atrosinenko updated this revision to Diff 288599.
atrosinenko added a comment.
This update is expected to be completely NFC w.r.t. code behavior and
significantly clarify the proof up to the end of half-width iterations.
Particularly, the reasoning about possible overflow of intermediate results
turned out to be actually unclear/incorrect.
@sepavloff could you take a look on the new version in case it clarifies some
of your questions? Another update for the second half of function may follow
slightly later.
Repository:
rG LLVM Github Monorepo
CHANGES SINCE LAST ACTION
https://reviews.llvm.org/D85031/new/
https://reviews.llvm.org/D85031
Files:
compiler-rt/lib/builtins/divdf3.c
compiler-rt/lib/builtins/divsf3.c
compiler-rt/lib/builtins/divtf3.c
compiler-rt/lib/builtins/fp_div_impl.inc
compiler-rt/lib/builtins/fp_lib.h
compiler-rt/lib/builtins/int_util.h
compiler-rt/test/builtins/Unit/divdf3_test.c
Index: compiler-rt/test/builtins/Unit/divdf3_test.c
===================================================================
--- compiler-rt/test/builtins/Unit/divdf3_test.c
+++ compiler-rt/test/builtins/Unit/divdf3_test.c
@@ -92,5 +92,13 @@
if (test__divdf3(0x1.0p+0, 0x1.00000001p+0, UINT64_C(0x3fefffffffe00000)))
return 1;
+ // some misc test cases obtained by fuzzing against h/w implementation
+ if (test__divdf3(0x1.fdc239dd64735p-658, -0x1.fff9364c0843fp-948, UINT64_C(0xd20fdc8fc0ceffb1)))
+ return 1;
+ if (test__divdf3(-0x1.78abb261d47c8p+794, 0x1.fb01d537cc5aep+266, UINT64_C(0xe0e7c6148ffc23e3)))
+ return 1;
+ if (test__divdf3(-0x1.da7dfe6048b8bp-875, 0x1.ffc7ea3ff60a4p-610, UINT64_C(0xaf5dab1fe0269e2a)))
+ return 1;
+
return 0;
}
Index: compiler-rt/lib/builtins/int_util.h
===================================================================
--- compiler-rt/lib/builtins/int_util.h
+++ compiler-rt/lib/builtins/int_util.h
@@ -28,4 +28,20 @@
#define COMPILE_TIME_ASSERT2(expr, cnt) \
typedef char ct_assert_##cnt[(expr) ? 1 : -1] UNUSED
+// Force unrolling the code specified to be repeated N times.
+#define REPEAT_0_TIMES(code_to_repeat) /* do nothing */
+#define REPEAT_1_TIMES(code_to_repeat) code_to_repeat
+#define REPEAT_2_TIMES(code_to_repeat) \
+ REPEAT_1_TIMES(code_to_repeat) \
+ code_to_repeat
+#define REPEAT_3_TIMES(code_to_repeat) \
+ REPEAT_2_TIMES(code_to_repeat) \
+ code_to_repeat
+#define REPEAT_4_TIMES(code_to_repeat) \
+ REPEAT_3_TIMES(code_to_repeat) \
+ code_to_repeat
+
+#define REPEAT_N_TIMES_(N, code_to_repeat) REPEAT_##N##_TIMES(code_to_repeat)
+#define REPEAT_N_TIMES(N, code_to_repeat) REPEAT_N_TIMES_(N, code_to_repeat)
+
#endif // INT_UTIL_H
Index: compiler-rt/lib/builtins/fp_lib.h
===================================================================
--- compiler-rt/lib/builtins/fp_lib.h
+++ compiler-rt/lib/builtins/fp_lib.h
@@ -40,9 +40,12 @@
#if defined SINGLE_PRECISION
+typedef uint16_t half_rep_t;
typedef uint32_t rep_t;
+typedef uint64_t twice_rep_t;
typedef int32_t srep_t;
typedef float fp_t;
+#define HALF_REP_C UINT16_C
#define REP_C UINT32_C
#define significandBits 23
@@ -58,9 +61,11 @@
#elif defined DOUBLE_PRECISION
+typedef uint32_t half_rep_t;
typedef uint64_t rep_t;
typedef int64_t srep_t;
typedef double fp_t;
+#define HALF_REP_C UINT32_C
#define REP_C UINT64_C
#define significandBits 52
@@ -102,9 +107,11 @@
#elif defined QUAD_PRECISION
#if __LDBL_MANT_DIG__ == 113 && defined(__SIZEOF_INT128__)
#define CRT_LDBL_128BIT
+typedef uint64_t half_rep_t;
typedef __uint128_t rep_t;
typedef __int128_t srep_t;
typedef long double fp_t;
+#define HALF_REP_C UINT64_C
#define REP_C (__uint128_t)
// Note: Since there is no explicit way to tell compiler the constant is a
// 128-bit integer, we let the constant be casted to 128-bit integer
Index: compiler-rt/lib/builtins/fp_div_impl.inc
===================================================================
--- /dev/null
+++ compiler-rt/lib/builtins/fp_div_impl.inc
@@ -0,0 +1,378 @@
+//===-- fp_div_impl.inc - Floating point division -----------------*- C -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+//
+// This file implements soft-float division with the IEEE-754 default
+// rounding (to nearest, ties to even).
+//
+//===----------------------------------------------------------------------===//
+
+#include "fp_lib.h"
+
+// The __divXf3__ function implements Newton-Raphson floating point division.
+// It uses 3 iterations for float32, 4 for float64 and 5 for float128,
+// respectively. Due to number of significant bits being roughly doubled
+// every iteration, the two modes are supported: N full-width iterations (as
+// it is done for float32 by default) and (N-1) half-width iteration plus one
+// final full-width iteration. It is expected that half-width integer
+// operations (w.r.t rep_t size) can be performed faster for some hardware but
+// they require error estimations to be computed separately due to larger
+// computational errors caused by truncating intermediate results.
+
+// Half the bit-size of rep_t
+#define HW (typeWidth / 2)
+// rep_t-sized bitmask with lower half of bits set to ones
+#define loMask (REP_C(-1) >> HW)
+
+#define NUMBER_OF_ITERATIONS \
+ (NUMBER_OF_HALF_ITERATIONS + NUMBER_OF_FULL_ITERATIONS)
+
+#if NUMBER_OF_FULL_ITERATIONS < 1
+#error At least one full iteration is required
+#endif
+
+static __inline fp_t __divXf3__(fp_t a, fp_t b) {
+
+ const unsigned int aExponent = toRep(a) >> significandBits & maxExponent;
+ const unsigned int bExponent = toRep(b) >> significandBits & maxExponent;
+ const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit;
+
+ rep_t aSignificand = toRep(a) & significandMask;
+ rep_t bSignificand = toRep(b) & significandMask;
+ int scale = 0;
+
+ // Detect if a or b is zero, denormal, infinity, or NaN.
+ if (aExponent - 1U >= maxExponent - 1U ||
+ bExponent - 1U >= maxExponent - 1U) {
+
+ const rep_t aAbs = toRep(a) & absMask;
+ const rep_t bAbs = toRep(b) & absMask;
+
+ // NaN / anything = qNaN
+ if (aAbs > infRep)
+ return fromRep(toRep(a) | quietBit);
+ // anything / NaN = qNaN
+ if (bAbs > infRep)
+ return fromRep(toRep(b) | quietBit);
+
+ if (aAbs == infRep) {
+ // infinity / infinity = NaN
+ if (bAbs == infRep)
+ return fromRep(qnanRep);
+ // infinity / anything else = +/- infinity
+ else
+ return fromRep(aAbs | quotientSign);
+ }
+
+ // anything else / infinity = +/- 0
+ if (bAbs == infRep)
+ return fromRep(quotientSign);
+
+ if (!aAbs) {
+ // zero / zero = NaN
+ if (!bAbs)
+ return fromRep(qnanRep);
+ // zero / anything else = +/- zero
+ else
+ return fromRep(quotientSign);
+ }
+ // anything else / zero = +/- infinity
+ if (!bAbs)
+ return fromRep(infRep | quotientSign);
+
+ // One or both of a or b is denormal. The other (if applicable) is a
+ // normal number. Renormalize one or both of a and b, and set scale to
+ // include the necessary exponent adjustment.
+ if (aAbs < implicitBit)
+ scale += normalize(&aSignificand);
+ if (bAbs < implicitBit)
+ scale -= normalize(&bSignificand);
+ }
+
+ // Set the implicit significand bit. If we fell through from the
+ // denormal path it was already set by normalize( ), but setting it twice
+ // won't hurt anything.
+ aSignificand |= implicitBit;
+ bSignificand |= implicitBit;
+
+ int writtenExponent = (aExponent - bExponent + scale) + exponentBias;
+
+ const rep_t b_UQ1 = bSignificand << (typeWidth - significandBits - 1);
+
+ // Align the significand of b as a UQ1.(n-1) fixed-point number in the range
+ // [1.0, 2.0) and get a UQ0.n approximate reciprocal using a small minimax
+ // polynomial approximation: x0 = 3/4 + 1/sqrt(2) - b/2.
+ // Analytically for infinitely precise computations, for b in [1, 2):
+ // abs(x0(b) - 1/b) <= 3/4 - 1/sqrt(2)
+ // Computationally, the initial approximation is between x0(1.0)
+ // (about 0.9571) and x0(2.0) (about 0.4571).
+
+ // Then, refine the reciprocal estimate using a Newton-Raphson iteration:
+ // x_{n+1} = x_n * (2 - x_n * b)
+ //
+ // Let b be the original divisor considered "in infinite precision" and
+ // obtained from IEEE754 representation of function argument (with the
+ // implicit bit set). Corresponds to rep_t-sized b_UQ1 represented in
+ // UQ1.(W-1).
+ //
+ // Let b_hw be an infinitely precise number obtained from the highest (HW-1)
+ // bits of divisor significand (with the implicit bit set). Corresponds to
+ // half_rep_t-sized b_UQ1_hw represented in UQ1.(HW-1) that is a **truncated**
+ // version of b_UQ1.
+ //
+ // Let e_n := x_n - 1/b_hw
+ // E_n := x_n - 1/b
+ // abs(E_n) <= abs(e_n) + (1/b_hw - 1/b)
+ // = abs(e_n) + (b - b_hw) / (b*b_hw)
+ // <= abs(e_n) + 2 * 2^-HW
+
+ // rep_t-sized iterations may be slower than the corresponding half-width
+ // variant depending on the handware and whether single/double/quad precision
+ // is selected.
+ // NB: Using half-width iterations increases computation errors due to
+ // rounding, so error estimations have to be computed taking the selected
+ // mode into account!
+#if NUMBER_OF_HALF_ITERATIONS > 0
+ // Starting with (n-1) half-width iterations
+ const half_rep_t b_UQ1_hw = bSignificand >> (significandBits + 1 - HW);
+
+ // C is (3/4 + 1/sqrt(2)) - 1 truncated to W0 fractional bits as UQ0.HW
+ // with W0 being either 16 or 32 and W0 <= HW.
+ // That is, C is the aforementioned 3/4 + 1/sqrt(2) constant (from which
+ // b/2 is subtracted to obtain x0) wrapped to [0, 1) range.
+#if defined(SINGLE_PRECISION)
+ // Use 16-bit initial estimation in case we are using half-width iterations
+ // for float32 division. This is expected to be useful for some 16-bit
+ // targets. Not used by default as it requires performing more work during
+ // rounding and would hardly help on regular 32- or 64-bit targets.
+ const half_rep_t C_hw = HALF_REP_C(0x7504);
+#else
+ // HW is at least 32. Shifting into the highest bits if needed.
+ const half_rep_t C_hw = HALF_REP_C(0x7504F333) << (HW - 32);
+#endif
+
+ // b >= 1, thus an upper bound for 3/4 + 1/sqrt(2) - b/2 is about 0.9572,
+ // so x0 fits to UQ0.HW without wrapping.
+ half_rep_t x_UQ0_hw = C_hw - (b_UQ1_hw /* exact b_hw/2 as UQ0.HW */);
+ // An e_0 error is comprised of errors due to
+ // * x0 being an inherently imprecise first approximation of 1/b_hw
+ // * C_hw being some (irrational) number **truncated** to W0 bits
+ // Please note that e_0 is calculated against the infinitely precise
+ // reciprocal of b_hw (that is, **truncated** version of b).
+ //
+ // e_0 <= 3/4 - 1/sqrt(2) + 2^-W0
+
+ // By construction, 1 <= b < 2
+ // f(x) = x * (2 - b*x) = 2*x - b*x^2
+ // f'(x) = 2 * (1 - b*x)
+ //
+ // On the (0, 1) interval, f(0) = 0,
+ // then it increses until f(1/b) = 1 / b, maximum on (0, 1),
+ // then it decreses to f(1) = 2 - b
+ REPEAT_N_TIMES(NUMBER_OF_HALF_ITERATIONS, {
+ // corr_UQ1_hw can be **larger** than 2 - b*x by at most 1*Ulp of corr_UQ1_hw.
+ // "0.0 - (...)" is equivalent to "2.0 - (...)" in UQ1.(HW-1).
+ // On the other hand, corr_UQ1_hw should not overflow from 2.0 to 0.0 provided
+ // no overflow ocurred earlier: ((rep_t)x_UQ0_hw * b_UQ1_hw >> HW) is
+ // expected to be strictly positive because b_UQ1_hw has its highest bit set
+ // and x_UQ0_hw should be rather large (it converges to 1/2 < 1/b_hw <= 1).
+ half_rep_t corr_UQ1_hw = 0 - ((rep_t)x_UQ0_hw * b_UQ1_hw >> HW);
+
+ // Now, we should multiply UQ0.HW and UQ1.(HW-1) numbers, naturally
+ // obtaining an UQ1.(HW-1) number and proving its highest bit could be
+ // considered to be 0 to be able to represent it in UQ0.HW.
+ // From the above analysis of f(x), if corr_UQ1_hw would be represented
+ // without any intermediate loss of precision (that is, in twice_rep_t)
+ // x_UQ0_hw could be at most [1.]000... if b_hw is exactly 1.0 and strictly
+ // less otherwise. On the other hand, to obtain [1.]000..., one have to pass
+ // 1/b_hw == 1.0 to f(x), so this cannot occur at all without overflow.
+ // The fact corr_UQ1_hw was virtually round up (due to result of
+ // multiplication being **first** truncated, then negated) can increase
+ // x_UQ0_hw by up to 2*Ulp of x_UQ0_hw.
+ x_UQ0_hw = (rep_t)x_UQ0_hw * corr_UQ1_hw >> (HW - 1);
+ // Now, either no overflow occurred or x_UQ0_hw is 0 or 1 in its half_rep_t
+ // representation. In the latter case, x_UQ0_hw will be either 0 or 1 after
+ // any number of iterations, so just subtract 2 from the reciprocal
+ // approximation after last iteration.
+
+ // In infinite precision, with 0 <= eps1, eps2 <= U = 2^-HW:
+ // corr_UQ1_hw = 2 - (1/b_hw + e_n) * b_hw + 2*eps1
+ // = 1 - e_n * b_hw + 2*eps1
+ // x_UQ0_hw = (1/b_hw + e_n) * (1 - e_n*b_hw + 2*eps1) - eps2
+ // = 1/b_hw - e_n + 2*eps1/b_hw + e_n - e_n^2*b_hw + 2*e_n*eps1 - eps2
+ // = 1/b_hw + 2*eps1/b_hw - e_n^2*b_hw + 2*e_n*eps1 - eps2
+ // e_{n+1} = -e_n^2*b_hw + 2*eps1/b_hw + 2*e_n*eps1 - eps2
+ // = 2*e_n*eps1 - (e_n^2*b_hw + eps2) + 2*eps1/b_hw
+ // \------ >0 -------/ \-- >0 ---/
+ // abs(e_{n+1}) <= 2*abs(e_n)*U + max(2 * U, U + 2*e_n^2)
+ })
+ // For initial half-width iterations, U = 2^-HW
+ // Let abs(e_n) <= u_n * U,
+ // then abs(e_{n+1}) <= U * [2*u_n*U + max(2, 1 + 2*u_n^2*U)]
+ // u_{n+1} <= 2 * u_n * U + max(2, 1 + 2 * u_n^2 * U)
+
+ // Account for possible overflow (see above) before proceeding with full-width
+ // iterations because the condition b == 1.0 may become false here if b is
+ // *close enough* to 1.0.
+ x_UQ0_hw -= 1U;
+ rep_t x_UQ0 = (rep_t)x_UQ0_hw << HW;
+ x_UQ0 -= 1U;
+
+#else
+ // C is (3/4 + 1/sqrt(2)) - 1 truncated to 32 fractional bits as UQ0.n
+ const rep_t C = REP_C(0x7504F333) << (typeWidth - 32);
+ rep_t x_UQ0 = C - b_UQ1;
+ // E_0 <= 3/4 - 1/sqrt(2) + 2 * 2^-32
+#endif
+
+ // Error estimations for full-precision iterations are calculated
+ // just as above, but with Ulp := 2^-W. We need at least one such iteration.
+
+#ifdef USE_NATIVE_FULL_ITERATIONS
+ REPEAT_N_TIMES(NUMBER_OF_FULL_ITERATIONS, {
+ rep_t corr_UQ1 = 0 /* = 2 */ - ((twice_rep_t)x_UQ0 * b_UQ1 >> typeWidth);
+ x_UQ0 = (twice_rep_t)x_UQ0 * corr_UQ1 >> (typeWidth - 1);
+ })
+#else
+#if NUMBER_OF_FULL_ITERATIONS != 1
+#error Only a single emulated full iteration is supported
+#endif
+#if !(NUMBER_OF_HALF_ITERATIONS > 0)
+ // Cannot normally reach here: only one full-width iteration is requested and
+ // the total number of iterations should be at least 3 even for float32.
+#error Check NUMBER_OF_HALF_ITERATIONS, NUMBER_OF_FULL_ITERATIONS and USE_NATIVE_FULL_ITERATIONS.
+#endif
+ rep_t blo = b_UQ1 & loMask;
+ // x_UQ0 = x_UQ0_hw * 2^HW - 1
+ // x_UQ0 * b_UQ1 = (x_UQ0_hw * 2^HW) * (b_UQ1_hw * 2^HW + blo) - b_UQ1
+ rep_t corr_UQ1 = 0 - ( (rep_t)x_UQ0_hw * b_UQ1_hw
+ + ((rep_t)x_UQ0_hw * blo >> HW)
+ - REP_C(1)); // to account for *possible* carry due to "- b_UQ1"
+ rep_t lo_corr = corr_UQ1 & loMask;
+ rep_t hi_corr = corr_UQ1 >> HW;
+ // x_UQ0 * corr_UQ1 = (x_UQ0_hw * 2^HW) * (hi_corr * 2^HW + lo_corr) - corr_UQ1
+ x_UQ0 = ((rep_t)x_UQ0_hw * hi_corr << 1)
+ + ((rep_t)x_UQ0_hw * lo_corr >> (HW - 1))
+ - REP_C(2); // 1 to account for the highest bit of corr_UQ1 can be 1 and
+ // plus 1 to account for possible carry
+ // Just like the case of half-width iterations but with possibility
+ // of overflowing to 2
+ x_UQ0 -= 1U;
+ // ... and then traditional fixup by 2 should work
+
+ // On error estimates, analogously to the half-width iterations:
+ // abs(E_{N-1}) <= (u_{N-1} + 2 /* due to conversion e_n -> E_n */) * 2^-HW + (2^-HW + 2^-W))
+ // abs(E_{N-1}) <= (u_{N-1} + 3.01) * 2^-HW
+
+ // With 0 <= eps1 < 2^-W
+ // E_N = -E_{N-1}^2*b + 4*eps1/b + 4*E_{N-1}*eps1 - (1+2+1)*eps2
+ // = 4*E_{N-1}*eps1 - (E_{N-1}^2 * b + 4 * eps2) + 4*eps1/b
+ // E_N <= 2^-W * [ 4*(u_{N-1} + 3.01) * 2^-HW + 4 + 2 * (e_{N-1} + 3.01)^2 ]
+#endif
+
+ // Finally, account for possible overflow, as explained above.
+ x_UQ0 -= 2U;
+
+ // u_n for different precisions (with N-1 half-width iterations):
+ // W0 is the precision of C
+ // u_0 = (3/4 - 1/sqrt(2) + 2^-W0) * 2^HW + 1
+
+ // | f32 | f64 | f128
+ // Iterations | | |
+ // 0 | < 2813.1 | < 184224974 | < 791240229949381501
+ // 1 | < 242.7 | < 15804007 | < 67877680634450550
+ // 2 | < 2.81 | < 116308 | < 499533089406164
+ // 3 | | < 7.31 | < 27054455403
+ // 4 | | | < 80.4
+
+ // Final error | < 74 / 2^32 | < 220 / 2^64 | < 13921 * 2^-128
+
+#if defined(SINGLE_PRECISION) && NUMBER_OF_HALF_ITERATIONS == 2
+#define RECIPROCAL_PRECISION REP_C(74)
+#elif defined(SINGLE_PRECISION) && NUMBER_OF_ITERATIONS == 3
+#define RECIPROCAL_PRECISION REP_C(8)
+#elif defined(DOUBLE_PRECISION) && NUMBER_OF_ITERATIONS == 4
+#define RECIPROCAL_PRECISION REP_C(220)
+#elif defined(QUAD_PRECISION) && NUMBER_OF_ITERATIONS == 5
+#define RECIPROCAL_PRECISION REP_C(13921)
+#else
+#error Invalid number of iterations
+#endif
+
+ // Suppose 1/b - P * 2^-W < x < 1/b + P * 2^-W
+ x_UQ0 -= RECIPROCAL_PRECISION;
+ // Now 1/b - (2*P) * 2^-W < x < 1/b
+ // FIXME Is x_UQ0 still >= 0.5?
+
+ rep_t quotient_UQ1, dummy;
+ wideMultiply(x_UQ0, aSignificand << 1, "ient_UQ1, &dummy);
+ // Now, a/b - 4*P * 2^-W < q < a/b
+
+ // quotient_UQ1 is in [0.5, 2.0) as UQ1.(SB+1), adjust it to be in [1.0, 2.0) as UQ1.SB
+ rep_t residualLo;
+ if (quotient_UQ1 < (implicitBit << 1)) {
+ residualLo = (aSignificand << (significandBits + 1)) - quotient_UQ1 * bSignificand;
+ writtenExponent -= 1;
+
+ // the error is doubled
+ } else {
+ quotient_UQ1 >>= 1;
+ residualLo = (aSignificand << significandBits) - quotient_UQ1 * bSignificand;
+ }
+ // Now, q cannot be greater than a/b and can differ by at most 8*P * 2^-W + 2^-SB
+ // Each NextAfter() increments the floating point value by at least 2^-SB
+ // (more, if exponent was incremented).
+ // Different cases (<---> is of 2^-SB length, * = a/b that is shown as a midpoint):
+ // q
+ // | | * | | | | |
+ // <---> 2^t
+ // | | | | | * | |
+ // q
+ // To require at most one NextAfter(), an error should be less than 1.5 * 2^-SB.
+ // (8*P) * 2^-W + 2^-SB < 1.5 * 2^-SB <=> (8*P) * 2^-W < 0.5 * 2^-SB <=> P < 2^(W-4-SB)
+ // Generally, for at most R NextAfter(), the P < (2*R - 1) * 2^(W-4-SB)
+ // For f32: 8 < 32 (OK) but 71 > 32 (but two NextAfter() are enough)
+ // For f64: 220 < 256 (OK)
+ // For f128: 14120 > 4096 (*three* NextAfter() are required)
+
+ // If we have overflowed the exponent, return infinity
+ if (writtenExponent >= maxExponent)
+ return fromRep(infRep | quotientSign);
+
+ // Now, quotient_UQ1_SB <= the correctly-rounded result
+ // and may need taking NextAfter() up to 3 times (see error estimates above)
+ // r = a - b * q
+
+ if (writtenExponent < 0) {
+ // Result is definitely subnormal, flushing to zero
+ return fromRep(quotientSign);
+ }
+
+ // Clear the implicit bit
+ rep_t absResult = quotient_UQ1 & significandMask;
+ // Insert the exponent
+ absResult |= (rep_t)writtenExponent << significandBits;
+
+ // Round
+ residualLo <<= 1;
+ residualLo += absResult & 1; // tie to even
+ absResult += residualLo > bSignificand;
+#if defined(QUAD_PRECISION) || (defined(SINGLE_PRECISION) && NUMBER_OF_HALF_ITERATIONS > 0)
+ // Do not round Infinity to NaN
+ absResult += absResult < infRep && residualLo > (2 + 1) * bSignificand;
+#endif
+#if defined(QUAD_PRECISION)
+ absResult += absResult < infRep && residualLo > (4 + 1) * bSignificand;
+#endif
+
+ if ((absResult & ~significandMask) == 0) {
+ // Result is subnormal, flushing to zero
+ return fromRep(quotientSign);
+ }
+ // Result is normal, insert the sign and return
+ return fromRep(absResult | quotientSign);
+}
Index: compiler-rt/lib/builtins/divtf3.c
===================================================================
--- compiler-rt/lib/builtins/divtf3.c
+++ compiler-rt/lib/builtins/divtf3.c
@@ -9,213 +9,18 @@
// This file implements quad-precision soft-float division
// with the IEEE-754 default rounding (to nearest, ties to even).
//
-// For simplicity, this implementation currently flushes denormals to zero.
-// It should be a fairly straightforward exercise to implement gradual
-// underflow with correct rounding.
-//
//===----------------------------------------------------------------------===//
#define QUAD_PRECISION
#include "fp_lib.h"
#if defined(CRT_HAS_128BIT) && defined(CRT_LDBL_128BIT)
-COMPILER_RT_ABI fp_t __divtf3(fp_t a, fp_t b) {
-
- const unsigned int aExponent = toRep(a) >> significandBits & maxExponent;
- const unsigned int bExponent = toRep(b) >> significandBits & maxExponent;
- const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit;
-
- rep_t aSignificand = toRep(a) & significandMask;
- rep_t bSignificand = toRep(b) & significandMask;
- int scale = 0;
-
- // Detect if a or b is zero, denormal, infinity, or NaN.
- if (aExponent - 1U >= maxExponent - 1U ||
- bExponent - 1U >= maxExponent - 1U) {
-
- const rep_t aAbs = toRep(a) & absMask;
- const rep_t bAbs = toRep(b) & absMask;
-
- // NaN / anything = qNaN
- if (aAbs > infRep)
- return fromRep(toRep(a) | quietBit);
- // anything / NaN = qNaN
- if (bAbs > infRep)
- return fromRep(toRep(b) | quietBit);
-
- if (aAbs == infRep) {
- // infinity / infinity = NaN
- if (bAbs == infRep)
- return fromRep(qnanRep);
- // infinity / anything else = +/- infinity
- else
- return fromRep(aAbs | quotientSign);
- }
-
- // anything else / infinity = +/- 0
- if (bAbs == infRep)
- return fromRep(quotientSign);
-
- if (!aAbs) {
- // zero / zero = NaN
- if (!bAbs)
- return fromRep(qnanRep);
- // zero / anything else = +/- zero
- else
- return fromRep(quotientSign);
- }
- // anything else / zero = +/- infinity
- if (!bAbs)
- return fromRep(infRep | quotientSign);
-
- // One or both of a or b is denormal. The other (if applicable) is a
- // normal number. Renormalize one or both of a and b, and set scale to
- // include the necessary exponent adjustment.
- if (aAbs < implicitBit)
- scale += normalize(&aSignificand);
- if (bAbs < implicitBit)
- scale -= normalize(&bSignificand);
- }
-
- // Set the implicit significand bit. If we fell through from the
- // denormal path it was already set by normalize( ), but setting it twice
- // won't hurt anything.
- aSignificand |= implicitBit;
- bSignificand |= implicitBit;
- int quotientExponent = aExponent - bExponent + scale;
-
- // Align the significand of b as a Q63 fixed-point number in the range
- // [1, 2.0) and get a Q64 approximate reciprocal using a small minimax
- // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2. This
- // is accurate to about 3.5 binary digits.
- const uint64_t q63b = bSignificand >> 49;
- uint64_t recip64 = UINT64_C(0x7504f333F9DE6484) - q63b;
- // 0x7504f333F9DE6484 / 2^64 + 1 = 3/4 + 1/sqrt(2)
-
- // Now refine the reciprocal estimate using a Newton-Raphson iteration:
- //
- // x1 = x0 * (2 - x0 * b)
- //
- // This doubles the number of correct binary digits in the approximation
- // with each iteration.
- uint64_t correction64;
- correction64 = -((rep_t)recip64 * q63b >> 64);
- recip64 = (rep_t)recip64 * correction64 >> 63;
- correction64 = -((rep_t)recip64 * q63b >> 64);
- recip64 = (rep_t)recip64 * correction64 >> 63;
- correction64 = -((rep_t)recip64 * q63b >> 64);
- recip64 = (rep_t)recip64 * correction64 >> 63;
- correction64 = -((rep_t)recip64 * q63b >> 64);
- recip64 = (rep_t)recip64 * correction64 >> 63;
- correction64 = -((rep_t)recip64 * q63b >> 64);
- recip64 = (rep_t)recip64 * correction64 >> 63;
-
- // The reciprocal may have overflowed to zero if the upper half of b is
- // exactly 1.0. This would sabatoge the full-width final stage of the
- // computation that follows, so we adjust the reciprocal down by one bit.
- recip64--;
-
- // We need to perform one more iteration to get us to 112 binary digits;
- // The last iteration needs to happen with extra precision.
- const uint64_t q127blo = bSignificand << 15;
- rep_t correction, reciprocal;
-
- // NOTE: This operation is equivalent to __multi3, which is not implemented
- // in some architechure
- rep_t r64q63, r64q127, r64cH, r64cL, dummy;
- wideMultiply((rep_t)recip64, (rep_t)q63b, &dummy, &r64q63);
- wideMultiply((rep_t)recip64, (rep_t)q127blo, &dummy, &r64q127);
-
- correction = -(r64q63 + (r64q127 >> 64));
-
- uint64_t cHi = correction >> 64;
- uint64_t cLo = correction;
-
- wideMultiply((rep_t)recip64, (rep_t)cHi, &dummy, &r64cH);
- wideMultiply((rep_t)recip64, (rep_t)cLo, &dummy, &r64cL);
-
- reciprocal = r64cH + (r64cL >> 64);
-
- // Adjust the final 128-bit reciprocal estimate downward to ensure that it
- // is strictly smaller than the infinitely precise exact reciprocal. Because
- // the computation of the Newton-Raphson step is truncating at every step,
- // this adjustment is small; most of the work is already done.
- reciprocal -= 2;
-
- // The numerical reciprocal is accurate to within 2^-112, lies in the
- // interval [0.5, 1.0), and is strictly smaller than the true reciprocal
- // of b. Multiplying a by this reciprocal thus gives a numerical q = a/b
- // in Q127 with the following properties:
- //
- // 1. q < a/b
- // 2. q is in the interval [0.5, 2.0)
- // 3. The error in q is bounded away from 2^-113 (actually, we have a
- // couple of bits to spare, but this is all we need).
-
- // We need a 128 x 128 multiply high to compute q, which isn't a basic
- // operation in C, so we need to be a little bit fussy.
- rep_t quotient, quotientLo;
- wideMultiply(aSignificand << 2, reciprocal, "ient, "ientLo);
-
- // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0).
- // In either case, we are going to compute a residual of the form
- //
- // r = a - q*b
- //
- // We know from the construction of q that r satisfies:
- //
- // 0 <= r < ulp(q)*b
- //
- // If r is greater than 1/2 ulp(q)*b, then q rounds up. Otherwise, we
- // already have the correct result. The exact halfway case cannot occur.
- // We also take this time to right shift quotient if it falls in the [1,2)
- // range and adjust the exponent accordingly.
- rep_t residual;
- rep_t qb;
- if (quotient < (implicitBit << 1)) {
- wideMultiply(quotient, bSignificand, &dummy, &qb);
- residual = (aSignificand << 113) - qb;
- quotientExponent--;
- } else {
- quotient >>= 1;
- wideMultiply(quotient, bSignificand, &dummy, &qb);
- residual = (aSignificand << 112) - qb;
- }
+#define NUMBER_OF_HALF_ITERATIONS 4
+#define NUMBER_OF_FULL_ITERATIONS 1
- const int writtenExponent = quotientExponent + exponentBias;
+#include "fp_div_impl.inc"
- if (writtenExponent >= maxExponent) {
- // If we have overflowed the exponent, return infinity.
- return fromRep(infRep | quotientSign);
- } else if (writtenExponent < 1) {
- if (writtenExponent == 0) {
- // Check whether the rounded result is normal.
- const bool round = (residual << 1) > bSignificand;
- // Clear the implicit bit.
- rep_t absResult = quotient & significandMask;
- // Round.
- absResult += round;
- if (absResult & ~significandMask) {
- // The rounded result is normal; return it.
- return fromRep(absResult | quotientSign);
- }
- }
- // Flush denormals to zero. In the future, it would be nice to add
- // code to round them correctly.
- return fromRep(quotientSign);
- } else {
- const bool round = (residual << 1) >= bSignificand;
- // Clear the implicit bit.
- rep_t absResult = quotient & significandMask;
- // Insert the exponent.
- absResult |= (rep_t)writtenExponent << significandBits;
- // Round.
- absResult += round;
- // Insert the sign and return.
- const fp_t result = fromRep(absResult | quotientSign);
- return result;
- }
-}
+COMPILER_RT_ABI fp_t __divtf3(fp_t a, fp_t b) { return __divXf3__(a, b); }
#endif
Index: compiler-rt/lib/builtins/divsf3.c
===================================================================
--- compiler-rt/lib/builtins/divsf3.c
+++ compiler-rt/lib/builtins/divsf3.c
@@ -9,181 +9,17 @@
// This file implements single-precision soft-float division
// with the IEEE-754 default rounding (to nearest, ties to even).
//
-// For simplicity, this implementation currently flushes denormals to zero.
-// It should be a fairly straightforward exercise to implement gradual
-// underflow with correct rounding.
-//
//===----------------------------------------------------------------------===//
#define SINGLE_PRECISION
-#include "fp_lib.h"
-
-COMPILER_RT_ABI fp_t __divsf3(fp_t a, fp_t b) {
-
- const unsigned int aExponent = toRep(a) >> significandBits & maxExponent;
- const unsigned int bExponent = toRep(b) >> significandBits & maxExponent;
- const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit;
-
- rep_t aSignificand = toRep(a) & significandMask;
- rep_t bSignificand = toRep(b) & significandMask;
- int scale = 0;
-
- // Detect if a or b is zero, denormal, infinity, or NaN.
- if (aExponent - 1U >= maxExponent - 1U ||
- bExponent - 1U >= maxExponent - 1U) {
-
- const rep_t aAbs = toRep(a) & absMask;
- const rep_t bAbs = toRep(b) & absMask;
-
- // NaN / anything = qNaN
- if (aAbs > infRep)
- return fromRep(toRep(a) | quietBit);
- // anything / NaN = qNaN
- if (bAbs > infRep)
- return fromRep(toRep(b) | quietBit);
-
- if (aAbs == infRep) {
- // infinity / infinity = NaN
- if (bAbs == infRep)
- return fromRep(qnanRep);
- // infinity / anything else = +/- infinity
- else
- return fromRep(aAbs | quotientSign);
- }
-
- // anything else / infinity = +/- 0
- if (bAbs == infRep)
- return fromRep(quotientSign);
-
- if (!aAbs) {
- // zero / zero = NaN
- if (!bAbs)
- return fromRep(qnanRep);
- // zero / anything else = +/- zero
- else
- return fromRep(quotientSign);
- }
- // anything else / zero = +/- infinity
- if (!bAbs)
- return fromRep(infRep | quotientSign);
-
- // One or both of a or b is denormal. The other (if applicable) is a
- // normal number. Renormalize one or both of a and b, and set scale to
- // include the necessary exponent adjustment.
- if (aAbs < implicitBit)
- scale += normalize(&aSignificand);
- if (bAbs < implicitBit)
- scale -= normalize(&bSignificand);
- }
-
- // Set the implicit significand bit. If we fell through from the
- // denormal path it was already set by normalize( ), but setting it twice
- // won't hurt anything.
- aSignificand |= implicitBit;
- bSignificand |= implicitBit;
- int quotientExponent = aExponent - bExponent + scale;
- // 0x7504F333 / 2^32 + 1 = 3/4 + 1/sqrt(2)
-
- // Align the significand of b as a Q31 fixed-point number in the range
- // [1, 2.0) and get a Q32 approximate reciprocal using a small minimax
- // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2. This
- // is accurate to about 3.5 binary digits.
- uint32_t q31b = bSignificand << 8;
- uint32_t reciprocal = UINT32_C(0x7504f333) - q31b;
-
- // Now refine the reciprocal estimate using a Newton-Raphson iteration:
- //
- // x1 = x0 * (2 - x0 * b)
- //
- // This doubles the number of correct binary digits in the approximation
- // with each iteration.
- uint32_t correction;
- correction = -((uint64_t)reciprocal * q31b >> 32);
- reciprocal = (uint64_t)reciprocal * correction >> 31;
- correction = -((uint64_t)reciprocal * q31b >> 32);
- reciprocal = (uint64_t)reciprocal * correction >> 31;
- correction = -((uint64_t)reciprocal * q31b >> 32);
- reciprocal = (uint64_t)reciprocal * correction >> 31;
-
- // Adust the final 32-bit reciprocal estimate downward to ensure that it is
- // strictly smaller than the infinitely precise exact reciprocal. Because
- // the computation of the Newton-Raphson step is truncating at every step,
- // this adjustment is small; most of the work is already done.
- reciprocal -= 2;
-
- // The numerical reciprocal is accurate to within 2^-28, lies in the
- // interval [0x1.000000eep-1, 0x1.fffffffcp-1], and is strictly smaller
- // than the true reciprocal of b. Multiplying a by this reciprocal thus
- // gives a numerical q = a/b in Q24 with the following properties:
- //
- // 1. q < a/b
- // 2. q is in the interval [0x1.000000eep-1, 0x1.fffffffcp0)
- // 3. The error in q is at most 2^-24 + 2^-27 -- the 2^24 term comes
- // from the fact that we truncate the product, and the 2^27 term
- // is the error in the reciprocal of b scaled by the maximum
- // possible value of a. As a consequence of this error bound,
- // either q or nextafter(q) is the correctly rounded.
- rep_t quotient = (uint64_t)reciprocal * (aSignificand << 1) >> 32;
-
- // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0).
- // In either case, we are going to compute a residual of the form
- //
- // r = a - q*b
- //
- // We know from the construction of q that r satisfies:
- //
- // 0 <= r < ulp(q)*b
- //
- // If r is greater than 1/2 ulp(q)*b, then q rounds up. Otherwise, we
- // already have the correct result. The exact halfway case cannot occur.
- // We also take this time to right shift quotient if it falls in the [1,2)
- // range and adjust the exponent accordingly.
- rep_t residual;
- if (quotient < (implicitBit << 1)) {
- residual = (aSignificand << 24) - quotient * bSignificand;
- quotientExponent--;
- } else {
- quotient >>= 1;
- residual = (aSignificand << 23) - quotient * bSignificand;
- }
-
- const int writtenExponent = quotientExponent + exponentBias;
- if (writtenExponent >= maxExponent) {
- // If we have overflowed the exponent, return infinity.
- return fromRep(infRep | quotientSign);
- }
+#define NUMBER_OF_HALF_ITERATIONS 0
+#define NUMBER_OF_FULL_ITERATIONS 3
+#define USE_NATIVE_FULL_ITERATIONS
- else if (writtenExponent < 1) {
- if (writtenExponent == 0) {
- // Check whether the rounded result is normal.
- const bool round = (residual << 1) > bSignificand;
- // Clear the implicit bit.
- rep_t absResult = quotient & significandMask;
- // Round.
- absResult += round;
- if (absResult & ~significandMask) {
- // The rounded result is normal; return it.
- return fromRep(absResult | quotientSign);
- }
- }
- // Flush denormals to zero. In the future, it would be nice to add
- // code to round them correctly.
- return fromRep(quotientSign);
- }
+#include "fp_div_impl.inc"
- else {
- const bool round = (residual << 1) > bSignificand;
- // Clear the implicit bit.
- rep_t absResult = quotient & significandMask;
- // Insert the exponent.
- absResult |= (rep_t)writtenExponent << significandBits;
- // Round.
- absResult += round;
- // Insert the sign and return.
- return fromRep(absResult | quotientSign);
- }
-}
+COMPILER_RT_ABI fp_t __divsf3(fp_t a, fp_t b) { return __divXf3__(a, b); }
#if defined(__ARM_EABI__)
#if defined(COMPILER_RT_ARMHF_TARGET)
Index: compiler-rt/lib/builtins/divdf3.c
===================================================================
--- compiler-rt/lib/builtins/divdf3.c
+++ compiler-rt/lib/builtins/divdf3.c
@@ -9,197 +9,16 @@
// This file implements double-precision soft-float division
// with the IEEE-754 default rounding (to nearest, ties to even).
//
-// For simplicity, this implementation currently flushes denormals to zero.
-// It should be a fairly straightforward exercise to implement gradual
-// underflow with correct rounding.
-//
//===----------------------------------------------------------------------===//
#define DOUBLE_PRECISION
-#include "fp_lib.h"
-
-COMPILER_RT_ABI fp_t __divdf3(fp_t a, fp_t b) {
-
- const unsigned int aExponent = toRep(a) >> significandBits & maxExponent;
- const unsigned int bExponent = toRep(b) >> significandBits & maxExponent;
- const rep_t quotientSign = (toRep(a) ^ toRep(b)) & signBit;
-
- rep_t aSignificand = toRep(a) & significandMask;
- rep_t bSignificand = toRep(b) & significandMask;
- int scale = 0;
-
- // Detect if a or b is zero, denormal, infinity, or NaN.
- if (aExponent - 1U >= maxExponent - 1U ||
- bExponent - 1U >= maxExponent - 1U) {
-
- const rep_t aAbs = toRep(a) & absMask;
- const rep_t bAbs = toRep(b) & absMask;
-
- // NaN / anything = qNaN
- if (aAbs > infRep)
- return fromRep(toRep(a) | quietBit);
- // anything / NaN = qNaN
- if (bAbs > infRep)
- return fromRep(toRep(b) | quietBit);
-
- if (aAbs == infRep) {
- // infinity / infinity = NaN
- if (bAbs == infRep)
- return fromRep(qnanRep);
- // infinity / anything else = +/- infinity
- else
- return fromRep(aAbs | quotientSign);
- }
-
- // anything else / infinity = +/- 0
- if (bAbs == infRep)
- return fromRep(quotientSign);
-
- if (!aAbs) {
- // zero / zero = NaN
- if (!bAbs)
- return fromRep(qnanRep);
- // zero / anything else = +/- zero
- else
- return fromRep(quotientSign);
- }
- // anything else / zero = +/- infinity
- if (!bAbs)
- return fromRep(infRep | quotientSign);
-
- // One or both of a or b is denormal. The other (if applicable) is a
- // normal number. Renormalize one or both of a and b, and set scale to
- // include the necessary exponent adjustment.
- if (aAbs < implicitBit)
- scale += normalize(&aSignificand);
- if (bAbs < implicitBit)
- scale -= normalize(&bSignificand);
- }
-
- // Set the implicit significand bit. If we fell through from the
- // denormal path it was already set by normalize( ), but setting it twice
- // won't hurt anything.
- aSignificand |= implicitBit;
- bSignificand |= implicitBit;
- int quotientExponent = aExponent - bExponent + scale;
-
- // Align the significand of b as a Q31 fixed-point number in the range
- // [1, 2.0) and get a Q32 approximate reciprocal using a small minimax
- // polynomial approximation: reciprocal = 3/4 + 1/sqrt(2) - b/2. This
- // is accurate to about 3.5 binary digits.
- const uint32_t q31b = bSignificand >> 21;
- uint32_t recip32 = UINT32_C(0x7504f333) - q31b;
- // 0x7504F333 / 2^32 + 1 = 3/4 + 1/sqrt(2)
-
- // Now refine the reciprocal estimate using a Newton-Raphson iteration:
- //
- // x1 = x0 * (2 - x0 * b)
- //
- // This doubles the number of correct binary digits in the approximation
- // with each iteration.
- uint32_t correction32;
- correction32 = -((uint64_t)recip32 * q31b >> 32);
- recip32 = (uint64_t)recip32 * correction32 >> 31;
- correction32 = -((uint64_t)recip32 * q31b >> 32);
- recip32 = (uint64_t)recip32 * correction32 >> 31;
- correction32 = -((uint64_t)recip32 * q31b >> 32);
- recip32 = (uint64_t)recip32 * correction32 >> 31;
-
- // The reciprocal may have overflowed to zero if the upper half of b is
- // exactly 1.0. This would sabatoge the full-width final stage of the
- // computation that follows, so we adjust the reciprocal down by one bit.
- recip32--;
-
- // We need to perform one more iteration to get us to 56 binary digits.
- // The last iteration needs to happen with extra precision.
- const uint32_t q63blo = bSignificand << 11;
- uint64_t correction, reciprocal;
- correction = -((uint64_t)recip32 * q31b + ((uint64_t)recip32 * q63blo >> 32));
- uint32_t cHi = correction >> 32;
- uint32_t cLo = correction;
- reciprocal = (uint64_t)recip32 * cHi + ((uint64_t)recip32 * cLo >> 32);
-
- // Adjust the final 64-bit reciprocal estimate downward to ensure that it is
- // strictly smaller than the infinitely precise exact reciprocal. Because
- // the computation of the Newton-Raphson step is truncating at every step,
- // this adjustment is small; most of the work is already done.
- reciprocal -= 2;
-
- // The numerical reciprocal is accurate to within 2^-56, lies in the
- // interval [0.5, 1.0), and is strictly smaller than the true reciprocal
- // of b. Multiplying a by this reciprocal thus gives a numerical q = a/b
- // in Q53 with the following properties:
- //
- // 1. q < a/b
- // 2. q is in the interval [0.5, 2.0)
- // 3. The error in q is bounded away from 2^-53 (actually, we have a
- // couple of bits to spare, but this is all we need).
-
- // We need a 64 x 64 multiply high to compute q, which isn't a basic
- // operation in C, so we need to be a little bit fussy.
- rep_t quotient, quotientLo;
- wideMultiply(aSignificand << 2, reciprocal, "ient, "ientLo);
-
- // Two cases: quotient is in [0.5, 1.0) or quotient is in [1.0, 2.0).
- // In either case, we are going to compute a residual of the form
- //
- // r = a - q*b
- //
- // We know from the construction of q that r satisfies:
- //
- // 0 <= r < ulp(q)*b
- //
- // If r is greater than 1/2 ulp(q)*b, then q rounds up. Otherwise, we
- // already have the correct result. The exact halfway case cannot occur.
- // We also take this time to right shift quotient if it falls in the [1,2)
- // range and adjust the exponent accordingly.
- rep_t residual;
- if (quotient < (implicitBit << 1)) {
- residual = (aSignificand << 53) - quotient * bSignificand;
- quotientExponent--;
- } else {
- quotient >>= 1;
- residual = (aSignificand << 52) - quotient * bSignificand;
- }
-
- const int writtenExponent = quotientExponent + exponentBias;
- if (writtenExponent >= maxExponent) {
- // If we have overflowed the exponent, return infinity.
- return fromRep(infRep | quotientSign);
- }
+#define NUMBER_OF_HALF_ITERATIONS 3
+#define NUMBER_OF_FULL_ITERATIONS 1
- else if (writtenExponent < 1) {
- if (writtenExponent == 0) {
- // Check whether the rounded result is normal.
- const bool round = (residual << 1) > bSignificand;
- // Clear the implicit bit.
- rep_t absResult = quotient & significandMask;
- // Round.
- absResult += round;
- if (absResult & ~significandMask) {
- // The rounded result is normal; return it.
- return fromRep(absResult | quotientSign);
- }
- }
- // Flush denormals to zero. In the future, it would be nice to add
- // code to round them correctly.
- return fromRep(quotientSign);
- }
+#include "fp_div_impl.inc"
- else {
- const bool round = (residual << 1) > bSignificand;
- // Clear the implicit bit.
- rep_t absResult = quotient & significandMask;
- // Insert the exponent.
- absResult |= (rep_t)writtenExponent << significandBits;
- // Round.
- absResult += round;
- // Insert the sign and return.
- const double result = fromRep(absResult | quotientSign);
- return result;
- }
-}
+COMPILER_RT_ABI fp_t __divdf3(fp_t a, fp_t b) { return __divXf3__(a, b); }
#if defined(__ARM_EABI__)
#if defined(COMPILER_RT_ARMHF_TARGET)
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