Semi-Automatic Floating-Point Implementation of Special Functions

Transcription

1 Semi-Automatic Floating-Point Implementation of Special Functions Christoph Lauter 1 Marc Mezzarobba 1,2 Pequan group 1 Université Paris 6 2 CNRS ARITH 22, Lyon,

2 }main() { int temp; float celsius; char repeat; char flag; do { flag='n"; do { if(flag=='n') printf("input a valid temperature :"); else printf("input a valid temperature,stupid:"); scanf("%d",&temp); flag='y'; } while (temp<0 temp >100); celsius=(5.0/9.0)*(temp-32); printf("%d degrees F is %6.2f degrees celsius\n",temp,cel printf("do you have another temperature?"); repeat=getchar(); putchar('\n'); Code Generation for Mathematical Functions explicit function expression equation black box f = log(x) f = e sin(x) Φ(f ) = 0 x f (x) double fun(double x) {... }

3 }main() { int temp; float celsius; char repeat; char flag; do { flag='n"; do { if(flag=='n') printf("input a valid temperature :"); else printf("input a valid temperature,stupid:"); scanf("%d",&temp); flag='y'; } while (temp<0 temp >100); celsius=(5.0/9.0)*(temp-32); printf("%d degrees F is %6.2f degrees celsius\n",temp,cel printf("do you have another temperature?"); repeat=getchar(); putchar('\n'); Code Generation for Mathematical Functions explicit function expression equation black box f = log(x) f = e sin(x) Φ(f ) = 0 x f (x) double fun(double x) {... }

4 Our Focus: Special Functions Ai(x) Ci(x) erf(x) J 0 (x) D-finite Functions = sol ns of linear ODEs with polynomial coeff ts p r (x) f (r) (x) + + p 1 (x) f (x) + p 0 (x) f (x) = 0 p 0, p 1,..., p r R[x] Example: f (x) = arctan(x) (x 2 + 1) f (x) + 2x f (x) = 0, [ ] f (0) f = (0) [ ] 0 1

5 }main() { int temp; float celsius; char repeat; char flag; do { flag='n"; do { if(flag=='n') printf("input a valid temperature :"); else printf("input a valid temperature,stupid:"); scanf("%d",&temp); flag='y'; } while (temp<0 temp >100); celsius=(5.0/9.0)*(temp-32); printf("%d degrees F is %6.2f degrees celsius\n",temp,cel printf("do you have another temperature?"); repeat=getchar(); putchar('\n'); Problem Statement { pr f (r) + + p 0 f = 0, f (0),..., f (r 1) (0) D R, ε > 0 double fun(double x); x D double, fun(x) f (x) f (x) ε Semi-automatic : may require implementation hints Future goal: full automation Rigorous in principle, some shortcuts in current prototype

6 differential equation Overview sets of polynomial approximations NumGfun (Maple) rigorous multiple precision solver for ODE Sollya Metalibm-lutetia (cf. previous talk by O. Kupriianova)

7 differential equation The : Taylor Series p r (x) f (r) (x) + p 0 (x) f (x) = 0 Ansatz: f (x) = f (x) = x f (x) = n=0 c n x n n=0 (n + 1)c n+1 x n c n 1 x n n=1 c n = b 0 (n)c n 1 + b 1 (n)c n 2 + f (x) = c 0 + c 1 x + + c d 1 x d 1 +

8 differential equation The : Error Bounds f (x) = x ρ d 1 n=0 c n x n + n=d c n x n } {{ }? Majorizing series : using the ODE, find a simple ˆf (x) = ĉ n x n such that c n ĉ n for all n. f (x) d 1 c n x n ĉ n ρ n! ε. n=0 n=d

9 The : Analytic Continuation differential equation What if D [ ρ, ρ]? i i

10 The : Analytic Continuation differential equation What if D [ ρ, ρ]? i [ ] f (x1 ) f (x 1 ) i x 1 = [ ] [ ] f (0) f (0)

11 The : Analytic Continuation differential equation What if D [ ρ, ρ]? i [ ] f (x1 ) f (x 1 ) i x 1 = [ ] [ ] f (0) f (0)

12 The : Analytic Continuation differential equation What if D [ ρ, ρ]? i [ ] f (x1 ) f (x 1 ) [ ] f (x2 ) f (x 2 ) i x 1 x 2 = [ = [ ] [ ] f (0) f (0) ] [ ] f (x 1 ) f (x 1 )

13 The : Analytic Continuation differential equation What if D [ ρ, ρ]? i [ ] f (x1 ) f (x 1 ) [ ] f (x2 ) f (x 2 )... i x 1 x 2 = [ = [ ] [ ] f (0) f (0) ] [ ] f (x 1 ) f (x 1 )

14 differential equation The : Economization Have: f (x) (c c d 1 x d 1 ) ε 2 x C, x 1 c c 7 x 7 + c 8 x 8 + c 9 x 9 c (1) c (1) 7 x7 + c (1) 8 x8 c (2) c (2) 7 x7 c T 8(x) c T 9(x). T n 1 over [ 1, 1] f (x) ( c (k) c (k) d 1 k xk ) ε x [ 1, 1]

15 The : Tight Approximation differential equation : D = D i x D i p i (x) f (x) ε deg p i = O(500) : D = Di x Di pi (x) f (x) 1 ε deg pi = O(10)

16 The : Tight Approximation differential equation : leveraging Metalibm Reuse domain splitting algorithm sketched in last talk D = D i Reuse minimax implementation with relative error bounds x Di pi (x) f (x) 1 ε Addressing issues Pure black-box interface too slow Zeros of f in domain D?

17 differential equation The : FP Polynomials Implementation need: p i F[x] Leverage existing FP minimax heuristics

18 differential equation The : FP Polynomials Implementation need: p i F[x] Leverage existing FP minimax heuristics unless f has a zero in the domain

19 differential equation The : Zeros of f Suppose f (c) = 0 for c D We want pi (x) f (x) 1 ε We need pi (c) = 0 No if pi F[x] but c R\F

20 differential equation The : Zeros of f Suppose f (c) = 0 for c D Actually we want x F. pi (x) f (x) 1 ε p i (x) never needs to be 0 exactly Okay but we need to compute pi Express c as a symbol, evaluated with Newton-Raphson on f Give f (x c)/x k to minimax algorithm, yielding q Deduce pi from x k q(x)

21 differential equation The : Evaluation scheme Reuse Metalibm-Lutetia to generate Horner scheme We could also use Metalibm-Lugdunum here

22 Examples: erfc Consider erfc(x) = 1 2 x e t2 dt π over D = [ 2; 2] at ε = x Describe erfc with f (x) + 2xf (x) = 0, f (0) = 1, f (0) = 2 π Generated has domain split into 16 subdomains Code runs in 110 to 350 cycles, libm exponential in 80 cycles Code is bitwise the same if we take MPFR s erfc instead of ODE 1e-19 8e-20 6e-20 4e-20 2e e-20-4e-20-6e-20-8e-20-1e e

23 Examples: J I I I x 30 I Consider Bessel function J0 over D = [0.5; 42] at ε = 2 45 I J0 is given as 40 xf 00 (x) + f 0 (x) + xf (x) = 0, Code generation takes about 30 minutes Generator has to cope with 13 zeros in the domain Code runs in 60 to 500 cycles, libm exponential in 80 cycles f (x) 1, x 0 1.5e-14 1e-14 5e e-15-1e e-14-2e

24 Conclusion and Outlook Don t. Have your special functions generated! Process starts from the very basic definition Only small domains handled fully automatically Full range implementions require manual intervention -backend interface not satisfactory Code generation performance unpredictable

25 Thanks! Thank you for your attention! Questions?