[libc][math] Improve the performance of sqrtf128.

Use a combination of polynomial approximation and Newton-Raphson iterations
in 64-bit and 128-bit integers to improve the performance of sqrtf128.  The
correct rounding is provided by squaring the result and comparing it with
the argument.

Co-authored-by: Alexei Sibidanov <[email protected]>

Author: lntue

libc/src/__support/big_int.h

@@ -241,7 +241,7 @@ LIBC_INLINE constexpr void quick_mul_hi(cpp::array<word, N> &dst,
 }
 
 template <typename word, size_t N>
-LIBC_INLINE constexpr bool is_negative(cpp::array<word, N> &array) {
+LIBC_INLINE constexpr bool is_negative(const cpp::array<word, N> &array) {
   using signed_word = cpp::make_signed_t<word>;
   return cpp::bit_cast<signed_word>(array.back()) < 0;
 }

libc/src/math/generic/CMakeLists.txt

@@ -3379,8 +3379,12 @@ add_entrypoint_object(
   HDRS
     ../sqrtf128.h
   DEPENDS
+    libc.src.__support.CPP.bit
+    libc.src.__support.FPUtil.fenv_impl
+    libc.src.__support.FPUtil.fp_bits
+    libc.src.__support.FPUtil.rounding_mode
+    libc.src.__support.macros.optimization
     libc.src.__support.macros.properties.types
-    libc.src.__support.FPUtil.sqrt
   COMPILE_OPTIONS
     ${libc_opt_high_flag}
   )

libc/src/math/generic/sqrtf128.cpp

@@ -1,20 +1,355 @@
 //===-- Implementation of sqrtf128 function -------------------------------===//
 //
+// Copyright (c) 2024 Alexei Sibidanov <[email protected]>
+//
 // 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
 //
 //===----------------------------------------------------------------------===//
 
 #include "src/math/sqrtf128.h"
-#include "src/__support/FPUtil/sqrt.h"
+#include "src/__support/CPP/bit.h"
+#include "src/__support/FPUtil/FEnvImpl.h"
+#include "src/__support/FPUtil/FPBits.h"
+#include "src/__support/FPUtil/rounding_mode.h"
 #include "src/__support/common.h"
-#include "src/__support/macros/config.h"
+#include "src/__support/macros/optimization.h"
+#include "src/__support/uint128.h"
 
 namespace LIBC_NAMESPACE_DECL {
 
+using FPBits = fputil::FPBits<float128>;
+
+namespace {
+
+template <typename T, typename U = T> static inline constexpr T prod_hi(T, U);
+
+// Get high part of integer multiplications.
+// Use template to prevent implicit conversion.
+template <>
+inline constexpr uint64_t prod_hi<uint64_t>(uint64_t x, uint64_t y) {
+  return static_cast<uint64_t>(
+      (static_cast<UInt128>(x) * static_cast<UInt128>(y)) >> 64);
+}
+
+// Get high part of unsigned 128x64 bit multiplication.
+template <>
+inline constexpr UInt128 prod_hi<UInt128, uint64_t>(UInt128 y, uint64_t x) {
+  uint64_t y_lo = static_cast<uint64_t>(y);
+  uint64_t y_hi = static_cast<uint64_t>(y >> 64);
+  UInt128 xyl = static_cast<UInt128>(x) * static_cast<UInt128>(y_lo);
+  UInt128 xyh = static_cast<UInt128>(x) * static_cast<UInt128>(y_hi);
+  return xyh + (xyl >> 64);
+}
+
+// Get high part of signed 64x64 bit multiplication.
+template <> inline constexpr int64_t prod_hi<int64_t>(int64_t x, int64_t y) {
+  return static_cast<int64_t>(
+      (static_cast<Int128>(x) * static_cast<Int128>(y)) >> 64);
+}
+
+// Get high 128-bit part of unsigned 128x128 bit multiplication.
+template <> inline constexpr UInt128 prod_hi<UInt128>(UInt128 x, UInt128 y) {
+  uint64_t x_lo = static_cast<uint64_t>(x);
+  uint64_t x_hi = static_cast<uint64_t>(x >> 64);
+  uint64_t y_lo = static_cast<uint64_t>(y);
+  uint64_t y_hi = static_cast<uint64_t>(y >> 64);
+
+  UInt128 xh_yh = static_cast<UInt128>(x_hi) * static_cast<UInt128>(y_hi);
+  UInt128 xh_yl = static_cast<UInt128>(x_hi) * static_cast<UInt128>(y_lo);
+  UInt128 xl_yh = static_cast<UInt128>(x_lo) * static_cast<UInt128>(y_hi);
+
+  xh_yh += xh_yl >> 64;
+
+  return xh_yh + (xl_yh >> 64);
+}
+
+// Get high 128-bit part of mixed sign 128x128 bit multiplication.
+template <>
+inline constexpr Int128 prod_hi<Int128, UInt128>(Int128 x, UInt128 y) {
+  UInt128 mask = static_cast<UInt128>(x >> 127);
+  UInt128 negative_part = y & mask;
+  UInt128 prod = prod_hi(static_cast<UInt128>(x), y);
+  return static_cast<Int128>(prod - negative_part);
+}
+
+constexpr uint32_t RSQRT_COEFFS[64][4] = {
+    {0xffffffff, 0xfffff780, 0xbff55815, 0x9bb5b6e7},
+    {0xfc0bd889, 0xfa1d6e7d, 0xb8a95a89, 0x938bf8f0},
+    {0xf82ec882, 0xf473bea9, 0xb1bf4705, 0x8bed0079},
+    {0xf467f280, 0xeefff2a1, 0xab309d4a, 0x84cdb431},
+    {0xf0b6848c, 0xe9bf46f4, 0xa4f76232, 0x7e24037b},
+    {0xed19b75e, 0xe4af2628, 0x9f0e1340, 0x77e6ca62},
+    {0xe990cdad, 0xdfcd2521, 0x996f9b96, 0x720db8df},
+    {0xe61b138e, 0xdb16ffde, 0x94174a00, 0x6c913cff},
+    {0xe2b7dddf, 0xd68a967b, 0x8f00c812, 0x676a6f92},
+    {0xdf6689b7, 0xd225ea80, 0x8a281226, 0x62930308},
+    {0xdc267bea, 0xcde71c63, 0x8589702c, 0x5e05343e},
+    {0xd8f7208e, 0xc9cc6948, 0x81216f2e, 0x59bbbcf8},
+    {0xd5d7ea91, 0xc5d428ee, 0x7cecdb76, 0x55b1c7d6},
+    {0xd2c8534e, 0xc1fccbc9, 0x78e8bb45, 0x51e2e592},
+    {0xcfc7da32, 0xbe44d94a, 0x75124a0a, 0x4e4b0369},
+    {0xccd6045f, 0xbaaaee41, 0x7166f40f, 0x4ae66284},
+    {0xc9f25c5c, 0xb72dbb69, 0x6de45288, 0x47b19045},
+    {0xc71c71c7, 0xb3cc040f, 0x6a882804, 0x44a95f5f},
+    {0xc453d90f, 0xb0849cd4, 0x67505d2a, 0x41cae1a0},
+    {0xc1982b2e, 0xad566a85, 0x643afdc8, 0x3f13625c},
+    {0xbee9056f, 0xaa406113, 0x6146361f, 0x3c806169},
+    {0xbc46092e, 0xa7418293, 0x5e70506d, 0x3a0f8e8e},
+    {0xb9aedba5, 0xa458de58, 0x5bb7b2b1, 0x37bec572},
+    {0xb72325b7, 0xa1859022, 0x591adc9a, 0x358c09e2},
+    {0xb4a293c2, 0x9ec6bf52, 0x569865a7, 0x33758476},
+    {0xb22cd56d, 0x9c1b9e36, 0x542efb6a, 0x31797f8a},
+    {0xafc19d86, 0x9983695c, 0x51dd5ffb, 0x2f96647a},
+    {0xad60a1d1, 0x96fd66f7, 0x4fa2687c, 0x2dcab91f},
+    {0xab099ae9, 0x9488e64b, 0x4d7cfbc9, 0x2c151d8a},
+    {0xa8bc441a, 0x92253f20, 0x4b6c1139, 0x2a7449ef},
+    {0xa6785b42, 0x8fd1d14a, 0x496eaf82, 0x28e70cc3},
+    {0xa43da0ae, 0x8d8e042a, 0x4783eba7, 0x276c4900},
+    {0xa20bd701, 0x8b594648, 0x45aae80a, 0x2602f493},
+    {0x9fe2c315, 0x89330ce4, 0x43e2d382, 0x24aa16ec},
+    {0x9dc22be4, 0x871ad399, 0x422ae88c, 0x2360c7af},
+    {0x9ba9da6c, 0x85101c05, 0x40826c88, 0x22262d7b},
+    {0x99999999, 0x83126d70, 0x3ee8af07, 0x20f97cd2},
+    {0x97913630, 0x81215480, 0x3d5d0922, 0x1fd9f714},
+    {0x95907eb8, 0x7f3c62ef, 0x3bdedce0, 0x1ec6e994},
+    {0x93974369, 0x7d632f45, 0x3a6d94a9, 0x1dbfacbb},
+    {0x91a55615, 0x7b955498, 0x3908a2be, 0x1cc3a33b},
+    {0x8fba8a1c, 0x79d2724e, 0x37af80bf, 0x1bd23960},
+    {0x8dd6b456, 0x781a2be4, 0x3661af39, 0x1aeae458},
+    {0x8bf9ab07, 0x766c28ba, 0x351eb539, 0x1a0d21a2},
+    {0x8a2345cc, 0x74c813dd, 0x33e61feb, 0x19387676},
+    {0x88535d90, 0x732d9bdc, 0x32b7823a, 0x186c6f3e},
+    {0x8689cc7e, 0x719c7297, 0x3192747d, 0x17a89f21},
+    {0x84c66df1, 0x70144d19, 0x30769424, 0x16ec9f89},
+    {0x83091e6a, 0x6e94e36c, 0x2f63836f, 0x16380fbf},
+    {0x8151bb87, 0x6d1df079, 0x2e58e925, 0x158a9484},
+    {0x7fa023f1, 0x6baf31de, 0x2d567053, 0x14e3d7ba},
+    {0x7df43758, 0x6a4867d3, 0x2c5bc811, 0x1443880e},
+    {0x7c4dd664, 0x68e95508, 0x2b68a346, 0x13a958ab},
+    {0x7aace2b0, 0x6791be86, 0x2a7cb871, 0x131500ee},
+    {0x79113ebc, 0x66416b95, 0x2997c17a, 0x12863c29},
+    {0x777acde8, 0x64f825a1, 0x28b97b82, 0x11fcc95c},
+    {0x75e9746a, 0x63b5b822, 0x27e1a6b4, 0x11786b03},
+    {0x745d1746, 0x6279f081, 0x2710061d, 0x10f8e6da},
+    {0x72d59c46, 0x61449e06, 0x26445f86, 0x107e05ac},
+    {0x7152e9f4, 0x601591be, 0x257e7b4d, 0x10079327},
+    {0x6fd4e793, 0x5eec9e6b, 0x24be2445, 0x0f955da9},
+    {0x6e5b7d16, 0x5dc9986e, 0x24032795, 0x0f273620},
+    {0x6ce6931d, 0x5cac55b7, 0x234d5496, 0x0ebcefdb},
+    {0x6b7612ec, 0x5b94adb2, 0x229c7cbc, 0x0e56606e},
+};
+
+// Approximate rsqrt with cubic polynomials.
+// The range [1,2] is splitted into 64 equal sub-ranges and the reciprocal
+// square root is approximated by a cubic polynomial by the minimax method in
+// each subrange. The approximation accuracy fits into 32-33 bits and thus it is
+// natural to round coefficients into 32 bit. The constant coefficient can be
+// rounded to 33 bits since the most significant bit is always 1 and implicitly
+// assumed in the table.
+LIBC_INLINE uint64_t rsqrt_approx(uint64_t m) {
+  // ULP(m) = 2^-64.
+  // Use the top 6 bits as index for looking up polynomial coeffs.
+  uint64_t indx = m >> 58;
+
+  uint64_t c0 = static_cast<uint64_t>(RSQRT_COEFFS[indx][0]);
+  c0 <<= 31;        // to 64 bit with the space for the implicit bit
+  c0 |= 1ull << 63; // add implicit bit
+
+  uint64_t c1 = static_cast<uint64_t>(RSQRT_COEFFS[indx][1]);
+  c1 <<= 25; // to 64 bit format
+
+  uint64_t c2 = static_cast<uint64_t>(RSQRT_COEFFS[indx][2]);
+  uint64_t c3 = static_cast<uint64_t>(RSQRT_COEFFS[indx][3]);
+
+  uint64_t d = (m << 6) >> 32; // local coordinate in the subrange [0, 2^32]
+  uint64_t d2 = (d * d) >> 32; // square of the local coordinate
+  uint64_t re = c0 + (d2 * c2 >> 13); // even part of the polynomial (positive)
+  uint64_t ro = d * ((c1 + ((d2 * c3) >> 19)) >> 26) >>
+                6;      // odd part of the polynomial (negative)
+  uint64_t r = re - ro; // maximal error < 1.55e-10 and it is less than 2^-32
+
+  // Newton-Raphson first order step to improve accuracy of the result to almost
+  // 64 bits:
+  // For the initial approximation r0 ~ 1/sqrt(x), let
+  //   h = r0^2 * x - 1
+  // be its scaled error.  Then the first-order Newton-Raphson iteration is:
+  //   r1 = r0 - r0 * h / 2
+  // which has error bounded by:
+  //   |r1 - 1/sqrt(x)| < h^2 / 2.
+  uint64_t r2 = prod_hi<uint64_t>(r, r);
+  // h = r0^2*x - 1.
+  int64_t h = static_cast<int64_t>(prod_hi<uint64_t>(m, r2) + r2);
+  // hr = r * h / 2
+  int64_t hr = prod_hi<int64_t>(h, static_cast<int64_t>(r >> 1));
+  r -= hr;
+  // Adjust in the unlucky case x~1;
+  if (LIBC_UNLIKELY(!r))
+    --r;
+  return r;
+}
+
+} // anonymous namespace
+
 LLVM_LIBC_FUNCTION(float128, sqrtf128, (float128 x)) {
-  return fputil::sqrt<float128>(x);
+  using FPBits = fputil::FPBits<float128>;
+  // Get rounding mode.
+  uint32_t rm = fputil::get_round();
+
+  FPBits xbits(x);
+  UInt128 x_u = xbits.uintval();
+  // Bring leading bit of the mantissa to the highest bit.
+  //   ulp(x_frac) = 2^-128.
+  UInt128 x_frac = xbits.get_mantissa() << (FPBits::EXP_LEN + 1);
+
+  int sign_exp = static_cast<int>(x_u >> FPBits::FRACTION_LEN);
+
+  if (LIBC_UNLIKELY(sign_exp == 0 || sign_exp >= 0x7fff)) {
+    // Special cases: NAN, inf, negative numbers
+    if (sign_exp >= 0x7fff) {
+      // x = -0 or x = inf
+      if (xbits.is_zero() || xbits == xbits.inf())
+        return x;
+      // x is nan
+      if (xbits.is_nan()) {
+        // pass through quiet nan
+        if (xbits.is_quiet_nan())
+          return x;
+        // transform signaling nan to quiet and return
+        return xbits.quiet_nan().get_val();
+      }
+      // x < 0 or x = -inf
+      fputil::set_errno_if_required(EDOM);
+      fputil::raise_except_if_required(FE_INVALID);
+      return xbits.quiet_nan().get_val();
+    }
+    // x is subnormal or x=+0
+    if (x == 0)
+      return x;
+
+    // Normalize subnormal inputs.
+    sign_exp = -cpp::countl_zero(x_frac);
+    int normal_shifts = 1 - sign_exp;
+    x_frac <<= normal_shifts;
+  }
+
+  // For sign_exp = biased exponent of x = real_exponent + 16383,
+  // let f be the real exponent of the output:
+  //   f = floor(real_exponent / 2)
+  // Then:
+  //   floor((sign_exp + 1) / 2) = f + 8192
+  // Hence, the biased exponent of the final result is:
+  //   f + 16383 = floor((sign_exp + 1) / 2) + 8191.
+  // Since the output mantissa will include the hidden bit, we can define the
+  // output exponent part:
+  //   e2 = floor((sign_exp + 1) / 2) + 8190
+  unsigned i = static_cast<unsigned>(1 - (sign_exp & 1));
+  uint32_t q2 = (sign_exp + 1) >> 1;
+  // Exponent of the final result
+  uint32_t e2 = q2 + 8190;
+
+  constexpr uint64_t RSQRT_2[2] = {~0ull,
+                                   0xb504f333f9de6484 /* 2^64/sqrt(2) */};
+
+  // Approximate 1/sqrt(1 + x_frac)
+  // Error: |r_1 - 1/sqrt(x)| < 2^-63.
+  uint64_t r1 = rsqrt_approx(static_cast<uint64_t>(x_frac >> 64));
+  // Adjust for the even/odd exponent.
+  uint64_t r2 = prod_hi(r1, RSQRT_2[i]);
+  unsigned shift = 2 - i;
+
+  // Normalized input:
+  //   1 <= x_reduced < 4
+  UInt128 x_reduced = (x_frac >> shift) | (UInt128(1) << (126 + i));
+  // With r2 ~ 1/sqrt(x) up to 2^-63, we perform another round of Newton-Raphson
+  // iteration:
+  //   r3 = r2 - r2 * h / 2,
+  // for h = r2^2 * x - 1.
+  // Then:
+  //   sqrt(x) = x * (1 / sqrt(x))
+  //           ~ x * r3
+  //           = x * (r2 - r2 * h / 2)
+  //           = (x * r2) - (x * r2) * h / 2
+  UInt128 sx = prod_hi(x_reduced, r2);
+  UInt128 h = prod_hi(sx, r2) << 2;
+  UInt128 ds = static_cast<UInt128>(prod_hi(static_cast<Int128>(h), sx));
+  UInt128 v = (sx << 1) - ds;
+
+  uint32_t nrst = rm == FE_TONEAREST;
+  // The result lies within (-2,5) of true square root so we now
+  // test that we can correctly round the result taking into account
+  // the rounding mode
+  // check the lowest 14 bits.
+  int dd = (static_cast<int>(v) << 18) >> 18;
+
+  if (LIBC_UNLIKELY(dd < 4 && dd >= -8)) { // can round correctly?
+    // m is almost the final result it can be only 1 ulp off so we
+    // just need to test both possibilities. We square it and
+    // compare with the initial argument.
+    UInt128 m = v >> 15;
+    UInt128 m2 = m * m;
+    Int128 t0, t1;
+    // The difference of the squared result and the argument
+    t0 = static_cast<Int128>(m2 - (x_reduced << 98));
+    if (t0 == 0) {
+      // the square root is exact
+      v = m << 15;
+    } else {
+      // Add +-1 ulp to m depend on the sign of the difference. Here
+      // we do not need to square again since (m+1)^2 = m^2 + 2*m +
+      // 1 so just need to add shifted m and 1.
+      t1 = t0;
+      Int128 sgn = t0 >> 127; // sign of the difference
+      t1 -= (m << 1) ^ sgn;
+      t1 += 1 + sgn;
+
+      Int128 sgn1 = t1 >> 127;
+      if (LIBC_UNLIKELY(sgn == sgn1)) {
+        t0 = t1;
+        v -= sgn << 15;
+        t1 -= (m << 1) ^ sgn;
+        t1 += 1 + sgn;
+      }
+
+      if (t1 == 0) {
+        // 1 ulp offset brings again an exact root
+        v = (m - (2 * sgn + 1)) << 15;
+      } else {
+        t1 += t0;
+        Int128 side = t1 >> 127; // select what is closer m or m+-1
+        v &= ~UInt128(0) << 15;  // wipe the fractional bits
+        v -= ((sgn & side) | (~sgn & 1)) << (15 + side);
+        v |= 1; // add sticky bit since we cannot have an exact mid-point
+                // situation
+      }
+    }
+  }
+
+  unsigned frac = static_cast<unsigned>(v) & 0x7fff; // fractional part
+  unsigned rnd;                                      // round bit
+  if (LIBC_LIKELY(nrst != 0)) {
+    rnd = frac >> 14; // round to nearest tie to even
+  } else if (rm == FE_UPWARD) {
+    rnd = !!frac; // round up
+  } else if (rm == FE_DOWNWARD) {
+    rnd = 0; // round down
+  } else {
+    rnd = 0; // round to zero
+  }
+
+  v >>= 15; // position mantissa
+  v += rnd; // round
+
+  // // Set inexact flag only if square root is inexact
+  // // TODO: We will have to raise FE_INEXACT most of the time, but this
+  // // operation is very costly, especially in x86-64, since technically, it
+  // // needs to synchronize both SSE and x87 flags.  Need to investigate
+  // // further to see how we can make this performant.
+  // if(frac) fputil::raise_except_if_required(FE_INEXACT);
+
+  v += static_cast<UInt128>(e2) << FPBits::FRACTION_LEN; // place exponent
+  return cpp::bit_cast<float128>(v);
 }
 
 } // namespace LIBC_NAMESPACE_DECL