Reducing Code Execution Time

Multiplication And Division

Fixed multiplication and division can converted to bit shifting to optimise code. The bit shifting operations can be found by doing a Taylor’s expansion of the equation. The error in the result depends on how many bit shifting operations you wish to perform. The below example uses bit shifting instead of fixed-number multiplication to reduces the number of execution cycles, and shows the difference and error between the result and the actual value.

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// Want to calculate output = 2*sqrt(2)*input
// This bit-shifting performs output = 2*input + input/2 + input/4 + input/16 + input/64,
// which is the Taylor's expansion of the above equation
output = (input<<1) + (input>>1) + (input>>2) + (input>>4)+ (input>>6);

// input = 1 -> output = 2.828125
// 2*sqrt(2)*1 = 2.828427125
// difference = 0.000302125
// error = 0.011%

Powers And Square Roots

pow(), a function provided with most IDEs, is a slow function. If raising to the power of an integer, multiply the variable by itself rather than use the pow() function.

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// Want to calculate output = 2*sqrt(2)*input
// This bit-shifting performs output = 2*input + input/2 + input/4 + input/16 + input/64,
// which is the Taylor's expansion of the above equation
output = (input<<1) + (input>>1) + (input>>2) + (input>>4)+ (input>>6);

// input = 1 -> output = 2.828125
// 2*sqrt(2)*1 = 2.828427125
// difference = 0.000302125
// error = 0.011%

There is a brilliant article on square root optimisation: http://www.azillionmonkeys.com/qed/sqroot.html

Fixed Point

mbedded.ninja has an open-source fixed-point library (MFixedPoint) that is hosted on GitLab.

Trigonometry

Fast and accurate sin/cos approximations can be found here.

The CORDIC algorithm can be used, which is a simple and efficient algorithm for calculating trigonometry and hyperbolic functions.