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Verilog Transcendental Functions for Numerical Testbenches

Verilog Transcendental Functions for Numerical Testbenches. Mark G. Arnold University of Manchester Institute of Science & Technology, UK Colin Walter University of Manchester Institute of Science & Technology, UK Freddy Engineer

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Verilog Transcendental Functions for Numerical Testbenches

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  1. Verilog Transcendental Functions for Numerical Testbenches Mark G. Arnold University of Manchester Institute of Science & Technology, UK Colin Walter University of Manchester Institute of Science & Technology, UK Freddy Engineer Xilinx, Inc., San Jose, CA

  2. Numeric-intensive embedded hardware systems • use transcendental functions • FFT: sin(x),cos(x) • Neural Nets: 1/(1+exp(-x)) • Graphics: sqrt(x*x+y*y+z*z), trig • How to test such designs in Verilog? • Need testbench aware of math functions • Problem: 1993 Verilog lacks these functions • Problem: Access to C functions needs PLI • Problem: Synthesizable Verilog uses reg • Numeric algorithm uses real

  3. Ways to use reg as real Fixed-point (FX) cheap but hard to design Floating point (FP) easier but more expensive Logarithmic number system (LNS) cheaper than FX, easy as FP, but weird see www.xlnsresearch.com for more

  4. We need log(x) and exp(x) for LNS, so why not design something more general: a transcendental package for Verilog? Useful for testbenches that verifies hardware involving transcendental functions. Available: www.cs.uwyo.edu/~marnold/verilogmath.html module math; `include “math.v” endmodule

  5. Black-Box Testbench Testbench Verilog does not know about sin(x),log(x) Only can compare against expected overall behaviour Embedded Hardware sin(x) log(x) etc… other hard- ware )

  6. White-Box Testbench Testbench Verilog sin(x) Verilog log(x) Can test individual function units Embedded Hardware sin(x) log(x) etc… other hard- ware

  7. sin(x) • Syntax: math.sin(x) • Computed by: c1x+c3 x3+c5 x5+c7 x7 • in the range: -/2 < x < /2 • Range reduction: x<0, sin(x) = -sin(-x) • x>/2, sin(x) = -sin(x-) • Errors: none

  8. cos(x) Syntax: math.cos(x) Computed by: sin(x + /2) Range reduction: like sin(x) Errors: none

  9. tan(x) Syntax: math.tan(x) Computed by: sin(x)/cos(x) Errors: x = /2

  10. tan-1(x) • Syntax: math.atan(x) • Computed by: b0+a1 / (x2+b1-a2 / • (x2+b2 – a3/(x2+b3))) • in the range: 0 < x < 1 • Range reduction: x < 0, tan-1(x) = tan-1(-x) • x > 1, tan-1(x) = /2 - tan-1(1/x) • Errors: none

  11. cos-1(x) Syntax: math.acos(x) _____ Computed by: tan-1( 1.0-x2 / x ); in the range: 0 < x < 1 Errors: x<0, x>1

  12. sin-1(x) • Syntax: math.asin(x) • ______ • Computed by: tan-1( x / 1.0 - x2 ) • in the range: 0 < x < 1 • Errors: x < 0, x > 1

  13. ex Syntax: math.exp(x) Computed by: 2x ln(2) Range reduction: e-x = 1/ex Errors: x>177

  14. xy Syntax: math.pow(x,y) Computed by: e y ln(x) in the range: x>0 Errors: x<=0, y ln(x)>177

  15. __ x Syntax: math.sqrt(x) Computed by: eln(x) / 2 in the range: x > 0 Errors: x < 0

  16. ln(x) Syntax: math.log(x) Computed by: log2(x) / log2(e) in the range: x > 0 Errors: x <= 0

  17. 2x Syntax: N/A Computed by: products of rootof2(i) Errors: x > 255

  18. For k bits of precision, rootof2(-k) starts with 2k root of two Our function uses k = 23 bits of precision, rootof2(-23) = 83886082 = 1.000000082629586 Simpler example: k = 2, rootof2(-2) = 42 = 1.1892 When ith bit of x is one, multiply product by corresponding rootof2(i)

  19. rootof2(i) squares on each iteration: i=-2, ( 42 )1 = 1.1892 Example: x = 5.75 = 101. 112, 2x1.1892

  20. rootof2(i) squares on each iteration: i=-2, ( 42 )1 = 1.1892 i=-1, ( 42 )2 = 1.18922 = 1.4142 Example: x = 5.75 = 101. 112, 2x1.4142*1.1892

  21. rootof2(i) squares on each iteration: i=-2, ( 42 )1 = 1.1892 i=-1, ( 42 )2 = 1.18922 = 1.4142 i= 0, ( 42 )4 = 1.41422 = 2.0 Example: x = 5.75 = 101. 112, 2x2.0*1.4142*1.1892

  22. rootof2(i) squares on each iteration: i=-2, ( 42 )1 = 1.1892 i=-1, ( 42 )2 = 1.18922 = 1.4142 i= 0, ( 42 )4 = 1.41422 = 2.0 i=+1, ( 42 )8 = 2.02 = 4.0 Example: x = 5.75 = 101. 112, 2x 2.0*1.4142*1.1892

  23. rootof2(i) squares on each iteration: i=-2, ( 42 )1 = 1.1892 i=-1, ( 42 )2 = 1.18922 = 1.4142 i= 0, ( 42 )4 = 1.41422 = 2.0 i=+1, ( 42 )8 = 2.02 = 4.0 i=+2, ( 42 )16 = 4.02 = 16.0 Example: x = 5.75 = 101. 112, 2x16.0*2.0*1.4142*1.1892 = 53.8165

  24. Code that computes 2x1 prod = 1.0; power = 128.0; for (i = 7; i >= -23; i = i-1) begin if (x1 > power) begin prod = prod * rootof2(i); x1 = x1 - power; end power = power / 2.0; end

  25. log2(x) Syntax: N/A Computed by: iteration involving rootof2(i) Errors: x < 0

  26. Example x= 53.8165 • 53.8165 >= 16.0? • log2(x) ???. ??2

  27. Example x= 53.8165 • 53.8165 >= 16.0? • yes, 53.8165/16.0 = 3.3635 • 3.3635 >= 4.0? • log2(x) 1??. ??2

  28. Example x= 53.8165 • 53.8165 >= 16.0? • yes, 53.8165/16.0 = 3.3635 • 3.3635 >= 4.0? • no • 3.3635 >= 2.0? • log2(x) 10?. ??2

  29. Example x= 53.8165 • 53.8165 >= 16.0? • yes, 53.8165/16.0 = 3.3635 • 3.3635 >= 4.0? • no • 3.3635 >= 2.0? • yes, 3.3635/2.0 = 1.68175 • 1.68175 >= 1.4142? • log2(x) 101. ??2

  30. Example x= 53.8165 • 53.8165 >= 16.0? • yes, 53.8165/16.0 = 3.3635 • 3.3635 >= 4.0? • no • 3.3635 >= 2.0? • yes, 3.3635/2.0 = 1.68175 • 1.68175 >= 1.4142? • yes, 1.68175/1.4142 = 1.1892 • 1.1892 >= 1.1892? • log2(x) 101. 1?2

  31. Example x= 53.8165 • 53.8165 >= 16.0? • yes, 53.8165/16.0 = 3.3635 • 3.3635 >= 4.0? • no • 3.3635 >= 2.0? • yes, 3.3635/2.0 = 1.68176 • 1.68176 >= 1.4142? • yes, 1.68176/1.4142 = 1.1892 • 1.1892 >= 1.1892? • yes, … • log2(x) 101. 112 = 5.75

  32. Code that computes log2(re) log2 = 0.0; for (i=7; i>=-23; i=i-1) begin if (re > rootof2(i)) begin re = re/rootof2(i); log2 = 2.0*log2 + 1.0; end else log2 = log2*2; end

  33. Conclusions We provide 23-bit functions for testbench use We have trig functions useful for FFT and graphics We have exp, log, pow and sqrt functions useful for LNS and neural nets Our functions are compact and portable don’t need PLI encourage white-box testing

  34. IEEE 754 standard for Floating Point (FP): Mantissa: reg with significant bits Exponent: reg that determines scaling Each has a separate datapath Simulator supports real type real p,t,a,b; OK: t=a+b; OK: p=a*b; Synthesis does not support real WRONG: p=a*b; WRONG: t=a+b;

  35. Have to design by hand: Both + and * are expensive Most embedded systems avoid FP Debugging: $display($bitstoreal(a));

  36. Fixed point (FX) reg scaled by a fixed power of two Simulator supports FX only partially: OK t=a+b; WRONG: p=a*b; Synthesis is same Have to design by hand: Rounding for * Scaling for * + is same as integer: cheap Scaling difficult: FX delays time to market But, FX less expensive than FP Most embedded systems use FX

  37. Debugging: $display(`SCALE * a); `SCALE = constant designer determines

  38.  LNS reg has fixed-point logarithm  Neither synthesis nor simulation supports  Multiply is as cheap as +  Automatic scaling like FP  Lower Cost, Power than FX  LNS is seldom used  Debugging: 1993 Verilog lacks log/exp

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