#### Strange things happen No. 121

##### Summing up in a different order.
 Q: We may reorder summing up within our power series defining $sin ⁡ ( x )$: Equation 3. Reordering of terms with respect to an implementation. $sin ⁡ ( x ) = x + ( - x 3 3 ! + x 5 5 ! ) + ( - x 7 7 ! + x 9 9 ! ) + ...$ From a mathematical point of view there is no difference to Equation 2, “Power series definition of $f ⁡ x = sin ⁡ ( x )$”. Nevertheless: Rename your current sine implementation to double sinOld(double). Implement a new method double sin(double) using the above summing reordering. Compare the results of double sinOld(double) and double sin(double) for seven terms. What do you observe? Do you have an explanation? A: Maven module source code available at P/Sd1/math/V3. The following results result from this test: Old Implementation:+++++++++++++++++++++++++++++++++++++++ sinOld(pi/2)=0.9999999999939768, difference=-6.023181953196399E-12 sinOld(pi)=-7.727858895155385E-7, difference=-7.727858895155385E-7 sinOld(4 * PI)=-9143.306026012957, difference=-9143.306026012957 New reorder Implementation:+++++++++++++++++++++++++++++++++++++ sin(pi/2)=1.0000000000000435, difference=4.3520742565306136E-14 sin(pi)=2.2419510618081458E-8, difference=2.2419510618081458E-8 sin(4 * PI)=4518.2187229323445, difference=4518.2187229323445Comparing corresponding values our reordered implementation is more than 100 times more precise. For larger values we still have a factor of two. This is due to limited arithmetic precision: Each double value can only be approximated by an in memory representation of eight bytes. Consider the following example: double one = 1., a = 0.000000000000200, b = 0.000000000000201; System.out.println("(1 + (a - b)) - 1:" + ((one + (a - b)) - one)); System.out.println("((1 + a) - b) - 1:" + (((one + a) - b) - one));This produces the following output:(1 + (a - b)) - 1:-9.992007221626409E-16 ((1 + a) - b) - 1:-8.881784197001252E-16Errors like this sum up in a power series giving rise to reasonable deviations especially if higher powers are involved.