problem : multiprecision corrections very expensive William Hart Faster multiprecision integer division

Outline Multiprecision integer division Improvement No. 2 Problem : multiprecision corrections very expensive • Idea : use 3 × 2 divisions to reduce probability of MP-correction William Hart Faster multiprecision integer division

Outline Multiprecision integer division Improvement No. 2 Problem : multiprecision corrections very expensive • Idea : use 3 × 2 divisions to reduce probability of MP-correction • M¨ oller-Granlund give 3 × 2 version of their algorithm William Hart Faster multiprecision integer division

Outline Multiprecision integer division Our precomputed inverse • Use ν = ⌊ B 2 / ( d n − 1 + 1 ) ⌋ − B William Hart Faster multiprecision integer division

Outline Multiprecision integer division Our precomputed inverse • Use ν = ⌊ B 2 / ( d n − 1 + 1 ) ⌋ − B • Because we used d n − 1 + 1, quotient is never too large William Hart Faster multiprecision integer division

Outline Multiprecision integer division Our precomputed inverse • Use ν = ⌊ B 2 / ( d n − 1 + 1 ) ⌋ − B • Because we used d n − 1 + 1, quotient is never too large • Approximate quotient can be done with no corrections William Hart Faster multiprecision integer division