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NumericalCompliance
The floating point operations of the VideoCore IV QPU processors round using the IEEE 754 rount-to-zero rounding mode.
The rounding mode corresponds with the OpenCL CL_FP_ROUND_TO_ZERO mode and is allowed as only supported rounding mode for OpenCL 1.2 embedded profiles.
[Source]
NOTE: The CPU uses the IEEE 754 round-to-nearest-even rounding mode, which means that the results for a CPU and GPU floating-point operation might not match! To be exact, the result of the GPU operation is either the same or 1 ULP closer to zero as the result of the corresponding CPU operation.
As an example, consider this calculation: 358.6662292480469 - 3.0502657890319824:
| Calculation | Rounding mode | Result | Bit-cast integer |
|---|---|---|---|
| "Exact" | double precision | 355.6159634590149 | |
| CPU | round-to-nearest-even | 355.615966796875 | 0x43b1ced8 |
| GPU | round-to-zero | 355.6159362792969 | 0x43b1ced7 |
Neither of those results match the "exact" one, but both are "the closest value that can be represented" with single-precision floating point values using the corresponding rounding modes. Using the bit-cast integer representation, the difference of exactly 1 digit (corresponding to 1 ULP in floating-point) can be seen clearly.
- Inf is supported (at least by SFU)
- NaN is not supported
If x is a real number that lies between two finite consecutive floating-point numbers a and b, without being equal to one of them, then ulp(x) = |b - a|, otherwise ulp(x) is the distance between the two non-equal finite floating-point numbers nearest x. Moreover, ulp(NaN) is NaN.
The relative ULP is 2-23 ≈ 1.19e-07 [1] for single-precision floating-point values. So e.g. nextafter(1, 2) will return 1 + 2-23 ≈ 1.000000119 [2].
| Function | Allowed (in ULP) | Maximal error |
|---|---|---|
| x + y | correctly rounded (round-to-zero) | 0 |
| x - y | correctly rounded (round-to-zero) | 0 |
| x * y | correctly rounded (round-to-zero) | 0 |
| 1.0 / x | 3 | |
| x / y | 3 | |
| acos | 4 | |
| acospi | 5 | |
| asin | 4 | |
| asinpi | 5 | |
| atan | 5 | 1 |
| atan2 | 6 | |
| atanpi | 5 | |
| atan2pi | 6 | |
| acosh | 4 | |
| asinh | 4 | |
| atanh | 5 | |
| cbrt | 4 | |
| ceil | correctly rounded | 0 |
| clamp | 0 | 0 |
| copysign | 0 | 0 |
| cos | 4 | 2 |
| cosh | 4 | 2 |
| cospi | 4 | |
| cross | 3 | 0 |
| degrees | 2 | 2 |
| distance | 5.5 + 2 * len(vector) | 4 |
| dot | 2 * len(vector) - 1 | 0 |
| erfc | 16 | |
| erf | 16 | 1 |
| exp | 4 | 1 |
| exp2 | 4 | |
| exp10 | 4 | |
| expm1 | 4 | |
| fabs | 0 | 0 |
| fdim | correctly rounded | |
| floor | correctly rounded | 0 |
| fma | correctly rounded | 0 |
| fmax | 0 | 0 |
| fmin | 0 | 0 |
| fmod | 0 | |
| fract | correctly rounded | |
| frexp | 0 | |
| hypot | 4 | |
| ilogb | 0 | |
| length | 5.5 + len(vector) | 4 |
| ldexp | correctly rounded | |
| log | 4 | 4 |
| log2 | 4 | |
| log10 | 4 | |
| log1p | 4 | |
| logb | 0 | |
| mad | infinite | |
| max | 0 | 0 |
| maxmag | 0 | |
| min | 0 | 0 |
| minmag | 0 | |
| mix | absolute error of 1e-3 | 0 |
| modf | 0 | |
| nan | 0 | |
| nextafter | 0 | 0 |
| normalize | 4.5 + len(vector) | 7 |
| pow | 16 | |
| pown | 16 | |
| powr | 16 | |
| radians | 2 | 2 |
| remainder | 0 | |
| remquo | 0 | |
| rint | correclty rounded | 0 |
| rootn | 16 | |
| round | correclty rounded | 0 |
| rsqrt | 4 | 1 |
| sign | 0 | 0 |
| sin | 4 | 1 |
| sincos | 4 (both) | 2 |
| sinh | 4 | 2 |
| sinpi | 4 | |
| smoothstep | absolute error of 1e-5 | |
| sqrt | 4 | 1 |
| step | 0 | 0 |
| tan | 5 | |
| tanh | 5 | |
| tanpi | 6 | |
| tgamma | 16 | |
| trunc | correctly rounded | 0 |
| half_cos | 8192 | |
| half_divide | 8192 | 8192 |
| half_exp | 8192 | 8192 |
| half_exp2 | 8192 | 8192 |
| half_exp10 | 8192 | 8192 |
| half_log | 8192 | 8192 |
| half_log2 | 8192 | 8192 |
| half_log10 | 8192 | 8192 |
| half_powr | 8192 | 8192 |
| half_recip | 8192 | 8192 |
| half_rsqrt | 8192 | 8192 |
| half_sin | 8192 | |
| half_sqrt | 8192 | |
| half_tan | 8192 | |
| fast_distance | 8192 + 2 * len(vector) | |
| fast_length | 8192 + len(vector) | |
| fast_normalize | 8192 + len(vector) | |
| native_cos | impl.-defined | |
| native_divide | impl.-defined | 8192 |
| native_exp | impl.-defined | 8192 |
| native_exp2 | impl.-defined | 8192 |
| native_exp10 | impl.-defined | 8192 |
| native_log | impl.-defined | 8192 |
| native_log2 | impl.-defined | 8192 |
| native_log10 | impl.-defined | 8192 |
| native_powr | impl.-defined | 8192 |
| native_recip | impl.-defined | 8192 |
| native_rsqrt | impl.-defined | 8192 |
| native_sin | impl.-defined | |
| native_sqrt | impl.-defined | 8192 |
| native_tan | impl.-defined |
Sources: OpenCL 1.2 FULL PROFILE OpenCL 1.2 EMBEDDED PROFILE
Calculations of ULP are done via one of the following methods:
- Plotting the difference between the original function and the approximation with kmplot
- Calculating the result for the functions with the native C implementation and the custom approximation and checking the difference ( On host only)
Currently not supported