Simplifying output in Matlab

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Aleem Andrew on 7 Nov 2021
Answered: Jan on 7 Nov 2021
If you get an expression like this as output
syms x
fx = -(1.0*(1.82e+39*x^2 + 1.81e+32*x - 3.04e+39))/(6.35e+41*x + 4.27e+37)
How do you simplify the output and use smaller numbers, for example
fx = -(1.0*(1.82e+7*x^2 + 1.81*x - 3.04e+7))/(6.35e+9*x + 4.27e+5)
Jan on 7 Nov 2021
The computation of smaller numbers is easier for human, but does not change the speed when performing the calculations on a computer.

Jan on 7 Nov 2021
The magnitude of the variables does not matter. For a computer, the addition of pi + 1.23456789, pi + 1.0 and pi+ 1.23456789e37 takes exactly the same time. Therefore the simplification does not accelerate the evaluation. And by the way, this is very fast in Matlab at all:
x = rand;
tic;
for k = 1:1e9
fx = -(1.0*(1.82e+39*x^2 + 1.81e+32*x - 3.04e+39))/(6.35e+41*x + 4.27e+37);
end
toc
Elapsed time is 0.777027 seconds.
tic;
for k = 1:1e9
fx = -(1.0*(1.82e+7*x^2 + 1.81*x - 3.04e+7))/(6.35e+9*x + 4.27e+5);
end
toc
Elapsed time is 0.706243 seconds.
Both take about 0.4 seconds on my i5-mobile Matlab R2018b. A tiny acceleration is measurable, if you omit the meaningless multiplication by 1.0.
The slow computation has another cause: ODE45 is a solver for non-stiff equations. Use a stiff solver for this equation instead:
tic
tspan = [0 10];
x0 = 0;
[t,x] = ode23s(@(t,x) -(1.0*(6.84e+45*x^2 + 5.24e+32*x - 2.49e+42))/(2.47e+39*x + 7.12e+37), tspan, x0);
plot(t,x,'b'); toc
Elapsed time is 0.207868 seconds.
There is a very sharp knee at the start. ODE45 struggles massively with it, because the stepsize controller drives crazy.