[libc-commits] [libc] [llvm] [libc][math][c23] Improve rsqrtf16() function (PR #160639)

Anton Shepelev via libc-commits libc-commits at lists.llvm.org
Sun Apr 26 19:39:21 PDT 2026


https://github.com/amemov updated https://github.com/llvm/llvm-project/pull/160639

>From eb3025d7bbe0d07af0efd7d4b64e848e878baa5e Mon Sep 17 00:00:00 2001
From: Anton Shepelev <shepelev777 at gmail.com>
Date: Wed, 24 Sep 2025 20:59:06 -0700
Subject: [PATCH 1/5] - Draft of math approximation for targets that have only
 int hardware - Fixed typo in +inf case. Should return +0 according to
 F.10.4.9

---
 libc/src/__support/math/rsqrtf16.h | 109 +++++++++++++++++++++++++++--
 1 file changed, 105 insertions(+), 4 deletions(-)

diff --git a/libc/src/__support/math/rsqrtf16.h b/libc/src/__support/math/rsqrtf16.h
index ab7529682950d..551aa9710ea70 100644
--- a/libc/src/__support/math/rsqrtf16.h
+++ b/libc/src/__support/math/rsqrtf16.h
@@ -58,12 +58,11 @@ LIBC_INLINE constexpr float16 rsqrtf16(float16 x) {
       return FPBits::quiet_nan().get_val();
     }
 
-    // x = +inf => rsqrt(x) = 0
-    return FPBits::zero().get_val();
+    // x = +inf => rsqrt(x) = +0
+    return FPBits::zero(xbits.sign()).get_val();
   }
 
-  // TODO: add integer based implementation when LIBC_TARGET_CPU_HAS_FPU_FLOAT
-  // is not defined
+#ifdef LIBC_TARGET_CPU_HAS_FPU_FLOAT
   float result = 1.0f / fputil::sqrt<float>(fputil::cast<float>(x));
 
   // Targeted post-corrections to ensure correct rounding in half for specific
@@ -76,6 +75,108 @@ LIBC_INLINE constexpr float16 rsqrtf16(float16 x) {
   }
 
   return fputil::cast<float16>(result);
+
+#else
+  // Range reduction:
+  // x can be expressed as m*2^e, where e - int exponent and m - mantissa
+  // rsqrtf16(x) = rsqrtf16(m*2^e)
+  // rsqrtf16(m*2^e) = 1/sqrt(m) * 1/sqrt(2^e) = 1/sqrt(m) * 1/2^(e/2)
+  // 1/sqrt(m) * 1/2^(e/2) = 1/sqrt(m) * 2^(-e/2)
+
+  // Compute reduction directly from half bits to avoid frexp/ldexp overhead.
+  int exponent = 0;
+  int signifcand = 0; // same as mantissa, but int
+  uint16_t eh = static_cast<uint16_t>((x_abs >> 10) & 0x1F);
+  uint16_t frac = static_cast<uint16_t>(x_abs & 0x3FF);
+
+  int result;
+  if (eh != 0) {
+    // ((2^-1 + frac/2^11) * 2) * 2^(eh-15)
+
+    // Normal: x = (1 + frac/2^10) * 2^(eh-15) = ((0.5 + frac/2^11) * 2) *
+    // 2^(eh-15)
+    // => mantissa in [0.5,1): m = 0.5 + frac/2^11, exponent = (eh - 15) + 1 =
+    // eh - 14
+    exponent = static_cast<int>(eh) - 14;
+    mantissa = 0.5f + static_cast<float>(frac) * 0x1.0p-11f;
+  } else {
+    // Subnormal: x = (frac/2^10) * 2^(1-15) = frac * 2^-24.
+    // Normalize frac so that bit 9 becomes 1; then mantissa m = (frac <<
+    // t)/2^10 ∈ [0.5,1) and exponent E = -14 - t so that x = m * 2^E.
+    if (LIBC_UNLIKELY(frac == 0)) {
+      // Should have been handled by zero check above, but keep safe.
+      return FPBits::inf(Sign::POS).get_val();
+    }
+    int shifts = 0;
+    while ((frac & 0x200u) == 0u) { // bring into [0x200, 0x3FF]
+      frac <<= 1;
+      ++shifts;
+    }
+    exponent = -14 - shifts;
+    mantissa = static_cast<float>(frac) * 0x1.0p-10f;
+  }
+
+  float result = 0.0f;
+  int exp_floored = -(exponent >> 1);
+
+  if (mantissa == 0.5f) {
+    // When mantissa is 0.5f, x was a power of 2 (or subnormal that normalizes
+    // this way). 1/sqrt(0.5f) = sqrt(2.0f).
+    // If exponent is odd (exponent = 2k + 1):
+    //   rsqrt(x) = (1/sqrt(0.5)) * 2^(-(2k+1)/2) = sqrt(2) * 2^(-k-0.5)
+    //            = sqrt(2) * 2^(-k) * (1/sqrt(2)) = 2^(-k)
+    //   exp_floored = -((2k+1)>>1) = -(k) = -k
+    //   So result = ldexp(1.0f, exp_floored)
+    // If exponent is even (exponent = 2k):
+    //   rsqrt(x) = (1/sqrt(0.5)) * 2^(-2k/2) = sqrt(2) * 2^(-k)
+    //   exp_floored = -((2k)>>1) = -(k) = -k
+    //   So result = ldexp(sqrt(2.0f), exp_floored)
+    if (exponent & 1) {
+      result = fputil::ldexp(1.0f, exp_floored);
+    } else {
+      constexpr float SQRT_2_F = 0x1.6a09e6p0f; // sqrt(2.0f)
+      result = fputil::ldexp(SQRT_2_F, exp_floored);
+    }
+  } else {
+    // 4 Degree minimax polynomial (single-precision coefficients) generated
+    // with Sollya: P = fpminimax(1/sqrt(x), 4,
+    // [|single,single,single,single,single|], [0.5;1])
+    float y = fputil::polyeval(mantissa,
+                               0x1.771256p1f,  // c0
+                               -0x1.5e7c4ap2f, // c1
+                               0x1.b3851cp2f,  // c2
+                               -0x1.1a27ep2f,  // c3
+                               0x1.265c66p0f); // c4
+
+    // Newton-Raphson iteration in float (use multiply_add to leverage FMA when
+    // available):
+    float y2 = y * y;
+    float factor = fputil::multiply_add(-0.5f * mantissa, y2, 1.5f);
+    y = y * factor;
+
+    result = fputil::ldexp(y, exp_floored);
+    if (exponent & 1) {
+      constexpr float ONE_OVER_SQRT2 = 0x1.6a09e6p-1f; // 1/sqrt(2)
+      result *= ONE_OVER_SQRT2;
+    }
+
+    // Targeted post-correction: for the specific half-precision mantissa
+    // pattern M == 0x011F we observe a consistent -1 ULP bias across exponents.
+    // Apply a tiny upward nudge to cross the rounding boundary in all modes.
+    const uint16_t half_mantissa = static_cast<uint16_t>(x_abs & 0x3ff);
+    if (half_mantissa == 0x011F) {
+      // Nudge up to fix consistent -1 ULP at that mantissa boundary
+      result = fputil::multiply_add(result, 0x1.0p-21f,
+                                    result); // result *= (1 + 2^-21)
+    } else if (half_mantissa == 0x0313) {
+      // Nudge down to fix +1 ULP under upward rounding at this mantissa
+      // boundary
+      result = fputil::multiply_add(result, -0x1.0p-21f,
+                                    result); // result *= (1 - 2^-21)
+    }
+  }
+  return fputil::cast<float16>(result);
+#endif
 }
 
 } // namespace math

>From ed1e4d394de1e1b47ec2e12d34dbc6583a39f4d7 Mon Sep 17 00:00:00 2001
From: amemov <shepelev777 at gmail.com>
Date: Mon, 17 Nov 2025 22:00:28 -0800
Subject: [PATCH 2/5] Added approximation from previous PR

---
 libc/src/__support/math/rsqrtf16.h | 45 +++++-------------------------
 1 file changed, 7 insertions(+), 38 deletions(-)

diff --git a/libc/src/__support/math/rsqrtf16.h b/libc/src/__support/math/rsqrtf16.h
index 551aa9710ea70..4b237950b0b17 100644
--- a/libc/src/__support/math/rsqrtf16.h
+++ b/libc/src/__support/math/rsqrtf16.h
@@ -16,6 +16,7 @@
 #include "src/__support/FPUtil/FEnvImpl.h"
 #include "src/__support/FPUtil/FPBits.h"
 #include "src/__support/FPUtil/ManipulationFunctions.h"
+#include "src/__support/FPUtil/PolyEval.h"
 #include "src/__support/FPUtil/cast.h"
 #include "src/__support/FPUtil/multiply_add.h"
 #include "src/__support/FPUtil/sqrt.h"
@@ -77,44 +78,10 @@ LIBC_INLINE constexpr float16 rsqrtf16(float16 x) {
   return fputil::cast<float16>(result);
 
 #else
-  // Range reduction:
-  // x can be expressed as m*2^e, where e - int exponent and m - mantissa
-  // rsqrtf16(x) = rsqrtf16(m*2^e)
-  // rsqrtf16(m*2^e) = 1/sqrt(m) * 1/sqrt(2^e) = 1/sqrt(m) * 1/2^(e/2)
-  // 1/sqrt(m) * 1/2^(e/2) = 1/sqrt(m) * 2^(-e/2)
+  float xf = fputil::cast<float>(x);
 
-  // Compute reduction directly from half bits to avoid frexp/ldexp overhead.
   int exponent = 0;
-  int signifcand = 0; // same as mantissa, but int
-  uint16_t eh = static_cast<uint16_t>((x_abs >> 10) & 0x1F);
-  uint16_t frac = static_cast<uint16_t>(x_abs & 0x3FF);
-
-  int result;
-  if (eh != 0) {
-    // ((2^-1 + frac/2^11) * 2) * 2^(eh-15)
-
-    // Normal: x = (1 + frac/2^10) * 2^(eh-15) = ((0.5 + frac/2^11) * 2) *
-    // 2^(eh-15)
-    // => mantissa in [0.5,1): m = 0.5 + frac/2^11, exponent = (eh - 15) + 1 =
-    // eh - 14
-    exponent = static_cast<int>(eh) - 14;
-    mantissa = 0.5f + static_cast<float>(frac) * 0x1.0p-11f;
-  } else {
-    // Subnormal: x = (frac/2^10) * 2^(1-15) = frac * 2^-24.
-    // Normalize frac so that bit 9 becomes 1; then mantissa m = (frac <<
-    // t)/2^10 ∈ [0.5,1) and exponent E = -14 - t so that x = m * 2^E.
-    if (LIBC_UNLIKELY(frac == 0)) {
-      // Should have been handled by zero check above, but keep safe.
-      return FPBits::inf(Sign::POS).get_val();
-    }
-    int shifts = 0;
-    while ((frac & 0x200u) == 0u) { // bring into [0x200, 0x3FF]
-      frac <<= 1;
-      ++shifts;
-    }
-    exponent = -14 - shifts;
-    mantissa = static_cast<float>(frac) * 0x1.0p-10f;
-  }
+  float mantissa = fputil::frexp(xf, exponent);
 
   float result = 0.0f;
   int exp_floored = -(exponent >> 1);
@@ -139,8 +106,9 @@ LIBC_INLINE constexpr float16 rsqrtf16(float16 x) {
     }
   } else {
     // 4 Degree minimax polynomial (single-precision coefficients) generated
-    // with Sollya: P = fpminimax(1/sqrt(x), 4,
-    // [|single,single,single,single,single|], [0.5;1])
+    // with Sollya:
+    //   P = fpminimax(1/sqrt(x), 4,
+    //       [|single,single,single,single,single|], [0.5;1])
     float y = fputil::polyeval(mantissa,
                                0x1.771256p1f,  // c0
                                -0x1.5e7c4ap2f, // c1
@@ -175,6 +143,7 @@ LIBC_INLINE constexpr float16 rsqrtf16(float16 x) {
                                     result); // result *= (1 - 2^-21)
     }
   }
+
   return fputil::cast<float16>(result);
 #endif
 }

>From a214bd1a221666f0d00c552922d9dd453905765f Mon Sep 17 00:00:00 2001
From: amemov <shepelev777 at gmail.com>
Date: Sat, 25 Apr 2026 13:37:01 -0700
Subject: [PATCH 3/5] new approximation draft

---
 libc/src/__support/math/CMakeLists.txt        |  14 ++
 libc/src/__support/math/rsqrtf16.h            | 237 +++++++++++++-----
 .../llvm-project-overlay/libc/BUILD.bazel     |   1 -
 3 files changed, 182 insertions(+), 70 deletions(-)

diff --git a/libc/src/__support/math/CMakeLists.txt b/libc/src/__support/math/CMakeLists.txt
index 095028c644d37..6f75baef8bf96 100644
--- a/libc/src/__support/math/CMakeLists.txt
+++ b/libc/src/__support/math/CMakeLists.txt
@@ -126,6 +126,20 @@ add_header_library(
     libc.src.__support.macros.properties.types
 )
 
+add_header_library(
+  rsqrtf16
+  HDRS
+    rsqrtf16.h
+  DEPENDS
+    libc.src.__support.FPUtil.cast
+    libc.src.__support.FPUtil.fenv_impl
+    libc.src.__support.FPUtil.fp_bits
+    libc.src.__support.FPUtil.multiply_add
+    libc.src.__support.FPUtil.sqrt
+    libc.src.__support.macros.optimization
+    libc.src.__support.macros.properties.types
+)
+
 add_header_library(
   asin_utils
   HDRS
diff --git a/libc/src/__support/math/rsqrtf16.h b/libc/src/__support/math/rsqrtf16.h
index 4b237950b0b17..23c74a8515c15 100644
--- a/libc/src/__support/math/rsqrtf16.h
+++ b/libc/src/__support/math/rsqrtf16.h
@@ -15,8 +15,6 @@
 
 #include "src/__support/FPUtil/FEnvImpl.h"
 #include "src/__support/FPUtil/FPBits.h"
-#include "src/__support/FPUtil/ManipulationFunctions.h"
-#include "src/__support/FPUtil/PolyEval.h"
 #include "src/__support/FPUtil/cast.h"
 #include "src/__support/FPUtil/multiply_add.h"
 #include "src/__support/FPUtil/sqrt.h"
@@ -25,6 +23,173 @@
 namespace LIBC_NAMESPACE_DECL {
 namespace math {
 
+namespace rsqrtf16_internal {
+
+LIBC_INLINE_VAR constexpr int RSQRT_FRACTION_BITS = 29;
+LIBC_INLINE_VAR constexpr int64_t ONE = int64_t(1) << RSQRT_FRACTION_BITS;
+LIBC_INLINE_VAR constexpr int64_t THREE_HALVES = 3 * (ONE >> 1);
+
+// Degree-4 minimax polynomial generated with Sollya:
+//   P = fpminimax(1/sqrt(x), 4,
+//       [|single,single,single,single,single|], [0.5;1])
+// Coefficients are stored in Q29 fixed-point format.
+LIBC_INLINE_VAR constexpr int64_t COEFFS[5] = {
+    1'573'164'416, -2'940'085'504, 3'653'406'208, -2'366'894'080, 617'319'616,
+};
+LIBC_INLINE_VAR constexpr int64_t ONE_OVER_SQRT2 = 0x16a09e60;
+
+LIBC_INLINE constexpr int floor_log2(uint64_t x) {
+  int result = -1;
+  while (x) {
+    x >>= 1;
+    ++result;
+  }
+  return result;
+}
+
+LIBC_INLINE constexpr int64_t eval_polynomial(uint32_t m) {
+  int64_t y = COEFFS[4];
+  y = COEFFS[3] + ((y * m) >> RSQRT_FRACTION_BITS);
+  y = COEFFS[2] + ((y * m) >> RSQRT_FRACTION_BITS);
+  y = COEFFS[1] + ((y * m) >> RSQRT_FRACTION_BITS);
+  y = COEFFS[0] + ((y * m) >> RSQRT_FRACTION_BITS);
+  return y;
+}
+
+LIBC_INLINE constexpr int64_t newton_raphson(uint32_t m, int64_t y) {
+  int64_t y2 = (y * y) >> RSQRT_FRACTION_BITS;
+  int64_t my2 = (static_cast<int64_t>(m) * y2) >> RSQRT_FRACTION_BITS;
+  int64_t factor = THREE_HALVES - (my2 >> 1);
+  return (y * factor) >> RSQRT_FRACTION_BITS;
+}
+
+LIBC_INLINE constexpr uint16_t fixed_to_half_bits(uint64_t y, int scale_exp) {
+  int y_log2 = floor_log2(y);
+  int out_exp = scale_exp + y_log2 - RSQRT_FRACTION_BITS;
+  int biased_exp = out_exp + 15;
+
+  uint32_t out_sig = y_log2 >= 10 ? static_cast<uint32_t>(y >> (y_log2 - 10))
+                                  : static_cast<uint32_t>(y << (10 - y_log2));
+
+  if (biased_exp <= 0)
+    return 0x0400;
+  if (biased_exp >= 31)
+    return 0x7bff;
+
+  return static_cast<uint16_t>((biased_exp << 10) | (out_sig & 0x3ff));
+}
+
+LIBC_INLINE constexpr uint16_t approximate_rsqrt(uint16_t x_abs) {
+  uint32_t x_mant = x_abs & 0x03ff;
+  int exponent = 0;
+
+  if (x_abs >= 0x0400) {
+    x_mant |= 0x0400;
+    exponent = static_cast<int>(x_abs >> 10) - 14;
+  } else {
+    exponent = -13;
+    while ((x_mant & 0x0400) == 0) {
+      x_mant <<= 1;
+      --exponent;
+    }
+  }
+
+  uint32_t m = x_mant << (RSQRT_FRACTION_BITS - 11);
+  int64_t y = newton_raphson(m, eval_polynomial(m));
+
+  int scale_exp = 0;
+  if (exponent & 1) {
+    y = (y * ONE_OVER_SQRT2) >> RSQRT_FRACTION_BITS;
+    scale_exp = -((exponent - 1) / 2);
+  } else {
+    scale_exp = -(exponent / 2);
+  }
+
+  return fixed_to_half_bits(static_cast<uint64_t>(y), scale_exp);
+}
+
+// Compare y = sig * 2^exp with 1 / sqrt(x_sig * 2^x_exp).
+// Return -1 if y is below the exact value, 0 if exact, and 1 if above.
+LIBC_INLINE constexpr int compare_with_rsqrt(uint32_t sig, int exp,
+                                             uint32_t x_sig, int x_exp) {
+  uint64_t lhs = static_cast<uint64_t>(sig) * sig * x_sig;
+  int scale = 2 * exp + x_exp;
+
+  if (scale >= 0)
+    return (scale == 0 && lhs == 1) ? 0 : 1;
+
+  int rshift = -scale;
+  if (rshift >= 64)
+    return -1;
+
+  uint64_t rhs = uint64_t(1) << rshift;
+  if (lhs < rhs)
+    return -1;
+  if (lhs > rhs)
+    return 1;
+  return 0;
+}
+
+LIBC_INLINE constexpr int compare_half_with_rsqrt(uint16_t y, uint32_t x_sig,
+                                                  int x_exp) {
+  uint32_t y_sig = 0x0400 | (y & 0x03ff);
+  int y_exp = static_cast<int>(y >> 10) - 25;
+  return compare_with_rsqrt(y_sig, y_exp, x_sig, x_exp);
+}
+
+LIBC_INLINE constexpr uint16_t floor_rsqrt(uint16_t approx, uint32_t x_sig,
+                                           int x_exp) {
+  uint16_t y = approx < 0x0400 ? 0x0400 : approx;
+  while (compare_half_with_rsqrt(y, x_sig, x_exp) > 0)
+    --y;
+  while (y < 0x7bff && compare_half_with_rsqrt(y + 1, x_sig, x_exp) <= 0)
+    ++y;
+  return y;
+}
+
+LIBC_INLINE constexpr uint16_t round_result(uint16_t y, uint32_t x_sig,
+                                            int x_exp) {
+  if (compare_half_with_rsqrt(y, x_sig, x_exp) == 0)
+    return y;
+
+  int rounding_mode = FE_TONEAREST;
+  if (!cpp::is_constant_evaluated())
+    rounding_mode = fputil::get_round();
+  if (rounding_mode == FE_UPWARD)
+    return y + 1;
+  if (rounding_mode != FE_TONEAREST)
+    return y;
+
+  uint32_t y_sig = 0x0400 | (y & 0x03ff);
+  int y_exp = static_cast<int>(y >> 10) - 25;
+  uint32_t midpoint_sig = (y_sig << 1) | 1;
+  int midpoint_cmp = compare_with_rsqrt(midpoint_sig, y_exp - 1, x_sig, x_exp);
+
+  if (midpoint_cmp < 0)
+    return y + 1;
+  if (midpoint_cmp > 0)
+    return y;
+  return (y & 1) ? static_cast<uint16_t>(y + 1) : y;
+}
+
+LIBC_INLINE constexpr float16 rsqrtf16_no_float(uint16_t x_abs) {
+  uint32_t x_sig = 0;
+  int x_exp = 0;
+  if (x_abs >= 0x0400) {
+    x_sig = 0x0400 | (x_abs & 0x03ff);
+    x_exp = static_cast<int>(x_abs >> 10) - 25;
+  } else {
+    x_sig = x_abs;
+    x_exp = -24;
+  }
+
+  uint16_t approx = approximate_rsqrt(x_abs);
+  uint16_t y = floor_rsqrt(approx, x_sig, x_exp);
+  return fputil::FPBits<float16>(round_result(y, x_sig, x_exp)).get_val();
+}
+
+} // namespace rsqrtf16_internal
+
 LIBC_INLINE constexpr float16 rsqrtf16(float16 x) {
   using FPBits = fputil::FPBits<float16>;
   FPBits xbits(x);
@@ -78,73 +243,7 @@ LIBC_INLINE constexpr float16 rsqrtf16(float16 x) {
   return fputil::cast<float16>(result);
 
 #else
-  float xf = fputil::cast<float>(x);
-
-  int exponent = 0;
-  float mantissa = fputil::frexp(xf, exponent);
-
-  float result = 0.0f;
-  int exp_floored = -(exponent >> 1);
-
-  if (mantissa == 0.5f) {
-    // When mantissa is 0.5f, x was a power of 2 (or subnormal that normalizes
-    // this way). 1/sqrt(0.5f) = sqrt(2.0f).
-    // If exponent is odd (exponent = 2k + 1):
-    //   rsqrt(x) = (1/sqrt(0.5)) * 2^(-(2k+1)/2) = sqrt(2) * 2^(-k-0.5)
-    //            = sqrt(2) * 2^(-k) * (1/sqrt(2)) = 2^(-k)
-    //   exp_floored = -((2k+1)>>1) = -(k) = -k
-    //   So result = ldexp(1.0f, exp_floored)
-    // If exponent is even (exponent = 2k):
-    //   rsqrt(x) = (1/sqrt(0.5)) * 2^(-2k/2) = sqrt(2) * 2^(-k)
-    //   exp_floored = -((2k)>>1) = -(k) = -k
-    //   So result = ldexp(sqrt(2.0f), exp_floored)
-    if (exponent & 1) {
-      result = fputil::ldexp(1.0f, exp_floored);
-    } else {
-      constexpr float SQRT_2_F = 0x1.6a09e6p0f; // sqrt(2.0f)
-      result = fputil::ldexp(SQRT_2_F, exp_floored);
-    }
-  } else {
-    // 4 Degree minimax polynomial (single-precision coefficients) generated
-    // with Sollya:
-    //   P = fpminimax(1/sqrt(x), 4,
-    //       [|single,single,single,single,single|], [0.5;1])
-    float y = fputil::polyeval(mantissa,
-                               0x1.771256p1f,  // c0
-                               -0x1.5e7c4ap2f, // c1
-                               0x1.b3851cp2f,  // c2
-                               -0x1.1a27ep2f,  // c3
-                               0x1.265c66p0f); // c4
-
-    // Newton-Raphson iteration in float (use multiply_add to leverage FMA when
-    // available):
-    float y2 = y * y;
-    float factor = fputil::multiply_add(-0.5f * mantissa, y2, 1.5f);
-    y = y * factor;
-
-    result = fputil::ldexp(y, exp_floored);
-    if (exponent & 1) {
-      constexpr float ONE_OVER_SQRT2 = 0x1.6a09e6p-1f; // 1/sqrt(2)
-      result *= ONE_OVER_SQRT2;
-    }
-
-    // Targeted post-correction: for the specific half-precision mantissa
-    // pattern M == 0x011F we observe a consistent -1 ULP bias across exponents.
-    // Apply a tiny upward nudge to cross the rounding boundary in all modes.
-    const uint16_t half_mantissa = static_cast<uint16_t>(x_abs & 0x3ff);
-    if (half_mantissa == 0x011F) {
-      // Nudge up to fix consistent -1 ULP at that mantissa boundary
-      result = fputil::multiply_add(result, 0x1.0p-21f,
-                                    result); // result *= (1 + 2^-21)
-    } else if (half_mantissa == 0x0313) {
-      // Nudge down to fix +1 ULP under upward rounding at this mantissa
-      // boundary
-      result = fputil::multiply_add(result, -0x1.0p-21f,
-                                    result); // result *= (1 - 2^-21)
-    }
-  }
-
-  return fputil::cast<float16>(result);
+  return rsqrtf16_internal::rsqrtf16_no_float(x_abs);
 #endif
 }
 
diff --git a/utils/bazel/llvm-project-overlay/libc/BUILD.bazel b/utils/bazel/llvm-project-overlay/libc/BUILD.bazel
index 9d1f4f9e6a493..2200cdc86c785 100644
--- a/utils/bazel/llvm-project-overlay/libc/BUILD.bazel
+++ b/utils/bazel/llvm-project-overlay/libc/BUILD.bazel
@@ -2997,7 +2997,6 @@ libc_support_library(
         ":__support_fputil_cast",
         ":__support_fputil_fenv_impl",
         ":__support_fputil_fp_bits",
-        ":__support_fputil_manipulation_functions",
         ":__support_fputil_multiply_add",
         ":__support_fputil_sqrt",
         ":__support_macros_optimization",

>From 5fa3ae6668bbf53443f7d311b970951c6e7b5693 Mon Sep 17 00:00:00 2001
From: amemov <shepelev777 at gmail.com>
Date: Sat, 25 Apr 2026 14:01:24 -0700
Subject: [PATCH 4/5] fix: remove duplicate target from cmake

---
 libc/src/__support/math/CMakeLists.txt | 16 ++--------------
 1 file changed, 2 insertions(+), 14 deletions(-)

diff --git a/libc/src/__support/math/CMakeLists.txt b/libc/src/__support/math/CMakeLists.txt
index 6f75baef8bf96..7eb6c97c3acfa 100644
--- a/libc/src/__support/math/CMakeLists.txt
+++ b/libc/src/__support/math/CMakeLists.txt
@@ -126,20 +126,6 @@ add_header_library(
     libc.src.__support.macros.properties.types
 )
 
-add_header_library(
-  rsqrtf16
-  HDRS
-    rsqrtf16.h
-  DEPENDS
-    libc.src.__support.FPUtil.cast
-    libc.src.__support.FPUtil.fenv_impl
-    libc.src.__support.FPUtil.fp_bits
-    libc.src.__support.FPUtil.multiply_add
-    libc.src.__support.FPUtil.sqrt
-    libc.src.__support.macros.optimization
-    libc.src.__support.macros.properties.types
-)
-
 add_header_library(
   asin_utils
   HDRS
@@ -2686,8 +2672,10 @@ add_header_library(
     libc.src.__support.FPUtil.cast
     libc.src.__support.FPUtil.fenv_impl
     libc.src.__support.FPUtil.fp_bits
+    libc.src.__support.FPUtil.multiply_add
     libc.src.__support.FPUtil.sqrt
     libc.src.__support.macros.optimization
+    libc.src.__support.macros.properties.types
 )
 
 add_header_library(

>From 1e8e5410e08dbfc1f8f3cb5e612bbb59c881bcc6 Mon Sep 17 00:00:00 2001
From: amemov <shepelev777 at gmail.com>
Date: Sun, 26 Apr 2026 19:39:05 -0700
Subject: [PATCH 5/5] reduced branching

---
 libc/src/__support/math/CMakeLists.txt        |  4 ++++
 libc/src/__support/math/rsqrtf16.h            | 21 +++++++------------
 .../llvm-project-overlay/libc/BUILD.bazel     |  1 +
 3 files changed, 13 insertions(+), 13 deletions(-)

diff --git a/libc/src/__support/math/CMakeLists.txt b/libc/src/__support/math/CMakeLists.txt
index 7eb6c97c3acfa..2b70cf31a4e77 100644
--- a/libc/src/__support/math/CMakeLists.txt
+++ b/libc/src/__support/math/CMakeLists.txt
@@ -2685,11 +2685,15 @@ add_header_library(
   DEPENDS
     libc.hdr.errno_macros
     libc.hdr.fenv_macros
+    libc.include.llvm-libc-macros.float16_macros
+    libc.src.__support.CPP.bit
     libc.src.__support.FPUtil.cast
     libc.src.__support.FPUtil.fenv_impl
     libc.src.__support.FPUtil.fp_bits
+    libc.src.__support.FPUtil.multiply_add
     libc.src.__support.FPUtil.sqrt
     libc.src.__support.macros.optimization
+    libc.src.__support.macros.properties.types
 )
 
 add_header_library(
diff --git a/libc/src/__support/math/rsqrtf16.h b/libc/src/__support/math/rsqrtf16.h
index 23c74a8515c15..67eaab1e0f652 100644
--- a/libc/src/__support/math/rsqrtf16.h
+++ b/libc/src/__support/math/rsqrtf16.h
@@ -13,6 +13,7 @@
 
 #ifdef LIBC_TYPES_HAS_FLOAT16
 
+#include "src/__support/CPP/bit.h"
 #include "src/__support/FPUtil/FEnvImpl.h"
 #include "src/__support/FPUtil/FPBits.h"
 #include "src/__support/FPUtil/cast.h"
@@ -39,12 +40,7 @@ LIBC_INLINE_VAR constexpr int64_t COEFFS[5] = {
 LIBC_INLINE_VAR constexpr int64_t ONE_OVER_SQRT2 = 0x16a09e60;
 
 LIBC_INLINE constexpr int floor_log2(uint64_t x) {
-  int result = -1;
-  while (x) {
-    x >>= 1;
-    ++result;
-  }
-  return result;
+  return 63 - cpp::countl_zero(x);
 }
 
 LIBC_INLINE constexpr int64_t eval_polynomial(uint32_t m) {
@@ -87,11 +83,9 @@ LIBC_INLINE constexpr uint16_t approximate_rsqrt(uint16_t x_abs) {
     x_mant |= 0x0400;
     exponent = static_cast<int>(x_abs >> 10) - 14;
   } else {
-    exponent = -13;
-    while ((x_mant & 0x0400) == 0) {
-      x_mant <<= 1;
-      --exponent;
-    }
+    int shift = cpp::countl_zero(x_mant) - (32 - 11);
+    x_mant <<= shift;
+    exponent = -13 - shift;
   }
 
   uint32_t m = x_mant << (RSQRT_FRACTION_BITS - 11);
@@ -140,9 +134,10 @@ LIBC_INLINE constexpr int compare_half_with_rsqrt(uint16_t y, uint32_t x_sig,
 LIBC_INLINE constexpr uint16_t floor_rsqrt(uint16_t approx, uint32_t x_sig,
                                            int x_exp) {
   uint16_t y = approx < 0x0400 ? 0x0400 : approx;
-  while (compare_half_with_rsqrt(y, x_sig, x_exp) > 0)
+  if (LIBC_UNLIKELY(compare_half_with_rsqrt(y, x_sig, x_exp) > 0))
     --y;
-  while (y < 0x7bff && compare_half_with_rsqrt(y + 1, x_sig, x_exp) <= 0)
+  if (LIBC_UNLIKELY(y < 0x7bff &&
+                    compare_half_with_rsqrt(y + 1, x_sig, x_exp) <= 0))
     ++y;
   return y;
 }
diff --git a/utils/bazel/llvm-project-overlay/libc/BUILD.bazel b/utils/bazel/llvm-project-overlay/libc/BUILD.bazel
index 2200cdc86c785..89f15f453cc2c 100644
--- a/utils/bazel/llvm-project-overlay/libc/BUILD.bazel
+++ b/utils/bazel/llvm-project-overlay/libc/BUILD.bazel
@@ -2994,6 +2994,7 @@ libc_support_library(
     name = "__support_math_rsqrtf16",
     hdrs = ["src/__support/math/rsqrtf16.h"],
     deps = [
+        ":__support_cpp_bit",
         ":__support_fputil_cast",
         ":__support_fputil_fenv_impl",
         ":__support_fputil_fp_bits",



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