diff --git a/include/fpu/softfloat-types.h b/include/fpu/softfloat-types.h
index e1732beba4..b1941384ae 100644
--- a/include/fpu/softfloat-types.h
+++ b/include/fpu/softfloat-types.h
@@ -333,8 +333,22 @@ typedef enum __attribute__((__packed__)) {
     /*
      * Are Pseudo-infinities (Inf with the Integer bit zero) valid?
      * If so, floatx80_is_infinity() will return true for them.
+     * If not, floatx80_invalid_encoding will return false for them,
+     * and using them as inputs to a float op will raise Invalid.
      */
     floatx80_pseudo_inf_valid = 2,
+    /*
+     * Are Pseudo-NaNs (NaNs where the Integer bit is zero) valid?
+     * If not, floatx80_invalid_encoding() will return false for them,
+     * and using them as inputs to a float op will raise Invalid.
+     */
+    floatx80_pseudo_nan_valid = 4,
+    /*
+     * Are Unnormals (0 < exp < 0x7fff, Integer bit zero) valid?
+     * If not, floatx80_invalid_encoding() will return false for them,
+     * and using them as inputs to a float op will raise Invalid.
+     */
+    floatx80_unnormal_valid = 8,
 } FloatX80Behaviour;
 
 /*
diff --git a/include/fpu/softfloat.h b/include/fpu/softfloat.h
index 1c8f3cbb78..c18ab2cb60 100644
--- a/include/fpu/softfloat.h
+++ b/include/fpu/softfloat.h
@@ -1073,41 +1073,45 @@ static inline bool floatx80_unordered_quiet(floatx80 a, floatx80 b,
 
 /*----------------------------------------------------------------------------
 | Return whether the given value is an invalid floatx80 encoding.
-| Invalid floatx80 encodings arise when the integer bit is not set, but
-| the exponent is not zero. The only times the integer bit is permitted to
-| be zero is in subnormal numbers and the value zero.
-| This includes what the Intel software developer's manual calls pseudo-NaNs,
-| pseudo-infinities and un-normal numbers. It does not include
-| pseudo-denormals, which must still be correctly handled as inputs even
-| if they are never generated as outputs.
+| Invalid floatx80 encodings may arise when the integer bit is not set
+| correctly; this is target-specific. In Intel terminology the
+| categories are:
+|  exp == 0, int = 0, mantissa == 0 : zeroes
+|  exp == 0, int = 0, mantissa != 0 : denormals
+|  exp == 0, int = 1 : pseudo-denormals
+|  0 < exp < 0x7fff, int = 0 : unnormals
+|  0 < exp < 0x7fff, int = 1 : normals
+|  exp == 0x7fff, int = 0, mantissa == 0 : pseudo-infinities
+|  exp == 0x7fff, int = 1, mantissa == 0 : infinities
+|  exp == 0x7fff, int = 0, mantissa != 0 : pseudo-NaNs
+|  exp == 0x7fff, int = 1, mantissa == 0 : NaNs
+|
+| The usual IEEE cases of zero, denormal, normal, inf and NaN are always valid.
+| x87 permits as input also pseudo-denormals.
+| m68k permits all those and also pseudo-infinities, pseudo-NaNs and unnormals.
+|
+| Since we don't have a target that handles floatx80 but prohibits
+| pseudo-denormals in input, we don't currently have a floatx80_behaviour
+| flag for that case, but instead always accept it. Conveniently this
+| means that all cases with either exponent 0 or the integer bit set are
+| valid for all targets.
 *----------------------------------------------------------------------------*/
 static inline bool floatx80_invalid_encoding(floatx80 a, float_status *s)
 {
-#if defined(TARGET_M68K)
-    /*-------------------------------------------------------------------------
-    | With m68k, the explicit integer bit can be zero in the case of:
-    | - zeros                (exp == 0, mantissa == 0)
-    | - denormalized numbers (exp == 0, mantissa != 0)
-    | - unnormalized numbers (exp != 0, exp < 0x7FFF)
-    | - infinities           (exp == 0x7FFF, mantissa == 0)
-    | - not-a-numbers        (exp == 0x7FFF, mantissa != 0)
-    |
-    | For infinities and NaNs, the explicit integer bit can be either one or
-    | zero.
-    |
-    | The IEEE 754 standard does not define a zero integer bit. Such a number
-    | is an unnormalized number. Hardware does not directly support
-    | denormalized and unnormalized numbers, but implicitly supports them by
-    | trapping them as unimplemented data types, allowing efficient conversion
-    | in software.
-    |
-    | See "M68000 FAMILY PROGRAMMER’S REFERENCE MANUAL",
-    |     "1.6 FLOATING-POINT DATA TYPES"
-    *------------------------------------------------------------------------*/
-    return false;
-#else
-    return (a.low & (1ULL << 63)) == 0 && (a.high & 0x7FFF) != 0;
-#endif
+    if ((a.low >> 63) || (a.high & 0x7fff) == 0) {
+        /* Anything with the Integer bit set or the exponent 0 is valid */
+        return false;
+    }
+
+    if ((a.high & 0x7fff) == 0x7fff) {
+        if (a.low) {
+            return !(s->floatx80_behaviour & floatx80_pseudo_nan_valid);
+        } else {
+            return !(s->floatx80_behaviour & floatx80_pseudo_inf_valid);
+        }
+    } else {
+        return !(s->floatx80_behaviour & floatx80_unnormal_valid);
+    }
 }
 
 #define floatx80_zero make_floatx80(0x0000, 0x0000000000000000LL)
diff --git a/target/m68k/cpu.c b/target/m68k/cpu.c
index 56b23de21f..505fa97a53 100644
--- a/target/m68k/cpu.c
+++ b/target/m68k/cpu.c
@@ -111,9 +111,35 @@ static void m68k_cpu_reset_hold(Object *obj, ResetType type)
      * m68k-specific floatx80 behaviour:
      *  * default Infinity values have a zero Integer bit
      *  * input Infinities may have the Integer bit either 0 or 1
+     *  * pseudo-denormals supported for input and output
+     *  * don't raise Invalid for pseudo-NaN/pseudo-Inf/Unnormal
+     *
+     * With m68k, the explicit integer bit can be zero in the case of:
+     * - zeros                (exp == 0, mantissa == 0)
+     * - denormalized numbers (exp == 0, mantissa != 0)
+     * - unnormalized numbers (exp != 0, exp < 0x7FFF)
+     * - infinities           (exp == 0x7FFF, mantissa == 0)
+     * - not-a-numbers        (exp == 0x7FFF, mantissa != 0)
+     *
+     * For infinities and NaNs, the explicit integer bit can be either one or
+     * zero.
+     *
+     * The IEEE 754 standard does not define a zero integer bit. Such a number
+     * is an unnormalized number. Hardware does not directly support
+     * denormalized and unnormalized numbers, but implicitly supports them by
+     * trapping them as unimplemented data types, allowing efficient conversion
+     * in software.
+     *
+     * See "M68000 FAMILY PROGRAMMER’S REFERENCE MANUAL",
+     *     "1.6 FLOATING-POINT DATA TYPES"
+     *
+     * Note though that QEMU's fp emulation does directly handle both
+     * denormal and unnormal values, and does not trap to guest software.
      */
     set_floatx80_behaviour(floatx80_default_inf_int_bit_is_zero |
-                           floatx80_pseudo_inf_valid,
+                           floatx80_pseudo_inf_valid |
+                           floatx80_pseudo_nan_valid |
+                           floatx80_unnormal_valid,
                            &env->fp_status);
 
     nan = floatx80_default_nan(&env->fp_status);