Julius Werner would like Shelley Chen, Hung-Te Lin and Yu-Ping Wu to review this change.
libpayload: Add simple 32.32 fixed-point math API
struct fraction is slooooooooooow. This patch adds a simple 64-bit
(32-bits integral, 32-bits fractional) fixed-point math API that is
*much* faster when doing intensive graphics operations. It is optimized
for speed over accuracy so some operations may lose a bit more precision
than expected, but overall it's still plenty of bits for most use cases.
Also includes support for basic trigonometric functions with a small
lookup table.
Signed-off-by: Julius Werner <jwerner@chromium.org>
Change-Id: Id0f9c23980e36ce0ac0b7c5cd0bc66153bca1fd0
---
A payloads/libpayload/include/fpmath.h
M payloads/libpayload/libc/Makefile.inc
A payloads/libpayload/libc/fpmath.c
3 files changed, 384 insertions(+), 0 deletions(-)
git pull ssh://review.coreboot.org:29418/coreboot refs/changes/93/42993/1
diff --git a/payloads/libpayload/include/fpmath.h b/payloads/libpayload/include/fpmath.h
new file mode 100644
index 0000000..db7cc21
--- /dev/null
+++ b/payloads/libpayload/include/fpmath.h
@@ -0,0 +1,234 @@
+/*
+ *
+ * Copyright (C) 2020 Google, Inc.
+ *
+ * Redistribution and use in source and binary forms, with or without
+ * modification, are permitted provided that the following conditions
+ * are met:
+ * 1. Redistributions of source code must retain the above copyright
+ * notice, this list of conditions and the following disclaimer.
+ * 2. Redistributions in binary form must reproduce the above copyright
+ * notice, this list of conditions and the following disclaimer in the
+ * documentation and/or other materials provided with the distribution.
+ * 3. The name of the author may not be used to endorse or promote products
+ * derived from this software without specific prior written permission.
+ *
+ * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
+ * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+ * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+ * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
+ * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+ * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
+ * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
+ * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
+ * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
+ * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
+ * SUCH DAMAGE.
+ */
+
+#include <stdint.h>
+
+/*
+ * This file implements operations for a simple 32.32 fixed-point math type.
+ * This is intended for speed-critical stuff (e.g. graphics) so there are
+ * intentionally no overflow checks or assertions, and operations are written
+ * to prefer speed over precision (e.g. multiplying by 1 may lose precision).
+ * For best results, only use for applications where 16.16 would fit.
+ */
+
+typedef struct { /* wrap in struct to prevent direct access */
+ int64_t v;
+} fpmath_t;
+
+#define FPMATH_SHIFT 32 /* define where to place the decimal point */
+
+/* Turn an integer into an fpmath_t. */
+static inline fpmath_t fp(int32_t a)
+{
+ return (fpmath_t){ .v = (int64_t)a << FPMATH_SHIFT };
+}
+
+/* Create an fpmath_t from a fraction. (numerator / denominator) */
+static inline fpmath_t fpfrac(int32_t numerator, int32_t denominator)
+{
+ return (fpmath_t){ .v = ((int64_t)numerator << FPMATH_SHIFT) / denominator };
+}
+
+/* Turn an fpmath_t back into an integer, rounding towards -INF. */
+static inline int32_t fpfloor(fpmath_t a)
+{
+ return a.v >> FPMATH_SHIFT;
+}
+
+/* Turn an fpmath_t back into an integer, rounding towards nearest. */
+static inline int32_t fpround(fpmath_t a)
+{
+ return (a.v + ((int64_t)1 << (FPMATH_SHIFT - 1))) >> FPMATH_SHIFT;
+}
+
+/* Turn an fpmath_t back into an integer, rounding towards +INF. */
+static inline int32_t fpceil(fpmath_t a)
+{
+ return (a.v + ((int64_t)1 << FPMATH_SHIFT) - 1) >> FPMATH_SHIFT;
+}
+
+/* Add two fpmath_t. (a + b) */
+static inline fpmath_t fpadd(fpmath_t a, fpmath_t b)
+{
+ return (fpmath_t){ .v = a.v + b.v };
+}
+
+/* Add an fpmath_t and an integer. (a + b) */
+static inline fpmath_t fpaddi(fpmath_t a, int32_t b)
+{
+ return (fpmath_t){ .v = a.v + ((int64_t)b << FPMATH_SHIFT) };
+}
+
+/* Subtract one fpmath_t from another. (a + b) */
+static inline fpmath_t fpsub(fpmath_t a, fpmath_t b)
+{
+ return (fpmath_t){ .v = a.v - b.v };
+}
+
+/* Subtract an integer from an fpmath_t. (a - b) */
+static inline fpmath_t fpsubi(fpmath_t a, int32_t b)
+{
+ return (fpmath_t){ .v = a.v - ((int64_t)b << FPMATH_SHIFT) };
+}
+
+/* Subtract an fpmath_t from an integer. (a - b) */
+static inline fpmath_t fpisub(int32_t a, fpmath_t b)
+{
+ return (fpmath_t){ .v = ((int64_t)a << FPMATH_SHIFT) - b.v };
+}
+
+/* Multiply two fpmath_t. (a * b)
+ Looses 16 bits fractional precision on each. */
+static inline fpmath_t fpmul(fpmath_t a, fpmath_t b)
+{
+ return (fpmath_t){ .v = (a.v >> (FPMATH_SHIFT/2)) * (b.v >> (FPMATH_SHIFT/2)) };
+}
+
+/* Multiply an fpmath_t and an integer. (a * b) */
+static inline fpmath_t fpmuli(fpmath_t a, int32_t b)
+{
+ return (fpmath_t){ .v = a.v * b };
+}
+
+/* Divide an fpmath_t by another. (a / b)
+ Looses 16 bits integral precision on each! Careful with this one! */
+static inline fpmath_t fpdiv(fpmath_t a, fpmath_t b)
+{
+ return (fpmath_t){ .v = (a.v << (FPMATH_SHIFT/2)) / (b.v >> (FPMATH_SHIFT/2)) };
+}
+
+/* Divide an fpmath_t by an integer. (a / b) */
+static inline fpmath_t fpdivi(fpmath_t a, int32_t b)
+{
+ return (fpmath_t){ .v = a.v / b };
+}
+
+/* Calculate absolute value of an fpmath_t. (ABS(a)) */
+static inline fpmath_t fpabs(fpmath_t a)
+{
+ return (fpmath_t){ .v = (a.v < 0 ? -a.v : a.v) };
+}
+
+/* Return true iff two fpmath_t are exactly equal. (a == b)
+ Like with floats, you probably don't want to use this most of the time. */
+static inline int fpequals(fpmath_t a, fpmath_t b)
+{
+ return a.v == b.v;
+}
+
+/* Return true iff one fpmath_t is less than another. (a < b) */
+static inline int fpless(fpmath_t a, fpmath_t b)
+{
+ return a.v < b.v;
+}
+
+/* Return true iff one fpmath_t is more than another. (a > b) */
+static inline int fpmore(fpmath_t a, fpmath_t b)
+{
+ return a.v > b.v;
+}
+
+/* Return the smaller of two fpmath_t. (MIN(a, b)) */
+static inline fpmath_t fpmin(fpmath_t a, fpmath_t b)
+{
+ if (a.v < b.v)
+ return a;
+ else
+ return b;
+}
+
+/* Return the larger of two fpmath_t. (MAX(a, b)) */
+static inline fpmath_t fpmax(fpmath_t a, fpmath_t b)
+{
+ if (a.v > b.v)
+ return a;
+ else
+ return b;
+}
+
+/* Return the constant PI as an fpmath_t (to as much precision as I was willing to bother). */
+static inline fpmath_t fppi(void)
+{
+ /* Can have one more digit when dividing unsigned, will be small enough to fit after. */
+ return (fpmath_t){ .v = ((uint64_t)3141592654 << FPMATH_SHIFT) / 1000000000 };
+}
+
+/*
+ * Returns the "one-based" sine of an fpmath_t, meaning the input is interpreted as if the range
+ * 0.0-1.0 corresponded to 0.0-PI/2 for radians. This is mostly here as the base primitives for
+ * the other trig stuff, but it may be useful to use directly if your input value already needs
+ * to be multiplied by some factor of PI and you want to save the instructions (and precision)
+ * for multiplying it in just so that the trig functions can divide it right out again.
+ */
+fpmath_t fpsin1(fpmath_t x);
+
+/* Returns the "one-based" cosine of an fpmath_t (analogous definition to fpsin1()). */
+static inline fpmath_t fpcos1(fpmath_t x)
+{
+ return fpsin1(fpaddi(x, 1));
+}
+
+/* Returns the sine of an fpmath_t interpreted as radians. */
+static inline fpmath_t fpsinr(fpmath_t radians)
+{
+ return fpsin1(fpdiv(radians, fpdivi(fppi(), 2)));
+}
+
+/* Returns the sine of an fpmath_t interpreted as degrees. */
+static inline fpmath_t fpsind(fpmath_t degrees)
+{
+ return fpsin1(fpdivi(degrees, 90));
+}
+
+/* Returns the cosine of an fpmath_t interpreted as radians. */
+static inline fpmath_t fpcosr(fpmath_t radians)
+{
+ return fpcos1(fpdiv(radians, fpdivi(fppi(), 2)));
+}
+
+/* Returns the cosine of an fpmath_t interpreted as degrees. */
+static inline fpmath_t fpcosd(fpmath_t degrees)
+{
+ return fpcos1(fpdivi(degrees, 90));
+}
+
+/* Returns the tangent of an fpmath_t interpreted as radians.
+ No safety rails, don't call this at the poles! */
+static inline fpmath_t fptanr(fpmath_t radians)
+{
+ fpmath_t one_based = fpdiv(radians, fpdivi(fppi(), 2));
+ return fpdiv(fpsin1(one_based), fpcos1(one_based));
+}
+
+/* Returns the tangent of an fpmath_t interpreted as degrees.
+ No safety rails, don't call this at the poles! */
+static inline fpmath_t fptand(fpmath_t degrees)
+{
+ fpmath_t one_based = fpdivi(degrees, 90);
+ return fpdiv(fpsin1(one_based), fpcos1(one_based));
+}
diff --git a/payloads/libpayload/libc/Makefile.inc b/payloads/libpayload/libc/Makefile.inc
index 2999023..f9006ae 100644
--- a/payloads/libpayload/libc/Makefile.inc
+++ b/payloads/libpayload/libc/Makefile.inc
@@ -38,3 +38,4 @@
libc-$(CONFIG_LP_LIBC) += die.c
libc-$(CONFIG_LP_LIBC) += coreboot.c
libc-$(CONFIG_LP_LIBC) += fmap.c
+libc-$(CONFIG_LP_LIBC) += fpmath.c
diff --git a/payloads/libpayload/libc/fpmath.c b/payloads/libpayload/libc/fpmath.c
new file mode 100644
index 0000000..4c2a6ac
--- /dev/null
+++ b/payloads/libpayload/libc/fpmath.c
@@ -0,0 +1,149 @@
+/*
+ *
+ * Copyright (C) 2020 Google, Inc.
+ *
+ * Redistribution and use in source and binary forms, with or without
+ * modification, are permitted provided that the following conditions
+ * are met:
+ * 1. Redistributions of source code must retain the above copyright
+ * notice, this list of conditions and the following disclaimer.
+ * 2. Redistributions in binary form must reproduce the above copyright
+ * notice, this list of conditions and the following disclaimer in the
+ * documentation and/or other materials provided with the distribution.
+ * 3. The name of the author may not be used to endorse or promote products
+ * derived from this software without specific prior written permission.
+ *
+ * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
+ * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+ * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+ * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
+ * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+ * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
+ * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
+ * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
+ * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
+ * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
+ * SUCH DAMAGE.
+ */
+
+#include <fpmath.h>
+
+/*
+ * This table represents one ascending arc of the sine curve, i.e. the values sin(x) for
+ * 0.0 <= x < PI/2. We divide that range into 256 equidistant points and store the corresponding
+ * sine values for those points. Since the values lie in the range 0.0 <= sin(x) < 1.0, in order
+ * to make the most use of the bytes we store, we map them to the range from 0 to UINT16_MAX.
+ *
+ * Generated with:
+ *
+ * for (i = 0; i < 256; i++) {
+ * double s = sin((double)i * M_PI / 2 / 256);
+ * uint16_t u = fmin(round(s * (1 << 16)), (1 << 16));
+ * printf("0x%04x,%s", u, i % 8 == 7 ? "\n" : " ");
+ * }
+ *
+ * In order to make sure the second access for linear interpolation (see below) cannot overrun
+ * the array, we stick a final 257th value 0xffff at the end. (It should really be 0x10000,
+ * but... this is good enough.)
+ */
+
+/* Table size as power of two. If we ever want to change the table size, updating this value
+ should make everything else fall back into place again (hopefully). */
+#define TP2 8
+
+static const uint16_t fpsin_table[(1 << TP2) + 1] = {
+ 0x0000, 0x0192, 0x0324, 0x04b6, 0x0648, 0x07da, 0x096c, 0x0afe,
+ 0x0c90, 0x0e21, 0x0fb3, 0x1144, 0x12d5, 0x1466, 0x15f7, 0x1787,
+ 0x1918, 0x1aa8, 0x1c38, 0x1dc7, 0x1f56, 0x20e5, 0x2274, 0x2402,
+ 0x2590, 0x271e, 0x28ab, 0x2a38, 0x2bc4, 0x2d50, 0x2edc, 0x3067,
+ 0x31f1, 0x337c, 0x3505, 0x368e, 0x3817, 0x399f, 0x3b27, 0x3cae,
+ 0x3e34, 0x3fba, 0x413f, 0x42c3, 0x4447, 0x45cb, 0x474d, 0x48cf,
+ 0x4a50, 0x4bd1, 0x4d50, 0x4ecf, 0x504d, 0x51cb, 0x5348, 0x54c3,
+ 0x563e, 0x57b9, 0x5932, 0x5aaa, 0x5c22, 0x5d99, 0x5f0f, 0x6084,
+ 0x61f8, 0x636b, 0x64dd, 0x664e, 0x67be, 0x692d, 0x6a9b, 0x6c08,
+ 0x6d74, 0x6edf, 0x7049, 0x71b2, 0x731a, 0x7480, 0x75e6, 0x774a,
+ 0x78ad, 0x7a10, 0x7b70, 0x7cd0, 0x7e2f, 0x7f8c, 0x80e8, 0x8243,
+ 0x839c, 0x84f5, 0x864c, 0x87a1, 0x88f6, 0x8a49, 0x8b9a, 0x8ceb,
+ 0x8e3a, 0x8f88, 0x90d4, 0x921f, 0x9368, 0x94b0, 0x95f7, 0x973c,
+ 0x9880, 0x99c2, 0x9b03, 0x9c42, 0x9d80, 0x9ebc, 0x9ff7, 0xa130,
+ 0xa268, 0xa39e, 0xa4d2, 0xa605, 0xa736, 0xa866, 0xa994, 0xaac1,
+ 0xabeb, 0xad14, 0xae3c, 0xaf62, 0xb086, 0xb1a8, 0xb2c9, 0xb3e8,
+ 0xb505, 0xb620, 0xb73a, 0xb852, 0xb968, 0xba7d, 0xbb8f, 0xbca0,
+ 0xbdaf, 0xbebc, 0xbfc7, 0xc0d1, 0xc1d8, 0xc2de, 0xc3e2, 0xc4e4,
+ 0xc5e4, 0xc6e2, 0xc7de, 0xc8d9, 0xc9d1, 0xcac7, 0xcbbc, 0xccae,
+ 0xcd9f, 0xce8e, 0xcf7a, 0xd065, 0xd14d, 0xd234, 0xd318, 0xd3fb,
+ 0xd4db, 0xd5ba, 0xd696, 0xd770, 0xd848, 0xd91e, 0xd9f2, 0xdac4,
+ 0xdb94, 0xdc62, 0xdd2d, 0xddf7, 0xdebe, 0xdf83, 0xe046, 0xe107,
+ 0xe1c6, 0xe282, 0xe33c, 0xe3f4, 0xe4aa, 0xe55e, 0xe610, 0xe6bf,
+ 0xe76c, 0xe817, 0xe8bf, 0xe966, 0xea0a, 0xeaab, 0xeb4b, 0xebe8,
+ 0xec83, 0xed1c, 0xedb3, 0xee47, 0xeed9, 0xef68, 0xeff5, 0xf080,
+ 0xf109, 0xf18f, 0xf213, 0xf295, 0xf314, 0xf391, 0xf40c, 0xf484,
+ 0xf4fa, 0xf56e, 0xf5df, 0xf64e, 0xf6ba, 0xf724, 0xf78c, 0xf7f1,
+ 0xf854, 0xf8b4, 0xf913, 0xf96e, 0xf9c8, 0xfa1f, 0xfa73, 0xfac5,
+ 0xfb15, 0xfb62, 0xfbad, 0xfbf5, 0xfc3b, 0xfc7f, 0xfcc0, 0xfcfe,
+ 0xfd3b, 0xfd74, 0xfdac, 0xfde1, 0xfe13, 0xfe43, 0xfe71, 0xfe9c,
+ 0xfec4, 0xfeeb, 0xff0e, 0xff30, 0xff4e, 0xff6b, 0xff85, 0xff9c,
+ 0xffb1, 0xffc4, 0xffd4, 0xffe1, 0xffec, 0xfff5, 0xfffb, 0xffff,
+ 0xffff,
+};
+
+/* x is in the "one-based" scale, so x == 1.0 is the top of the curve (PI/2 in radians). */
+fpmath_t fpsin1(fpmath_t x)
+{
+ /*
+ * When doing things like sin(x)/x, tiny errors can quickly become big problems, so just
+ * returning the nearest table value we have is not good enough (our fpmath_t has four
+ * times as much fractional precision as the sine table). A good and fast enough remedy
+ * is to linearly interpolate between the two nearest table values v1 and v2.
+ * (There are better but slower interpolations so we intentionally choose this one.)
+ *
+ * Most of this math can be done in 32 bits (because we're just operating on fractional
+ * parts in the 0.0-1.0 range anyway), so to help our 32-bit platforms a bit we keep it
+ * there as long as possible and only go back to an int64_t at the end.
+ */
+ uint32_t v1, v2;
+
+ /*
+ * Since x is "one-based" the part that maps to our curve (0.0 to PI/2 in radians) just
+ * happens to be exactly the fractional part of the fpmath_t, easy to extract.
+ */
+ int index = (x.v >> (FPMATH_SHIFT - TP2)) & ((1 << TP2) - 1);
+
+ /*
+ * In our one-based input domain, the period of the sine function is exactly 4.0. By
+ * extracting the first bit of the integral part of the fpmath_t we can check if it is
+ * odd-numbered (1.0-2.0, 3.0-4.0, etc.) or even numbered (0.0-1.0, 2.0-3.0, etc.), and
+ * that tells us whether we are in a "rising" (away from 0) or "falling" (towards 0)
+ * part of the sine curve. Our table curve is rising, so for the falling parts we have
+ * to mirror the curve horizontally by using the complement of our input index.
+ */
+ if (x.v & ((int64_t)1 << FPMATH_SHIFT)) {
+ v1 = fpsin_table[(1 << TP2) - index];
+ v2 = fpsin_table[(1 << TP2) - index - 1];
+ } else {
+ v1 = fpsin_table[index];
+ v2 = fpsin_table[index + 1];
+ }
+
+ /*
+ * Linear interpolation: sin(x) is interpolated as the closest number sin(x0) to the
+ * left of x we have in our table, plus the distance of that value to the closest number
+ * to the right of x (sin(x1)) times the fractional distance of x to x0. Since the table
+ * is conveniently exactly 256 values, x0 is exactly the upper 8 bits of the fractional
+ * part of x, meaning all fractional bits below that represent exactly the distance we
+ * need to interpolate over. (There are 24 of them but we need to multiply them with
+ * 16-bit table values to fit exactly 32 bits, so we discard the lowest 8 bits.)
+ */
+ uint32_t val = (v1 << (FPMATH_SHIFT - 16)) +
+ (v2 - v1) * ((x.v >> (16 - TP2)) & 0xffff);
+
+ /*
+ * Just like the first integral bit told us whether we need to mirror horizontally, the
+ * second can tell us to mirror vertically. In 2.0-4.0, 6.0-8.0, etc. of the input range
+ * the sine is negative, and in 0.0-2.0, 4.0-6.0, etc. it is positive.
+ */
+ if (x.v & ((int64_t)2 << FPMATH_SHIFT))
+ return (fpmath_t){ .v = -(int64_t)val };
+ else
+ return (fpmath_t){ .v = val };
+}
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