diff --git a/include/boost/decimal/detail/cmath/impl/trig_reduce.hpp b/include/boost/decimal/detail/cmath/impl/trig_reduce.hpp
index 5455f31b2..443588965 100644
--- a/include/boost/decimal/detail/cmath/impl/trig_reduce.hpp
+++ b/include/boost/decimal/detail/cmath/impl/trig_reduce.hpp
@@ -248,12 +248,8 @@ constexpr auto to_words(const boost::int128::uint128_t& m, std::uint64_t* out) n
     out[1] = lo / word_base;
     out[2] = hi % word_base;
     out[3] = hi / word_base;
-    int n {4};
-    while (n > 0 && out[n - 1] == 0U)
-    {
-        --n;
-    }
-    return n;
+    // Zero high words do not change the product.
+    return 4;
 }
 
 // m * 10^shift in words of 9 digits, low word first, for 0 <= shift < 9; returns the count of words.
@@ -436,17 +432,12 @@ constexpr auto trig_reduce(const Significand m, const int e, const bool xneg) no
     out.n = n;
     out.neg = rneg;
 
+    // The fraction is not zero: its worst case has 9, 19 and 37 leading zero digits (see trig_traits).
     int lead {};
     while (lead < window - 1 && f[lead] == 0U)
     {
         ++lead;
     }
-    if (lead == window - 1)
-    {
-        // Only for an exact multiple of pi/2, which no nonzero x is.
-        out.zero = true;
-        return out;
-    }
 
     std::uint64_t r[static_cast<std::size_t>(2 * frac + 1)] {};
     times_half_pi<frac, window - 1>(f, lead, r);
@@ -573,17 +564,12 @@ constexpr auto trig_reduce_binary(const Significand m, const int e, const bool x
     out.n = n;
     out.neg = rneg;
 
+    // The fraction is not zero: its worst case has 27, 58 and 117 leading zero bits (see trig_traits).
     int lead {};
     while (lead < frac && f[lead] == 0U)
     {
         ++lead;
     }
-    if (lead == frac)
-    {
-        // Only for an exact multiple of pi/2, which no nonzero x is.
-        out.zero = true;
-        return out;
-    }
 
     // r = (rw words of f from word lead) * pi/2 = (s from word rw) * 2^-64(lead + rw). The four spare
     // words take the scale by 10^k below.
diff --git a/test/test_trig_rounding.cpp b/test/test_trig_rounding.cpp
index 163e1ac76..9fb173b39 100644
--- a/test/test_trig_rounding.cpp
+++ b/test/test_trig_rounding.cpp
@@ -15,7 +15,7 @@ using namespace boost::decimal;
 using namespace boost::decimal::literals;
 
 // Each type has inputs near pi/2, pi and 1000*pi/2, the value nearest to a multiple of pi/2 in its exponent
-// range (4.327189e+48 for decimal32_t) and below 10^19 (76.96902), and the inputs on each side of 10^19.
+// range (4.327189e+48 for decimal32_t) and below 10^19 (76.96902), the inputs on each side of 10^19, -10^19 and 10^-80.
 template <typename T>
 void check(const T x, const T s, const T c, const T t)
 {
@@ -41,6 +41,8 @@ void test_nearest_decimal32_t()
     check(7.696902e+01_DF, 1.000000e0_DF, 1.294993e-8_DF, 7.722047e7_DF);
     check(9.999999e+18_DF, -9.628773e-1_DF, 2.699396e-1_DF, -3.567010e0_DF);
     check(1.000000e+19_DF, -9.270632e-1_DF, -3.749052e-1_DF, 2.472794e0_DF);
+    check(-1.000000e+19_DF, 9.270632e-1_DF, -3.749052e-1_DF, -2.472794e0_DF);
+    check(1.000000e-80_DF, 1.000000e-80_DF, 1.000000e0_DF, 1.000000e-80_DF);
 }
 
 void test_nearest_decimal_fast32_t()
@@ -60,6 +62,8 @@ void test_nearest_decimal_fast32_t()
     check(7.696902e+01_DFF, 1.000000e0_DFF, 1.294993e-8_DFF, 7.722047e7_DFF);
     check(9.999999e+18_DFF, -9.628773e-1_DFF, 2.699396e-1_DFF, -3.567010e0_DFF);
     check(1.000000e+19_DFF, -9.270632e-1_DFF, -3.749052e-1_DFF, 2.472794e0_DFF);
+    check(-1.000000e+19_DFF, 9.270632e-1_DFF, -3.749052e-1_DFF, -2.472794e0_DFF);
+    check(1.000000e-80_DFF, 1.000000e-80_DFF, 1.000000e0_DFF, 1.000000e-80_DFF);
 }
 
 void test_nearest_decimal64_t()
@@ -79,6 +83,8 @@ void test_nearest_decimal64_t()
     check(9.538513039607291e+06_DD, -6.771756623504492e-18_DD, -1.000000000000000e0_DD, 6.771756623504492e-18_DD);
     check(9.999999999999999e+18_DD, -2.113595126444045e-1_DD, -9.774083877349937e-1_DD, 2.162448320442599e-1_DD);
     check(1.000000000000000e+19_DD, -9.270631660486504e-1_DD, -3.749051695507178e-1_DD, 2.472793765846527e0_DD);
+    check(-1.000000000000000e+19_DD, 9.270631660486504e-1_DD, -3.749051695507178e-1_DD, -2.472793765846527e0_DD);
+    check(1.000000000000000e-80_DD, 1.000000000000000e-80_DD, 1.000000000000000e0_DD, 1.000000000000000e-80_DD);
 }
 
 void test_nearest_decimal_fast64_t()
@@ -98,6 +104,8 @@ void test_nearest_decimal_fast64_t()
     check(9.538513039607291e+06_DDF, -6.771756623504492e-18_DDF, -1.000000000000000e0_DDF, 6.771756623504492e-18_DDF);
     check(9.999999999999999e+18_DDF, -2.113595126444045e-1_DDF, -9.774083877349937e-1_DDF, 2.162448320442599e-1_DDF);
     check(1.000000000000000e+19_DDF, -9.270631660486504e-1_DDF, -3.749051695507178e-1_DDF, 2.472793765846527e0_DDF);
+    check(-1.000000000000000e+19_DDF, 9.270631660486504e-1_DDF, -3.749051695507178e-1_DDF, -2.472793765846527e0_DDF);
+    check(1.000000000000000e-80_DDF, 1.000000000000000e-80_DDF, 1.000000000000000e0_DDF, 1.000000000000000e-80_DDF);
 }
 
 void test_nearest_decimal128_t()
@@ -117,6 +125,8 @@ void test_nearest_decimal128_t()
     check(2.589477332551582209163675623249625e+08_DL, -1.000000000000000000000000000000000e0_DL, 1.532138639232766081410107492878833e-35_DL, -6.526824494816997196224318392388852e34_DL);
     check(9.999999999999999999999999999999999e+18_DL, -9.270631660486500103289527389466423e-1_DL, -3.749051695507187572163865946598085e-1_DL, 2.472793765846520245629178386203776e0_DL);
     check(1.000000000000000000000000000000000e+19_DL, -9.270631660486503852341222896649360e-1_DL, -3.749051695507178301532205460096107e-1_DL, 2.472793765846527360338186795636566e0_DL);
+    check(-1.000000000000000000000000000000000e+19_DL, 9.270631660486503852341222896649360e-1_DL, -3.749051695507178301532205460096107e-1_DL, -2.472793765846527360338186795636566e0_DL);
+    check(1.000000000000000000000000000000000e-80_DL, 1.000000000000000000000000000000000e-80_DL, 1.000000000000000000000000000000000e0_DL, 1.000000000000000000000000000000000e-80_DL);
 }
 
 void test_nearest_decimal_fast128_t()
@@ -136,6 +146,8 @@ void test_nearest_decimal_fast128_t()
     check(2.589477332551582209163675623249625e+08_DLF, -1.000000000000000000000000000000000e0_DLF, 1.532138639232766081410107492878833e-35_DLF, -6.526824494816997196224318392388852e34_DLF);
     check(9.999999999999999999999999999999999e+18_DLF, -9.270631660486500103289527389466423e-1_DLF, -3.749051695507187572163865946598085e-1_DLF, 2.472793765846520245629178386203776e0_DLF);
     check(1.000000000000000000000000000000000e+19_DLF, -9.270631660486503852341222896649360e-1_DLF, -3.749051695507178301532205460096107e-1_DLF, 2.472793765846527360338186795636566e0_DLF);
+    check(-1.000000000000000000000000000000000e+19_DLF, 9.270631660486503852341222896649360e-1_DLF, -3.749051695507178301532205460096107e-1_DLF, -2.472793765846527360338186795636566e0_DLF);
+    check(1.000000000000000000000000000000000e-80_DLF, 1.000000000000000000000000000000000e-80_DLF, 1.000000000000000000000000000000000e0_DLF, 1.000000000000000000000000000000000e-80_DLF);
 }
 
 void test_downward_decimal32_t()