``` ||||| _ _ _ _ = Z . = | |_ | | |_ __ _ _ _ _ ___| |_ = , = | ' \| | | ' \| ' \ _| ' \/ -_) _| = o ` = |_||_|_|_|_|_|_|_||_(_)_||_\___|\__| ||||| ``` # Revisiting The Fast Inverse Square Root - Is It Still Useful? _April 20, 2023_ This article has some discussion on [Hacker News][hn-comments]. [hn-comments]: https://news.ycombinator.com/item?id=35646315 In 2005, id Software released the source code for their 1999 game _Quake III Arena_ under the GPL-2 license. In the file [code/game/q\_math.c][q_rsqrt], there is a function for calculating the reciprocal square root of a number which at first glance seems to use a very peculiar algorithm: [q_rsqrt]: https://github.com/id-Software/Quake-III-Arena/blob/master/code/game/q_math.c#L552 ```c float Q_rsqrt( float number ) { long i; float x2, y; const float threehalfs = 1.5F; x2 = number * 0.5F; y = number; i = * ( long * ) &y; // evil floating point bit level hacking i = 0x5f3759df - ( i >> 1 ); // what the fuck? y = * ( float * ) &i; y = y * ( threehalfs - ( x2 * y * y ) ); // 1st iteration // y = y * ( threehalfs - ( x2 * y * y ) ); // 2nd iteration, this can be removed return y; } ``` Many articles have been written about this particular algorithm and it has its own well written Wikipedia [page][wikipage] where it is referred to as the _fast inverse square root_. The algorithm actually appeared on various forums before the Q3 source code was released. Ryszard of Beyond3D did some [investigating][rys20061] in 2004-2005 and [eventually][rys20062] tracked down the original author of the algorithm to be Greg Walsh at Ardent Computer who created it more than a decade earlier. [wikipage]: https://en.wikipedia.org/wiki/Fast_inverse_square_root [rys20061]: https://www.beyond3d.com/content/articles/8/ [rys20062]: https://www.beyond3d.com/content/articles/15 Table of contents - 1 [How does it work?][] - 1.1 [First approximation][] - 1.2 [Improving the approximation][] - 2 [How fast is it?][] - 2.1 [Initial testing][] - 2.2 [Disassembly][] - 2.3 [Broad sweep][] - 2.4 [Try it yourself][] - 3 [Final thoughts][] ## How does it work? So how does the method work, anyway? It is performed in two steps: 1. obtain a rough approximation `y` for the reciprocal square root of our `number`: ```c y = number; i = * ( long * ) &y; i = 0x5f3759df - ( i >> 1 ); y = * ( float * ) &i; ``` 2. improve the approximation using a single step of the Newton-Raphson (NR) method: ```c const float threehalfs = 1.5F; x2 = number * 0.5F; y = y * ( threehalfs - ( x2 * y * y ) ); ``` ### First approximation The most interesting part is the first one. It uses a seemingly magic number `0x5f3759df` and some bit shifting and somehow ends up with the reciprocal square root. The first line stores the 32-bit floating-point number `y` as a 32-bit integer `i` by taking a pointer to `y`, converting it to a `long` pointer and dereferencing it. So `y` and `i` hold two identical 32-bit vectors, but one is interpreted as a floating-point number and the other is interpreted as an integer number. Then, the integer number is shifted one step to the right, negated, and the constant `0x5f3759df` is added. Finally, the resulting value is interpreted as a floating number again by dereferencing a `float` pointer that points to the integer `i` value. Here, shifting, negation and addition is performed in the integer domain, how do these operations affect the number in the floating-point domain? In order to understand how this can yield an approximation of the reciprocal square root we must be familiar with how floating point numbers are represented in memory. A floating-point number consists of a sign $`s \in \{0,1\}`, exponent $`e\in \mathbb{Z}` and a fractional part $`0 \leq f \lt 1`. The value of the floating-point number is then[^float-normal] $$`y = (-1)^s \cdot (1 + f) \cdot 2^e.` In our case, we can assume that our `float` is in the IEEE 754 [binary32][] format[^cfloat], the bits are then ordered as shown below. ``` -31-30----23-22--------------------0 | S | E (8b) | F (23b) | --- -------- ----------------------- ``` The most significant bit is the sign bit $`S`, followed by 8 bits ($`E`) representing the exponent $`e` and the remaining 23 bits ($`F`) representing the fractional part $`f`. The number is negative when $`S=1`. The 8-bit number $`E` is not directly used as the exponent, it has an offset or bias of $`2^8-1 = 127`. So $`E=0` means that the exponent is $`e=-127`. $`F` is simply a fractional binary number with the decimal point before the first digit such that $`f=F\cdot2^{-23}`. We can write a simple C program `interp.c` to print both the integer and floating-point interpretations of a given number and also extract the different parts: ```c #include #include #include #include int main(int argc, char *args[]) { /* parse number from args */ uint32_t i; int ret; if (argc == 2) { ret = sscanf(args[1], "%u", &i); } else if (argc == 3 && strcmp(args[1], "-h") == 0) { ret = sscanf(args[2], "%x", &i); } else if (argc == 3 && strcmp(args[1], "-f") == 0) { float y; ret = sscanf(args[2], "%f", &y); i = *(uint32_t*)&y; } else { return EXIT_FAILURE; } if (ret != 1) return EXIT_FAILURE; /* print representations */ printf("hexadecimal: %x\n", i); printf("unsigned int: %u\n", i); printf("signed int: %d\n", i); printf("floating-point: %f\n", *(float*)&i); /* print components */ int S = i >> 31; int E = (i >> 23) & ((1 << 8)-1); int e = E - 127; int F = i & ((1 << 23)-1); float f = (float)F / (1 << 23); printf("S: %d\n", S); printf("E: %d (0x%x) <=> e: %d\n", E, E, e); printf("F: %d (0x%x) <=> f: %f\n", F, F, f); return EXIT_SUCCESS; } ``` We can for example look at the number `0x40b00000`: ```cmdline $ ./interp -h 40b00000 hexadecimal: 40b00000 unsigned int: 1085276160 signed int: 1085276160 floating-point: 5.500000 S: 0 E: 129 (0x81) <=> e: 2 F: 3145728 (0x300000) <=> f: 0.375000 ``` We can also extract the parts of a floating-point number: ```cmdline $ ./interp -f -32.1 hexadecimal: c2006666 unsigned int: 3254806118 signed int: -1040161178 floating-point: -32.099998 S: 1 E: 132 (0x84) <=> e: 5 F: 26214 (0x6666) <=> f: 0.003125 ``` Even now when we know how the floating-point numbers are represented in memory, it is not entirely obvious how performing operations in the integer domain would affect the floating-point domain. At first we can try to simply iterate over a range of floating-point number and see what integer values we get: ```c #include int main() { float x; for (x = 0.1; x <= 8.0; x += 0.1) { printf("%f\t%d\n", x, *(int*)&x); } } ``` We can then plot the floating-point values on the x-axis and the integer values on the y-axis with e.g. [gnuplot][] to get a plot like this: [gnuplot]: http://www.gnuplot.info/ ``` 41000000 +------------------------------------------------------------+ | + + + + + +********** | | *********** | 40800000 |-+ ********* +-| | ***** | | ***** | 40000000 |-+ ***** +-| | ** | | ** | 3f800000 |-+ *** +-| | * | | ** | | * | 3f000000 |-+ * +-| | * | | * | 3e800000 |-* +-| | * | | * + + + + + + + | 3e000000 +------------------------------------------------------------+ 0 1 2 3 4 5 6 7 8 floating-point ``` Well, this curve looks quite familiar. We can look further at some of the data points using our previous program: ```cmdline $ ./interp -f 1.0 hexadecimal: 3f800000 unsigned int: 1065353216 signed int: 1065353216 floating-point: 1.000000 S: 0 E: 127 (0x7f) <=> e: 0 F: 0 (0x0) <=> f: 0.000000 $ ./interp -f 2.0 hexadecimal: 40000000 unsigned int: 1073741824 signed int: 1073741824 floating-point: 2.000000 S: 0 E: 128 (0x80) <=> e: 1 F: 0 (0x0) <=> f: 0.000000 $ ./interp -f 3.0 hexadecimal: 40400000 unsigned int: 1077936128 signed int: 1077936128 floating-point: 3.000000 S: 0 E: 128 (0x80) <=> e: 1 F: 4194304 (0x400000) <=> f: 0.500000 ``` For 1.0 and 2.0 we get $`S=0`, $`F=0` and a non-zero biased exponent $`E`. If we remove the bias from this number (subtract by `127 << 23`) and then shift it to the far right we end up with the exponent $`e`, in other words the base 2 logarithm of the floating-point number. However, this only works when $`S=0` and $`F=0`, i.e. positive integers. If $`S=1` we have a negative number for which the logarithm is undefined. But if $`F\ne{}0` and we shift the exponent to the far right we will simply lose all of that data. We can instead convert it to a floating-point value and divide by $`2^{23}`, such that the fractional part scales our resulting value linearly: ```c (float) (*(int*)&x - (127 << 23)) / (1 << 23) ``` Then we don't exactly get the logarithm but we do get a linear approximation for all non power of two values. We can plot the approximation together with the actual logarithmic function: ``` 3 +----------------------------------------------------------------------+ | + + + + + ###*********** | | ##************ | 2 |-+ #********** +-| | #****** | | #****** | | ****** | 1 |-+ #*** +-| | #*** | | #** | 0 |........**............................................................| | ** | | * | -1 |-+ #* +-| | * | | #* | | * | -2 |-* +-| | * log2(x) ####### | |* + + + + + approximation+******* | -3 +----------------------------------------------------------------------+ 0 1 2 3 4 5 6 7 8 ``` This means that when we take a floating-point number and interpret it as an integer number, we obtain an approximation of the logarithm of that number, with some offset and scaling. And when we interpret an integer number as a floating-point number, we get the opposite, i.e. the exponential or antilogarithm of our integer value. This basically means that when we perform operations in the integer domain, it is as if we perform operations in the logarithmic domain. For example, if we remember our [logarithmic identities][logident], we know that if we take the logarithm of two numbers and add them together, we get the logarithm of their product: $$`\log{a} + \log{b} = \log{(a \cdot b)}.` In other words, if we perform addition in the integer domain we get multiplication in the floating-point domain --- approximately anyway. We can try this with another simple C program. One thing we need to consider is how our operation affects the exponent bias. When we add two numbers with biased exponents we get double bias: $$` \begin{align} E_1 + E_2 &=& (e_1 + B) + (e_2 + B) \\ &=& e_1 + e_2 + 2B. \end{align} ` We want our bias to remain as $`B` rather than $`2B` so in order to counter this we simply subtract the result by $`B`. Our C program that performs floating-point multiplication using integer addition may then look like this: ```c #include #include #include const uint32_t B = (127 << 23); int main(int argc, char *args[]) { /* parse factors from args */ float a, b; if (argc == 3) { int ret = sscanf(args[1], "%f", &a); ret += sscanf(args[2], "%f", &b); if (ret != 2) return EXIT_FAILURE; } else { return EXIT_FAILURE; } /* perform multiplication (integer addition) */ uint32_t sum = *(uint32_t*)&a + *(uint32_t*)&b - B; float y = *(float*)∑ /* compare with actual */ float y_actual = a*b; float rel_err = (y - y_actual) / y_actual; printf("%f =? %f (%.2f%%)\n", y, y_actual, 100*rel_err); } ``` Let's try it out: ```cmdline $ ./mul 3.14159 8.0 25.132721 =? 25.132721 (0.00%) $ ./mul 3.14159 0.2389047 0.741016 =? 0.750541 (-1.27%) $ ./mul -15.0 3.0 -44.000000 =? -45.000000 (-2.22%) $ ./mul 6.0 3.0 16.000000 =? 18.000000 (-11.11%) $ ./mul 0.0 10.0 0.000000 =? 0.000000 (inf%) ``` Most of the time it is not perfectly accurate, it is correct only if one of the factors is a power of two, and least accurate when both factors are right between two powers of 2. How about the reciprocal square root? The reciprocal square root $`\frac{1}{\sqrt{x}}` is equivalent to $`x^{-1/2}` so we will need another logarithmic identity: $$`p\log{x} = \log{x^p}` This means that if we perform multiplication in the integer domain, we get exponentiation in the floating-point domain. Depending on our exponent $`p` we can obtain several different functions, e.g: | $`p` | $`f(x)` | |-------|-----------------------| | 2 | $`x^2` | | 1/2 | $`\sqrt{x}` | | -1 | $`\frac{1}{x}` | | -1/2 | $`\frac{1}{\sqrt{x}}` | In order to get a first approximation of the reciprocal square root, we simply need to multiply by -1/2 in the integer domain and adjust for the bias. The bias will then be $`-B/2` and we want the bias to be $`B` so we simply need to add $`3B/2 = \texttt{0x5f400000}`. So, we will multiply by -1/2 by shifting right one step and negating, and then add the bias: ```c - (i << 1) + 0x5f400000; ``` This is now identical to the Q3 source code except that the constant value differs slightly. They used `0x5f3759df` while we currently have `0x5f400000`. We can see if it is possible to make improvements by looking at our error. We simply subtract our approximate value for the reciprocal square root by the exact value and plot the result for a certain range of numbers: ``` f4240 +--------------------------------------------------------------+ | **+ *** + *** + + | dbba0 |-+ * ** ****** ***** +-| | * ** ** *** | c3500 |-+ * * *** *** +-| | * *** ** | aae60 |-+ * ** +-| | * * | 927c0 |-+ * ** +-| 7a120 |-+* ** +-| | * ** | 61a80 |-+* ** +-| | * * | 493e0 |-* ** +-| | * ** | 30d40 |*+ * +-| |* ** | 186a0 |*+ **-| |* + + + + + *| 0 +--------------------------------------------------------------+ 1 1.5 2 2.5 3 3.5 4 floating-point number ``` The graph repeats horizontally in both directions (only in different scale) so we only need to look at this part to understand the error for all (normal) floating-point numbers. We can see that the approximate value is always overestimating, by simply subtracting a constant that is around half the maximum error we can make it symmetric and thus decrease the average absolute error. Looking at the graph, subtracting something like 0x7a120 might work. Our constant would then be 0x5f385ee0 which is closer to the constant used in Q3. In the integer domain, our error will simply center the error around the x-axis in the above diagram. In the floating-point domain, the error is affected similarly except when our subtraction borrows from the exponent: ``` 0.03 +--------------------------------------------------------------+ | **+ *** + + + + | | * ** ************ | 0.02 |-+ * ** *** *** +-| | * * *** *** | | * *** ** | 0.01 |-+ * ** +-| | * * | | * ** | 0 |..*..............................................***..........| | * ** | | * ** | | * * | -0.01 |-* ** +-| | * ******| |* | -0.02 |*+ +-| |* | |* + + + + + | -0.03 +--------------------------------------------------------------+ 1 1.5 2 2.5 3 3.5 4 floating-point number ``` We could potentially try to find an actual optimum for some reasonable [objective function][] but we will stop here. In the case of the original Q3 constant, it is not really clear how it was chosen, perhaps using trial and error. [binary32]: https://en.wikipedia.org/wiki/Single-precision_floating-point_format [repeating fractional part]: https://en.wikipedia.org/wiki/Repeating_decimal [logident]: https://en.wikipedia.org/wiki/List_of_logarithmic_identities [objective function]: https://en.wikipedia.org/wiki/Objective_function ### Improving the approximation The second part is less unconventional. When a first approximation has been obtained, one can improve it by using a method known as Newton-Raphson (NR). If you are unfamiliar with it, Wikipedia has a good [article][wiki-nr] on it. The NR method is used to improve an approximation for the root of an equation. Since we want the reciprocal square root we need an equation $`f(y)` that is zero when $`y` is exactly the reciprocal square root of $`x`: $$` \begin{align} y = \frac{1}{\sqrt{x}} \: \Leftrightarrow \: \frac{1}{y^2} = x \\ \Rightarrow f(y) = \frac{1}{y^2} - x = 0 \end{align} ` If we have an approximate value $`y_n` we can get a better approximation $`y_{n+1}` by calculating where the tangent of the function's graph at $`y=y_n` (i.e. the derivative) intersects $`f(y)=0$$`. That value can be expressed as $$` \begin{align} y_{n+1} &=& y_n - \frac{f(y_n)}{f'(y_n)} \\ &=& y_n \left( \frac{3}{2} - \frac{x}{2} \cdot {y_n}^2 \right) \end{align} ` which is the exact same expression that is used in the second part of the Q3 function. [wiki-nr]: https://en.wikipedia.org/wiki/Newton%27s_method ## How fast is it? Back in 2003 Chris Lomont wrote an [article][lomont2003] about his investigations of the algorithm. His testing yielded that the algorithm was four times faster than using the more straightforward way of simply using `sqrt(x)`[^sqrt] from the standard library and taking its reciprocal. In 2009, Elan Ruskin made a post, [Timing Square Root][ruskin2009], where he primarily looked at the square root function but also compared the fast inverse square root algorithm to other methods. On his Intel Core 2, the fast inverse square root was 4 times slower than using `rsqrtss`, or 30% slower than `rsqrtss` with a single NR step. [lomont2003]: https://www.lomont.org/papers/2003/InvSqrt.pdf [ruskin2009]: https://web.archive.org/web/20210208132927/http://assemblyrequired.crashworks.org/timing-square-root/ Since then, there has come several new extensions to the x86 instruction set. I have tried to sum up all square root instructions currently available: | Set | $`\sqrt{x}` | $`\frac{1}{\sqrt{x}}` | Width | |--------------------|-------------------------------------|--------------------------------------------------------|-------| | [x87][] (1980) | `fsqrt` | | 32 | | [3DNow!][] (1998) | | `pfrsqrt` | 128 | | [SSE][] (1999) | `sqrtps`, `sqrtss` | `rsqrtps`, `rsqrtss` | 128 | | [SSE2][] (2000) | `sqrtpd`, `sqrtsd` | | 128 | | [AVX][] (2011) | `vsqrtps`, `vsqrtpd`, `vsqrtps_nr`, | `vrsqrtps`, `vrsqrtps_nr` | 256 | | [AVX-512][] (2014) | | `vrsqrt14pd`, `vrsqrt14ps`, `vrsqrt14sd`, `vrsqrt14ss` | 512 | [x87]: https://en.wikipedia.org/wiki/8087 [3DNow!]: https://en.wikipedia.org/wiki/3DNow! [SSE]: https://en.wikipedia.org/wiki/Streaming_SIMD_Extensions [SSE2]: https://en.wikipedia.org/wiki/SSE2 [AVX]: https://en.wikipedia.org/wiki/Advanced_Vector_Extensions [AVX-512]: https://en.wikipedia.org/wiki/AVX-512 The `fsqrt` is quite obsolete by now. The 3DNow! extension has also been deprecated and is no longer supported. All x86-64 processors support at least SSE and SSE2. Most processors support AVX and some support AVX-512, but e.g. GCC currently chooses to not emit any AVX instructions by default. The `p` and `s` is short for "packed" and "scalar". The packed instructions are vector [SIMD][] instructions while the scalar ones only operate on a single value at a time. With a register width of e.g. 256 bits, the packed instruction can perform $`256/32=8` calculations in parallel. The `s` or `d` is short for "single" or "double" precision floating-point. Since we are considering approximations we will be using single precision floating-point numbers. We may then use either the `ps` or `ss` variants. [SIMD]: https://en.wikipedia.org/wiki/Single_instruction,_multiple_data The fast inverse square root method had a pretty hard time against the `rsqrtss` instruction back in 2009 already. And since then, multiple extensions with specialized SIMD instructions has been implemented in modern x86 processors. Surely, the fast inverse square root has no chance today and its time has passed? Why don't we give it a try ourselves right now, we can start by running some tests on my current machine which has a relatively modern processor: an AMD Zen 3 5950X from late 2020. ### Initial testing We will write a C program that tries to calculate the reciprocal square root using three different methods: - `exact`: simply `1.0 / sqrtf(x)`, using the `sqrtf` function from the C standard library, - `appr`: first approximation from the Q3 source as explained above, - `appr_nr`: the full Q3 method with one iteration of Newton-Raphson. For each method we perform the calculation for each value in a randomized input array and time how long it takes in total. We can use the `clock_gettime` function from libc (for POSIX systems) to get the time before and after we perform the calculations and calculate the difference. We will then repeat this many times to decrease the random variations. The C program looks like this: ```c #include #include #include #include #include #define N 4096 #define T 1000 #define E9 1000000000 #ifndef CLOCK_REALTIME #define CLOCK_REALTIME 0 #endif enum methods { EXACT, APPR, APPR_NR, M }; const char *METHODS[] = { "exact", "appr", "appr_nr" }; static inline float rsqrt_exact(float x) { return 1.0f / sqrtf(x); } static inline float rsqrt_appr(float x) { uint32_t i = *(uint32_t*)&x; i = -(i >> 1) + 0x5f3759df; return *(float*)&i; } static inline float rsqrt_nr(float x, float y) { return y * (1.5f - x*0.5f*y*y); } static inline float rsqrt_appr_nr(float x) { float y = rsqrt_appr(x); return rsqrt_nr(x, y); } int main() { srand(time(NULL)); float y_sum[M] = {0}; double t[M] = {0}; for (int trial = 0; trial < T; trial++) { struct timespec start, stop; float x[N], y[N]; for (int i = 0; i < N; i++) { x[i] = rand(); } clock_gettime(CLOCK_REALTIME, &start); for (int i = 0; i < N; i++) { y[i] = rsqrt_exact(x[i]); } clock_gettime(CLOCK_REALTIME, &stop); for (int i = 0; i < N; i++) { y_sum[EXACT] += y[i]; } t[EXACT] += ((stop.tv_sec-start.tv_sec)*E9 + stop.tv_nsec-start.tv_nsec); clock_gettime(CLOCK_REALTIME, &start); for (int i = 0; i < N; i++) { y[i] = rsqrt_appr(x[i]); } clock_gettime(CLOCK_REALTIME, &stop); for (int i = 0; i < N; i++) { y_sum[APPR] += y[i]; } t[APPR] += ((stop.tv_sec-start.tv_sec)*E9 + stop.tv_nsec-start.tv_nsec); clock_gettime(CLOCK_REALTIME, &start); for (int i = 0; i < N; i++) { y[i] = rsqrt_appr_nr(x[i]); } clock_gettime(CLOCK_REALTIME, &stop); for (int i = 0; i < N; i++) { y_sum[APPR_NR] += y[i]; } t[APPR_NR] += ((stop.tv_sec-start.tv_sec)*E9 + stop.tv_nsec-start.tv_nsec); } printf("rsqrt\tps/op\tratio\terr\n"); for (int m = 0; m < M; m++) { printf("%s\t%.0f\t%.2f\t%.4f\n", METHODS[m], t[m] * 1000.0f / N / T, (double) t[EXACT] / t[m], (y_sum[m] - y_sum[EXACT]) / y_sum[EXACT]); } return 0; } ``` At the end of the program we print three things for each method: - the average time to calculate a single operation in picoseconds -- the lower the better, - the ratio of the calculation time compared to the exact method -- the higher the faster, - the average error between the method and the exact method -- just to make sure the calculations are performed correctly. So, what do we expect? There are dedicated functions for calculating the reciprocal square root in the x86 instruction set that the compiler should be able to emit. The throughput may then be higther than in the approximate method where we perform multiple operations. Let's go ahead and try it, we'll compile it using GCC without any optimizations at first, explicitly with `-O0`. Since we are using `math.h` for the exact method we will also need to link the math library using `-lm`: ```cmdline $ gcc -lm -O0 rsqrt.c $ ./a.out rsqrt ps/op ratio err exact 3330 1.00 0.0000 appr 2020 1.65 0.0193 appr_nr 6115 0.54 -0.0010 ``` This seems reasonable. The error is noticeable for the first approximation but reduced after one iteration of NR. The first approximation is actually faster than the exact method but when done together with a step of NR it is twice as slow. The NR method requires more operations so this seems reasonable. Alright, but this is only a debug build, let's try adding optimizations using the `-O3` flag. This will enable all optimizations that do not disregard any standards compliance. ```cmdline $ gcc -lm -O3 rsqrt.c $ ./a.out rsqrt ps/op ratio err exact 1879 1.00 0.0000 appr 72 26.01 0.0193 appr_nr 178 10.54 -0.0010 ``` Hmm, now the approximations are actually a lot faster than before but the time of the exact method has only halved, making the approximation with NR more than ten times faster than the exact method. Perhaps the compiler failed to emit the reciprocal square root functions? Maybe it will improve if we use the `-Ofast` flag instead which is described by the [gcc(1)][] man page: > Disregard strict standards compliance. -Ofast enables all -O3 optimizations. > It also enables optimizations that are not valid for all standard- compliant > programs. It turns on -ffast-math, -fallow-store-data-races and the > Fortran-specific -fstack-arrays, unless -fmax-stack-var-size is specified, > and -fno-protect-parens. It turns off -fsemantic-interposition. [gcc(1)]: https://www.man7.org/linux/man-pages/man1/gcc.1.html Our exact method may no longer be as accurate as before, but it may be faster. ```cmdline $ gcc -lm -Ofast rsqrt.c $ ./a.out rsqrt ps/op ratio err exact 153 1.00 0.0000 appr 118 1.30 0.0137 appr_nr 179 0.85 -0.0009 ``` And it is indeed faster. The first approximation is still faster, but with a step of NR it is slower than the exact method. The error has decreased slightly for the approximations because we are still comparing against the "exact" method which now yields different results. Oddly enough, the first approximation has become half as fast. This seems to be a quirk of GCC, as Clang does not have this issue, otherwise it produces similar results: ```cmdline $ clang -lm -O0 rsqrt.c $ ./a.out rsqrt ps/op ratio err exact 3715 1.00 0.0000 appr 1933 1.92 0.0193 appr_nr 6001 0.62 -0.0010 $ clang -lm -O3 rsqrt.c $ ./a.out rsqrt ps/op ratio err exact 1900 1.00 0.0000 appr 61 31.26 0.0193 appr_nr 143 13.24 -0.0010 $ clang -lm -Ofast rsqrt.c $ ./a.out rsqrt ps/op ratio err exact 148 1.00 0.0000 appr 62 2.40 0.0144 appr_nr 145 1.02 -0.0009 ``` ### Disassembly For both compilers, there is quite a large difference between `-O3` and `-Ofast`. We can look at the disassembly to see what is going on. We will need to provide the `-g` flag to the compiler in order to get debug symbols in the binary that tell us which object code corresponds to which source code. Thereafter we can run `objdump -d` with the `-S` flag to see the disassembled instructions next to the source code: ```objdumpc $ gcc -lm -O3 -g rsqrt.c $ objdump -d -S a.out ... static inline float rsqrt_exact(float x) { return 1.0f / sqrtf(x); } 118e: 66 0f ef db pxor %xmm3,%xmm3 1192: 0f 2e d8 ucomiss %xmm0,%xmm3 1195: 0f 87 e1 02 00 00 ja 147c 119b: f3 0f 51 c0 sqrtss %xmm0,%xmm0 119f: f3 0f 10 0d 99 0e 00 movss 0xe99(%rip),%xmm1 # 2040 11a6: 00 ... 11ab: f3 0f 5e c8 divss %xmm0,%xmm1 ... 2040: 00 00 80 3f # 1.0f ``` In case you are unfamiliar, this is the AT&T syntax for x86-64 assembly. Note that the source operand is always before the destination operand. The parentheses indicate an address, for example `movss 0xecd(%rip),%xmm1` copies the value located 0xecd bytes ahead of the address in the `rip` register (instruction pointer, a.k.a. PC) to the `xmm1` register. The `xmmN` registers are 128 bits wide, or 4 words. However, the `ss` instructions are for scalar single-precision values, so it will only apply the operation on a single floating-point value in the least significant 32 bits. In the `-O3` case we use the scalar `sqrtss` followed by `divss`. There is also a compare `ucomiss` and a jump `ja` that will set [`errno`][errno] to `EDOM` in case the input is less than -0. We are not using `errno` at all so we can remove the setting of `errno` by providing the `-fno-math-errno` flag: ```cmdline $ gcc -lm -O3 -g -fno-math-errno rsqrt.c $ ./a.out rsqrt ps/op ratio err exact 479 1.00 0.0000 appr 116 4.13 0.0193 appr_nr 175 2.74 -0.0010 ``` ```objdumpc $ objdump -d -S a.out ... static inline float rsqrt_exact(float x) { return 1.0f / sqrtf(x); } 1170: 0f 51 0c 28 sqrtps (%rax,%rbp,1),%xmm1 1174: f3 0f 10 1d c4 0e 00 movss 0xec4(%rip),%xmm3 # 2040 117b: 00 117c: 48 83 c0 10 add $0x10,%rax 1180: 0f c6 db 00 shufps $0x0,%xmm3,%xmm3 1184: 0f 28 c3 movaps %xmm3,%xmm0 1187: 0f 5e c1 divps %xmm1,%xmm0 ... 2040: 00 00 80 3f # 1.0f ``` [errno]: https://en.wikipedia.org/wiki/Errno.h This prevents us from having to check every input value individually and thus allows us to use the packed variants of the instructions, performing 4 operations at a time. This improved the performance a lot. However, we still use [`sqrtps`][sqrtps] followed by [`divps`][divps]. We will have to also enable `-funsafe-math-optimizations`[^unsafe-math] and `-ffinite-math-only` in order to make GCC emit [`rsqrtps`][rsqrtps] instead. We then get identical code to when we used `-Ofast`: ```cmdline $ gcc -lm -O3 -g -fno-math-errno -funsafe-math-optimizations -ffinite-math-only rsqrt.c $ ./a.out rsqrt ps/op ratio err exact 155 1.00 0.0000 appr 120 1.29 0.0137 appr_nr 182 0.85 -0.0009 ``` ```objdumpc $ objdump -d -S a.out ... static inline float rsqrt_exact(float x) { return 1.0f / sqrtf(x); } 1170: 0f 52 0c 28 rsqrtps (%rax,%rbp,1),%xmm1 1174: 0f 28 04 28 movaps (%rax,%rbp,1),%xmm0 1178: 48 83 c0 10 add $0x10,%rax 117c: 0f 59 c1 mulps %xmm1,%xmm0 117f: 0f 59 c1 mulps %xmm1,%xmm0 1182: 0f 59 0d c7 0e 00 00 mulps 0xec7(%rip),%xmm1 # 2050 1189: 0f 58 05 b0 0e 00 00 addps 0xeb0(%rip),%xmm0 # 2040 1190: 0f 59 c1 mulps %xmm1,%xmm0 ... 2040: 00 00 40 c0 # -3.0f ... 2050: 00 00 00 bf # -0.5f ``` [sqrtps]: https://www.felixcloutier.com/x86/sqrtps [divps]: https://www.felixcloutier.com/x86/divps [rsqrtps]: https://www.felixcloutier.com/x86/rsqrtps Now it uses `rsqrtps`, but it also has several multiplication instructions as well as an addition. Why are these needed, isn't the reciprocal square root all we need? We can get a hint from looking at the disassembly of the `appr_nr` function: ```objdumpc static inline float rsqrt_nr(float x, float y) { return y * (1.5f - x*0.5f*y*y); } 12f8: f3 0f 10 1d 80 0d 00 movss 0xd80(%rip),%xmm3 # 2080 12ff: 00 ... 1304: 0f 59 05 65 0d 00 00 mulps 0xd65(%rip),%xmm0 # 2070 ... 1310: 0f c6 db 00 shufps $0x0,%xmm3,%xmm3 ... 1318: 0f 28 d1 movaps %xmm1,%xmm2 131b: 0f 59 d1 mulps %xmm1,%xmm2 131e: 0f 59 d0 mulps %xmm0,%xmm2 1321: 0f 28 c3 movaps %xmm3,%xmm0 1324: 0f 5c c2 subps %xmm2,%xmm0 1327: 0f 59 c1 mulps %xmm1,%xmm0 ... 2070: 00 00 00 3f # 0.5f ... 2080: 00 00 c0 3f # 1.5f ``` The last part looks quite similar, because it is actually doing the same thing: an iteration of Newton-Raphson[^alt-nr]. This is hinted in the man page of [gcc(1)][]: > This option enables use of the reciprocal estimate and reciprocal square root > estimate instructions with additional Newton-Raphson steps to increase > precision instead of doing a divide or square root and divide for > floating-point arguments. The `rsqrtps` instruction only guarantees a relative error smaller than $`1.5\cdot2^{-12}`, the NR iteration reduces it further just like in the Q3 code. If we do not need this extra precision, can we get a speedup by skipping the NR step? We can use built-in compiler [intrinsics][] in order to make the compiler only emit the `rsqrtps` instruction. The GCC manual has a [list][gcc-x86-builtins] of built-in functions for the x86 instruction set. There is a `__builtin_ia32_rsqrtps` function that will emit the `rsqrtps` instruction: ```c v4sf __builtin_ia32_rsqrtps (v4sf); ``` [intrinsics]: https://en.wikipedia.org/wiki/Intrinsic_function [gcc-x86-builtins]: https://gcc.gnu.org/onlinedocs/gcc-12.2.0/gcc/x86-Built-in-Functions.html The manual also has a [chapter][gcc-vector] about how to use these vector instructions with built-in functions. We need to add a `typedef` for the `v4sf` type which contains four floating point numbers. We will then use an array of $`N/4` of these vectors and simply provide one vector at a time to the built-in function. N is a multiple of four so there are no half full vectors. We can simply cast our previous `float` input array to a `vfs4` pointer. We will add these parts to our previous program: ```c typedef float v4sf __attribute__ ((vector_size(16))); v4sf rsqrt_intr(v4sf x) { return __builtin_ia32_rsqrtps(x); }; v4sf *xv = (v4sf*)x, *yv = (v4sf*)y; clock_gettime(CLOCK_REALTIME, &start); for (int i = 0; i < N/4; i++) { yv[i] = rsqrt_intr(xv[i]); } clock_gettime(CLOCK_REALTIME, &stop); for (int i = 0; i < N; i++) { y_sum[INTR] += y[i]; } t[INTR] += ((stop.tv_sec-start.tv_sec)*E9 + stop.tv_nsec-start.tv_nsec); ``` [gcc-vector]: https://gcc.gnu.org/onlinedocs/gcc/Vector-Extensions.html We can compile it in order to run and disassemble it: ```cmdline $ gcc -lm -O3 -g rsqrt_vec.c $ ./a.out rsqrt ps/op ratio err exact 1895 1.00 0.0000 appr 72 26.39 0.0193 appr_nr 175 10.81 -0.0010 rsqrtps 61 31.00 0.0000 ``` ```objdumpc $ objdump -d -S a.out ... v4sf rsqrt_intr(v4sf x) { return __builtin_ia32_rsqrtps(x); }; 1238: 41 0f 52 04 04 rsqrtps (%r12,%rax,1),%xmm0 ... ``` Now we are down to a single instruction and it is slightly faster than before. There are also extensions that not all processors support that we can try to use. We can tell the compiler to use any extensions that are available on our processor using `-march=native`. This may make the binary incompatible with other processors, though. ```cmdline $ gcc -lm -Ofast -g -march=native rsqrt_vec.c $ ./a.out rsqrt ps/op ratio err exact 78 1.00 0.0000 appr 40 1.96 0.0137 appr_nr 85 0.91 -0.0009 rsqrtps 62 1.25 0.0000 ``` Now we are down to almost as good as the first approximation. The intrinsic one is pretty much just as fast. The "exact" method got replaced by a 256-bit [`vrsqrtps`][rsqrtps] and a step of NR: ```objdumpc static inline float rsqrt_exact(float x) { return 1.0f / sqrtf(x); } 11d0: c5 fc 52 0c 18 vrsqrtps (%rax,%rbx,1),%ymm1 11d5: c5 f4 59 04 18 vmulps (%rax,%rbx,1),%ymm1,%ymm0 11da: 48 83 c0 20 add $0x20,%rax 11de: c4 e2 75 a8 05 79 0e vfmadd213ps 0xe79(%rip),%ymm1,%ymm0 11e5: 00 00 11e7: c5 f4 59 0d 91 0e 00 vmulps 0xe91(%rip),%ymm1,%ymm1 11ee: 00 11ef: c5 fc 59 c1 vmulps %ymm1,%ymm0,%ymm0 ``` The `__builtin_ia32_rsqrtps` is now using a single `vrsqrtps` and no NR step, however, it still uses only 128-bit registers. ### Broad sweep So, we did some testing on my machine and got some insight into what kind of instructions we can use to calculate the reciprocal square root and how they might perform. We will now try to run these benchmarks on several machines to give us an idea how well our findings apply in general. Those machines include all the ones that I happen to have convenient SSH access to. All resulting data can be downloaded from [here][bench_res], it also includes results for the inverse and square root functions, separately. Below is a list of the x86 machines that were tested along with their CPUs and their release date. All the previous tests were run on on the computer labeled as "igelkott". | Hostname | CPU Family | CPU Model | Year | Form factor | |------------|----------------|---------------------|------|----------------| | jackalope | [Core][] | Intel Celeron 550 | 2007 | i686 laptop | | narwhal | [Piledriver][] | AMD FX-6300 | 2012 | x86_64 desktop | | silverback | [Ivy Bridge][] | Intel Xeon E5-1410 | 2014 | x86_64 server | | bovinae | [Kaby Lake][] | Intel Core i5-8250U | 2017 | x86_64 laptop | | igelkott | [Zen 3][] | AMD Ryzen 5950X | 2020 | x86_64 desktop | | deck | [Zen 2][] | AMD APU 0405 | 2022 | x86_64 mobile | [Core]: https://en.wikipedia.org/wiki/Core_(microarchitecture) [Piledriver]: https://en.wikipedia.org/wiki/Piledriver_(microarchitecture) [Ivy Bridge]: https://en.wikipedia.org/wiki/Ivy_Bridge_(microarchitecture) [Kaby Lake]: https://en.wikipedia.org/wiki/Kaby_Lake [Zen 3]: https://en.wikipedia.org/wiki/Zen_3 [Zen 2]: https://en.wikipedia.org/wiki/Zen_2 Below is a plot of the performance ratio compared to the `exact` method, i.e. the time of each method divided by the time of the `exact` method. A higher ratio means higher performance, anything below 1 is slower than `exact` and anything above is faster. We use the `-Ofast` flag here, as it is the fastest option that can be used without sacrificing portability. ``` narwhal-gcc ***A*** bovinae-gcc %%%D%%%igelkott-clang ===G=== narwhal-clang ###B### bovinae-clang @@@E@@@ deck-gcc ***H*** silverback-gcc $$$C$$$ igelkott-gcc &&&F&&& deck-clang ###I### Inverse square root performance ratio, -Ofast 4 +---------------------------------------------------------------------+ | + + + + | 3.5 |-+ +-| | | 3 |-+ #I +-| | ##@E | 2.5 |-+ =G= ##@@=G +-| 2 |-+ ==== ==== ##==== +-| | === $=== ##==& | 1.5 |-+ === ####I## $==== @##=& **H +-| |----------===########&&&&F&&######--$$====---==##&&-******-----------|- 1 |-+ I####&&&&########B####&&&&######$$=G=##*****########B +-| | #####&###I#*###### | 0.5 |-+ +-| | + + + + | 0 +---------------------------------------------------------------------+ exact appr appr\_nr rsqrtps ``` The results are quite similar across all of the machines, the time of the methods are approximately ranked in the order `rsqrtps` <= `appr` < `exact` <= `appr_nr`. Using the `appr_nr` method is either slower or the same as the `exact` method, so it has no real benefit in this case. The "jackalope" machine was not included in the above plot because it had an extremely slow `exact` method. Especially when not using `-march=native` as the compiler then resorted to using the antique `fsqrt` instruction. Below is a table of the actual timings when using `-Ofast`, numbers in parenthesis uses `-march=native`. Each number is how long a single operation takes in picoseconds. |Machine/Compiler|exact|appr|appr_nr|rsqrtps| |-------|-----|----|-------|-------| |jackalope-clang|53634 (5363)|1500 (2733)|4971 (3996)|N/A| |narwhal-gcc|419 (363)|443 (418)|601 (343)|396 (231)| |narwhal-clang|389 (796)|340 (321)|445 (859)|349 (388)| |silverback-gcc|422 (294)|179 (199)|543 (543)|178 (189)| |bovinae-gcc|260 (127)|155 (81)|321 (119)|108 (105)| |bovinae-clang|255 (132)|108 (78)|272 (112)|95 (96)| |igelkott-gcc|141 (79)|111 (63)|168 (87)|58 (64)| |igelkott-clang|152 (76)|63 (40)|149 (70)|61 (62)| |deck-gcc|342 (160)|234 (114)|444 (172)|226 (120)| |deck-clang|297 (166)|189 (123)|332 (140)|101 (126)| The square root function yields slightly different results: ``` narwhal-gcc ***A*** bovinae-gcc %%%D%%%igelkott-clang ===G=== narwhal-clang ###B### bovinae-clang @@@E@@@ deck-gcc ***H*** silverback-gcc $$$C$$$ igelkott-gcc &&&F&&& deck-clang ###I### Square root performance ratio, -Ofast 4 +---------------------------------------------------------------------+ | + + + + | 3.5 |-+ E +-| | @@ @@ | 3 |-+ @@ @@ +-| | @@ =G= @@ | 2.5 |-+ @@ ===$C$== @@ +-| 2 |-+ @@ ===$$##I#$$=== @ +-| | @@ ==$####**H*####$== @@ | 1.5 |-+ @@==####**** ***###===@@ +-| |---------@=####**&&&&&&&&F&&&&--**####==@---------------%%%%D--------|- 1 |-+ I##&&&&&& &&&&&&&*####@ %%%%########I +-| | &&*#I########********H | 0.5 |-+ $C$$$$$$$$&&&&&&&&F +-| | + + + + | 0 +---------------------------------------------------------------------+ exact appr appr\_nr sqrtps ``` Oddly enough, the `sqrtps` built-in function is slower than the `exact` method, and the `appr` without NR is now faster instead. The `appr_nr` method still offers no advantage, it is instead consistently worse than `exact`. Here are the original timings for the square root function as well, with `-Ofast`. Again, numbers in parentheses use `-march=native`: |Machine/Compiler|exact|appr|appr_nr|sqrtps| |-------|-----|----|-------|-------| |jackalope-clang|35197 (5743)|1494 (2738)|19191 (4308)|N/A| |narwhal-gcc|505 (399)|399 (427)|659 (559)|796 (785)| |narwhal-clang|448 (823)|327 (319)|638 (847)|803 (780)| |silverback-gcc|625 (297)|271 (190)|958 (728)|1163 (1135)| |bovinae-gcc|301 (148)|155 (81)|408 (200)|225 (226)| |bovinae-clang|315 (244)|92 (60)|399 (159)|317 (227)| |igelkott-gcc|173 (95)|119 (38)|233 (124)|288 (296)| |igelkott-clang|168 (143)|63 (48)|234 (104)|170 (283)| |deck-gcc|419 (205)|215 (108)|519 (252)|575 (574)| |deck-clang|325 (244)|153 (88)|372 (180)|315 (458)| [bench_res]: https://git.sr.ht/~nhellman/hllmn/tree/master/item/content/blog/2023-04-20_rsqrt/bench/res As noted in [discussions][hn-comments] on HN, some usecases may require determinism, i.e. the guarantee that different machines will produce the exact same result. The [`rsqrtps`][rsqrtps] instruction is only required to have a relative error smaller than $`1.5 \cdot 2^{−12}` so this can be implemented differently on different machines[^rsqrt-diff]. However, the [IEEE 754][ieee-754] specification does have a specification for the square root function, which the [`sqrtps`][sqrtps] instruction is guaranteed to comply with. So, the best we may be able to do is to use `sqrtps` followed by `divps` --- which is exactly what we got when we used `-O3` and `-fno-math-errno`. It wasn't until we introduced `-funsafe-math-optimzations` that we lost reproducibility. If reproducibility is a requirement, a fair comparison would use only IEEE-754 compliant floating-point operations: ``` narwhal-gcc ***A*** bovinae-gcc %%%D%%%igelkott-clang ===G=== narwhal-clang ###B### bovinae-clang @@@E@@@ deck-gcc ***H*** silverback-gcc $$$C$$$ igelkott-gcc &&&F&&& deck-clang ###I### Inverse square root performance ratio, -O3 -fno-math-errno 12 +---------------------------------------------------------------------+ | + # + + | 10 |-+ # C$ +-| | # $$ $$ | | # $$ $$$ | 8 |-+ # $$ =G= $$ +-| | # $$ == ==== $$$ | 6 |-+ ## $$ === ==== $$$ +-| | # $$ == ==== $$ | | # $$ === ##I##### ====$$ | 4 |-+ # $$=== ####&&F&&&**############==== +-| | # $$== #####&&&&%%D%%%@@&&&&&&******#####I | 2 |-+ #$===#####&&&%%%%% %%%%%%@@@@@@&&***H +-| |------------$=####&&%%%-----------------------------%%@@@E-----------|--- | I##% + + | 0 +---------------------------------------------------------------------+ exact appr appr\_nr ``` [ieee-754]: https://en.wikipedia.org/wiki/IEEE_754 This tips the scales in favor of the approximations. Using the `appr_nr` method results in a 2-4 times faster implementation compared to the `exact` method. `narwhal-clang` is an outlier here because of a surprisingly slow `exact` method. ### Try it yourself You can try to run the benchmarks on your machine and see if you get similar results. There is a shell script `bench/run.sh` that will generate and run benchmarks using the `bench/bench.c.m4` file. These files can be found in [this blog's repo][repodir]. Simply run the script with no arguments and it will generate a `.tsv` file with all results: ```cmdline $ cd bench $ sh run.sh $ grep rsqrt bench.tsv | sort -nk3 | head rsqrt appr 40 1.91 0.0139 clang-Ofast-march=native rsqrt rsqrtps 56 32.08 0.0000 clang-O3 rsqrt appr 58 31.08 0.0193 clang-O3 rsqrt rsqrtps 58 2.48 0.0000 clang-O3-fno-math-errno-funsafe-math-optimizations-ffinite-math-only rsqrt rsqrtps 59 2.45 0.0000 gcc-Ofast rsqrt rsqrtps 59 2.48 0.0000 clang-Ofast rsqrt rsqrtps 59 31.07 0.0000 gcc-O3 rsqrt rsqrtps 59 7.83 0.0000 gcc-O3-fno-math-errno rsqrt appr 60 2.41 0.0144 clang-O3-fno-math-errno-funsafe-math-optimizations-ffinite-math-only rsqrt rsqrtps 60 8.09 0.0000 clang-O3-fno-math-errno ``` [repodir]: https://git.sr.ht/~nhellman/hllmn/tree/master/item/content/blog/2023-04-20_rsqrt ## Final thoughts To summarize, using simply `1/sqrtf(x)` on modern x86 processors can be both faster and more accurate than the _fast inverse square root_ method from Quake III's `Q_rsqrt` function. However, a key takeaway is that you have to order the compiler to make it faster. When simply compiling using `-O3`, the fast inverse square root method is actually _considerably_ faster than the naive implementation. We have to allow the compiler to violate some strict specification requirements in order to make it emit a faster implementation. As noted by [discussions][hn-comments] on HN, there are usecases where determinism, i.e. reproducibility between machines, is a requirement. In this case, we may only use floating-point operations that are standardized to be deterministic, which the `rsqrtps` instruction is _not_. For such usecases, the _fast inverse square root_ method can be 2-4 times faster than the naive implementation. If very low accuracy can be tolerated, it is possible to get a slightly faster implementation by skipping the Newton-Raphson step from the fast inverse square root method. Interestingly, the compiler also performs an NR step after using approximate implementations of the inverse square root. This can also be made slightly faster by skipping the NR step --- by only emitting the approximate instruction with the help of compiler intrinsics. In this post, we focused on x86, but how about other instructions sets?[^other-isas] The fast inverse square root method could perhaps still be useful for processors without dedicated square root instructions. How are the hardware implementations of approximate square roots typically implemented? Could an approximate hardware implementation potentially use something similar to the first approximation of the fast inverse square root method? _2023-04-21_: correct femtoseconds -> picoseconds, discuss determinism [^float-normal]: This only covers all _normal_ floating-point values, there are also two zeroes, subnormal numbers and special values such as `NaN` and `inf`. The algorithm does not take these into consideration. [^cfloat]: `float` is not necessarily binary32 according to the C standard, but on most architectures, e.g. modern x86, it is. The author of the `Q_rsqrt` function has made the assumption that binary32 is used. [^sqrt]: Lomont wrote `(float)(1.0/sqrt(x))`, however `sqrt` (rather than `sqrtf` works on a `double` rather than a `float`. This could perhaps have made the naive method slower as this may cause casting to and from `double`. [^unsafe-math]: This includes `-fno-signed-zeros`, `-fno-trapping-math`, `-fassociative-math` and `-freciprocal-math`. I tried to set all of these individually but I was not able to get the `rsqrtps` instruction unless I enabled `-funsafe-math-optimizations`. It seems to do additional things besides enabling those flags. [^alt-nr]: The compiler has performed the operations in a slightly different order. They appear to be calculating it as `-0.5f*y*(x*y*y+-3.0f)`. [^rsqrt-diff]: As per Robert O'Callahan's [investigations][ocallahan2021], AMD and Intel implementations of rsqrt instructions seem to differ in exactly 22 out of the $`2^{32}` possible input values. [ocallahan2021]: https://robert.ocallahan.org/2021/09/emulating-amd-rsqrtss-etc-on-intel.html [^other-isas]: I did some quick tests on an armv7 phone, `appr_nr` performed similarly to `exact`. I also attempted running on a MIPS32 machine but I ran into issues with the cross-compiling toolchain. For another time..