They are not storing π with unusual floating-point precision. They are using an incorrect value for π with double precision. To approximate 3.1415926536 in binary, at least 38 bits are required:
3.14159265359922… > 11.001001000011111101101010100010001001
Notice that 2^-36 is about 1.5e-11, which coincides with the trailing 99. Double-precision floating-point has a 52-bit significand. To evaluate
cos(pi/2) as -5e-12, the only other possible choice would be a 48-bit type, which would be very strange.
Near 0 and π, where the derivative is nearly zero, cos(θ) cannot be calculated very accurately:
cos(3.1415926536) ≈ -0.999999999999999999999947911
That differs from -1 by about 5.2e-23, which is smaller than ε for
cos(3.1415926536) is calculated as exactly -1... which is incorrect.
Near ±π/2, the derivative [-sin(θ)] is nearly ±1, so the error at the input becomes the output.
cos(1.57079632679961) ≈ -4.71338076867830836e-12
cos(1.57079632679962) ≈ -4.72338076867830836e-12
cos(1.57079632680000) ≈ -5.10338076867830836e-12
I happen to have a TI calculator that displays one less digit and calculates
cos(π/2) as -5.2e-12. However, it is very different electronically and was designed to give an exact value for
I would guess that in Spotlight,
cos(pi/2) is being calculated by retrieving a value for π, converting to a decimal string, storing that as the (exact, rational) binary value 11.00100100001111110110101010001000100100001101101111 (or 10000), dividing by 2, and then essentially subtracting that from the true value of π/2. You should find out whether
cos(pi/2 + cos(pi/2)) is closer to zero (it might be -2.2e-35).
Multiplication by a power of two should affect only the exponent, not the significand. It might be possible to determine how rounding is applied by repeated halving or doubling.