This benchmark is related to http://stackoverflow.com/questions/31555260/fast-vectorized-rsqrt-and-reciprocal-with-sse-avx-depending-on-precision.
It is known that division and square root are rather slow.
Instructions rcpps and rsqrtps allow to compute approximate results very fast for single precision floats.
Often they are immediately followed by Newton-Raphson iteration or some other numeric procedure with superlinear convergence.
Several approximate implementations of recip(x) = 1 / x and rsqrt(x) = 1 / sqrt(x) are gathered in this repo.
They are benchmarked for precision (maximum relative error is measured), and for throughput (number of CPU cycles per call).
Obtained with MSVC2013 C++ compiler in x64 mode on Intel Core i7-3770 (Ivy Bridge):
recip_float4_ieee: maximal error = 0
recip_float4_ieee: cycles per call = 7.30 (1.26497e+012)
recip_float4_fast: maximal error = 0.000300257
recip_float4_fast: cycles per call = 1.13 (1.26486e+012)
recip_float4_nr1: maximal error = 1.47033e-007
recip_float4_nr1: cycles per call = 3.69 (1.26497e+012)
recip_double2_ieee: maximal error = 0
recip_double2_ieee: cycles per call = 14.17 (6.34749e+011)
recip_double2_fast: maximal error = 0.000300175
recip_double2_fast: cycles per call = 3.36 (6.34688e+011)
recip_double2_nr1: maximal error = 9.01048e-008
recip_double2_nr1: cycles per call = 6.00 (6.34749e+011)
recip_double2_nr2: maximal error = 8.1719e-015
recip_double2_nr2: cycles per call = 9.42 (6.34749e+011)
recip_double2_r3: maximal error = 2.70471e-011
recip_double2_r3: cycles per call = 8.94 (6.34749e+011)
recip_double2_r4: maximal error = 8.33416e-015
recip_double2_r4: cycles per call = 10.09 (6.34749e+011)
recip_double2_r5: maximal error = 2.22045e-016
recip_double2_r5: cycles per call = 10.51 (6.34749e+011)
rsqrt_float4_ieee: maximal error = 0
rsqrt_float4_ieee: cycles per call = 14.46 (4.64887e+007)
rsqrt_float4_fast: maximal error = 0.000326109
rsqrt_float4_fast: cycles per call = 1.12 (4.6488e+007)
rsqrt_float4_nr1: maximal error = 3.10286e-007
rsqrt_float4_nr1: cycles per call = 5.52 (4.64887e+007)
rsqrt_double2_ieee: maximal error = 0
rsqrt_double2_ieee: cycles per call = 28.21 (2.73461e+007)
rsqrt_double2_fast: maximal error = 0.00032605
rsqrt_double2_fast: cycles per call = 3.38 (2.73475e+007)
rsqrt_double2_nr1: maximal error = 1.5948e-007
rsqrt_double2_nr1: cycles per call = 7.83 (2.7346e+007)
rsqrt_double2_nr2: maximal error = 3.8152e-014
rsqrt_double2_nr2: cycles per call = 13.18 (2.73461e+007)
rsqrt_double2_r2: maximal error = 1.5948e-007
rsqrt_double2_r2: cycles per call = 8.38 (2.7346e+007)
rsqrt_double2_r3: maximal error = 8.66757e-011
rsqrt_double2_r3: cycles per call = 10.65 (2.73461e+007)
rsqrt_double2_r4: maximal error = 4.95501e-014
rsqrt_double2_r4: cycles per call = 12.38 (2.73461e+007)
rsqrt_double2_r5: maximal error = 3.24557e-016
rsqrt_double2_r5: cycles per call = 13.51 (2.73461e+007)