16bit Simulation with GNU Octave Andeas Stahel Personal Context Compute on µC 16bit Simulation with GNU Octave Approximation An Example Approximation 16bit Arithmetic The Result Andreas Stahel In General Bern University of Applied Sciences Closing OctConf 2017, March 20{22, 2017, CERN, Geneva PersonalI 16bit Simulation with GNU Octave Andeas Stahel Personal Context Andreas Stahel Compute on µC Mathematics Approximation An Example Approximation 16bit Arithmetic Teaching: The Result Math at Bachelor level to mechanical and electrical engineers In General Numerical Methods for the Master Program of Biomedical Closing Engineering A class on how to use Octave to solve engineering problems As member of the Institute for Human Centered Engineering (HuCE): many industry projects in mathematical modeling Web: https://web.ti.bfh.ch/sha1/ E-mail: [email protected] PersonalII 16bit Simulation with GNU Octave Andeas Stahel Concerning Octave: Personal Octave is used regularly for teaching, project work and research. Context Compute on µC I teach a class on how to use Octave for engineering problems. Approximation An Example I started using Octave in 1993/94 and am addicted to it since. Approximation 16bit Arithmetic Octave replaces MATLAB for many reasons: open source, great The Result community support, platform independent, (legally) free. In General Closing My professional life would be different without Octave! Thank you guys Why computing on a µC? 16bit Simulation with GNU Octave Andeas Stahel Personal Context Compute on µC Approximation An Example Approximation 16bit Arithmetic The Result In General Closing Some µC are very affordable and thus used in many devices, not visible by the user. Functions can useful to calibrate sensors, or one might do a first step of the data treatment on the µC based sensor. Developing on a true µC can be tedious. Using a desktop and the power of Octave is convenient. Facts for Computing on µCI 16bit Simulation with GNU Octave Andeas Stahel Personal Context On most affordable µC only integer arithmetic is implemented in Compute on µC hardware, i.e. no FPU. Approximation An Example Floating point libraries are large, slow and the results are often Approximation 16bit Arithmetic overly accurate. The Result In General If you use a 12bit AD converter, there is no need for a 32bit Closing accuracy of the subsequent calculations. Since +, − and ∗ are available one can aim to implement polynomial functions. Facts for Computing on µCII 16bit Simulation with GNU Octave Andeas Stahel Personal Context Different approaches are possible to implement the evaluation of a Compute on µC given function. It is often a compromise between the amount or Approximation required storage and the computations needed. An Example Approximation 16bit Arithmetic more storage ! less storage The Result In General fewer computations ! more computations Closing piecewise interpolation one global look up table linear quadratic polynomial Facts for Computing on µC III 16bit Simulation with GNU Octave Andeas Stahel On a typical 16bit µC the arithmetic operations for integers are Personal Context implemented in harware. Compute on µC 16bit ± 16bit ! 16bit Approximation 16bit ∗ 16bit ! 32bit An Example Approximation 16 k 16bit Arithmetic Division by 2 is free (high double byte), division by 2 is cheap The Result (shift). In General Closing Use full the ranges available for the data types int16 or int32 to obtain optimal accuracy. For multiplications we aim to use the full range of 32bit results, but will only use the high double byte for further computations. Approximation by Polynomials 16bit Simulation with GNU Octave Andeas Stahel Personal Context To approximate a given function on a bounded interval by a Compute on µC polynomial, different mathematical tools might be useful: Approximation An Example Linear regression, i.e. least square approximation Approximation 16bit Arithmetic Chebyshev approximation The Result Optimization by using , based on maximum In General fminsearch() Closing norm, or L2-norm, or . A combination of the above. For the problem at hand I worked with Chebyshev approximations. Chebyshev ApproximationI 16bit Simulation with GNU Octave The Chebyshev polynomials on the interval [−1 ; +1] are given by Andeas Stahel Personal Tn(x) = cos(n arccos(x)) Context Compute on µC T0(x) = cos(0) = 1 Approximation T1(x) = cos(arccos(x)) = x An Example 2 Approximation T2(x) = 2 x − 1 16bit Arithmetic 2 The Result T3(x) = 4 x − 3 x In General T (x) = 8 x 4 − 8 x 2 + 1 Closing 4 . A recursive identity allows to determine the polynomials efficiently. Tn+1(x) = 2 x · Tn(x) − Tn−1(x) Chebyshev ApproximationII 16bit Simulation with GNU Octave Andeas Stahel A function defined on [−1 ; +1] is approximated by a polynomial Personal pN (x) of degree N. Context Compute on µC Z 1 Approximation 2 1 cn = f (x) Tn(x) p dx 2 An Example π −1 1 − x Approximation 16bit Arithmetic N The Result c0 X f (x) ≈ pN (x) = + cn Tn(x) In General 2 n=1 Closing This is easily implemented in Octave. If a function g(z) is defined on [a ; b] then move it to the standard b−a interval [−1 ; +1] by f (x) = g(a + (x + 1) · 2 ) Approximate arctan(x) by a Parabola 16bit Simulation with GNU Octave The above Chebyshev approximation can be used to approximate Andeas Stahel the function f (x) = arctan((1 + x)=2) by a parabola. Personal 2 Context f (x) ≈ p2(x) = −0:0709107 · x + 0:394737 · x + 0:4625339 Compute on µC Approximation = (−0:0709107 · x + 0:394737) · x + 0:4625339 An Example Approximation The relative error of this approximation p2(x) can be determined 16bit Arithmetic The Result in bits, use log2(), leading to 7:97 ≈ 8 correct bits. In General Function and Chebyshev approximation Error of Chebyshev approximation Closing 0.8 0.003 0.002 0.6 0.001 0.4 0 y y 0.2 -0.001 -0.002 0 -0.003 -0.2 -0.004 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 x x Setup for the 16bit Computation 16bit Simulation with GNU To determine the 16bit values of Octave Andeas Stahel p2(x) = (−0:0709107 · x + 0:394737) · x + 0:4625339 Personal Context aim for vector y16 and a factor yscale such that Compute on µC Approximation An Example yscale · y16 ≈ p2(x) Approximation 16bit Arithmetic The Result The goal is to implement this evaluation with a 16bit arithmetic, In General while keeping the result as accurate as possible and avoiding 15 Closing overflow, i.e. results exceeding ±2 . Start with a fine grid of x-values, e.g. x=linspace(-1,1,100000); Since −1 ≤ x ≤ 1 we multiply x by xscale and convert to int16 with x16 ≈ xscale · x, such that −MaxVal ≤ x16 ≤ +MaxVal = 215 − 1 = 32767 With this we use the full accuracy available on a 16bit arithmetic. Step 1: res1 = −0:070910677 · x + 0:3947365I 16bit Simulation with GNU Octave To perform the first Horner step proceed as follows: Andeas Stahel Personal y = −0:070910677 · x + 0:3947365 32767 Context y16 = -32767 (use full scale) yscale = 0:070910677 Compute on µC 16 yscale·xscale Approximation prod16 = y16 * x16/2 yscale = 216 k An Example if jprod16 + yscale · 0:395j > 32767 rescale, divide by 2 Approximation add16 = int16(yscale * 0.3947365) integer to be added 16bit Arithmetic The Result y16 = prod16 + add16 In General Closing With the above numbers rescaling by 1/4 is required and leads to add16 = 22800 . The result y16 satisfies yscale · y16 ≈ −0:070910677 ∗ x + 0:3947365 Step 1: res1 = −0:070910677 · x + 0:3947365II 16bit Simulation with GNU Octave Andeas Stahel The result can be visualized, using exact (double precision) and Personal approximate (16bit) computations. Context Compute on µC Result of step 1 40000 Approximation true 16bit An Example Approximation 20000 16bit Arithmetic The Result In General 0 Closing y scaled -20000 -40000 -1 -0.5 0 0.5 1 x Step 2: res2 = res1 · x + 0:4625339I 16bit Simulation with GNU Octave Andeas Stahel To perform the second Horner step proceed as follows: Personal Context y = res1 · x + 0:4625339 Compute on µC 16 yscale·xscale Approximation prod16 = y16 * x16/2 yscale = 216 k An Example if jprod16 + yscale · 0:4625j > 32767 rescale, divide by 2 Approximation add16 = int16(yscale * 0.4625339) integer to be added 16bit Arithmetic The Result y16 = prod16 + add16 In General Closing With the above numbers no rescaling is required, thus add16 = 13357 The result y16 satisfies yscale · y16 ≈ p2(x) Step 2: res2 = res1 · x + 0:4625339II 16bit Simulation with GNU Octave The result can be visualized, exact (double precision) and Andeas Stahel approximate (16bit) computations. Personal Context Result of step 2 40000 Compute on µC true Approximation 16bit An Example 20000 Approximation 16bit Arithmetic The Result 0 In General y scaled Closing -20000 -40000 -1 -0.5 0 0.5 1 x 1+x This is an approximation of the function arctan( 2 ). The Result 16bit Simulation with GNU Octave To examine the quality graph the difference of true function and its Andeas Stahel 16bit approximation. The accuracy is given by Personal 7.96 bit for difference to the arctan-function Context Compute on µC 13.6 bit for the difference to the polynomial p2(x) Approximation An Example The error is dominated by the Chebyshev approximation. Approximation 16bit Arithmetic Difference to 16bit approximation Difference to 16bit approximation The Result 0.003 -0.0012 In General 0.002 Closing 0.001 -0.0014 0 -0.0016 -0.001 difference difference -0.002 -0.0018 -0.003 -0.004 -1 -0.5 0 0.5 1 0.05 0.1 0.15 0.2 0.25 x x The Resulting C++ Code 16bit Simulation with GNU Octave Andeas Stahel #include <octave/oct.h> Personal DEFUN DLD (atan32,args ,nargout ,..