diff --git a/tests/unit/test-cutils.c b/tests/unit/test-cutils.c
index f781997aef7..1fb9d5323ab 100644
--- a/tests/unit/test-cutils.c
+++ b/tests/unit/test-cutils.c
@@ -2693,14 +2693,14 @@ static void test_qemu_strtosz_float(void)
     g_assert_cmpuint(res, ==, 1024);
     g_assert_true(endptr == str + 4);

-    /* FIXME An empty fraction head should be tolerated */
+    /* An empty fraction head is tolerated */
     str = " .5k";
     endptr = str;
     res = 0xbaadf00d;
     err = qemu_strtosz(str, &endptr, &res);
-    g_assert_cmpint(err, ==, -EINVAL /* FIXME 0 */);
-    g_assert_cmpuint(res, ==, 0 /* FIXME 512 */);
-    g_assert_true(endptr == str /* FIXME + 4 */);
+    g_assert_cmpint(err, ==, 0);
+    g_assert_cmpuint(res, ==, 512);
+    g_assert_true(endptr == str + 4);

     /* For convenience, we permit values that are not byte-exact */
     str = "12.345M";
@@ -2711,16 +2711,16 @@ static void test_qemu_strtosz_float(void)
     g_assert_cmpuint(res, ==, (uint64_t) (12.345 * MiB + 0.5));
     g_assert_true(endptr == str + 7);

-    /* FIXME Fraction tail should round correctly */
+    /* Fraction tail can round up */
     str = "1.9999999999999999999999999999999999999999999999999999k";
     endptr = str;
     res = 0xbaadf00d;
     err = qemu_strtosz(str, &endptr, &res);
     g_assert_cmpint(err, ==, 0);
-    g_assert_cmpint(res, ==, 1024 /* FIXME 2048 */);
+    g_assert_cmpuint(res, ==, 2048);
     g_assert_true(endptr == str + 55);

-    /* FIXME ERANGE underflow in the fraction tail should not matter for 'k' */
+    /* ERANGE underflow in the fraction tail does not matter for 'k' */
     str = "1."
         "00000000000000000000000000000000000000000000000000"
         "00000000000000000000000000000000000000000000000000"
@@ -2734,7 +2734,7 @@ static void test_qemu_strtosz_float(void)
     res = 0xbaadf00d;
     err = qemu_strtosz(str, &endptr, &res);
     g_assert_cmpint(err, ==, 0);
-    g_assert_cmpuint(res, ==, 1 /* FIXME 1024 */);
+    g_assert_cmpuint(res, ==, 1024);
     g_assert_true(endptr == str + 354);
 }

@@ -2826,16 +2826,16 @@ static void test_qemu_strtosz_invalid(void)
     g_assert_cmpuint(res, ==, 0);
     g_assert_true(endptr == str);

-    /* FIXME Fraction tail can cause ERANGE overflow */
+    /* Fraction tail can cause ERANGE overflow */
     str = "15.9999999999999999999999999999999999999999999999999999E";
     endptr = str;
     res = 0xbaadf00d;
     err = qemu_strtosz(str, &endptr, &res);
-    g_assert_cmpint(err, ==, 0 /* FIXME -ERANGE */);
-    g_assert_cmpuint(res, ==, 15ULL * EiB /* FIXME 0 */);
-    g_assert_true(endptr == str + 56 /* FIXME str */);
+    g_assert_cmpint(err, ==, -ERANGE);
+    g_assert_cmpuint(res, ==, 0);
+    g_assert_true(endptr == str + 56);

-    /* FIXME ERANGE underflow in the fraction tail should matter for 'B' */
+    /* ERANGE underflow in the fraction tail matters for 'B' */
     str = "1."
         "00000000000000000000000000000000000000000000000000"
         "00000000000000000000000000000000000000000000000000"
@@ -2848,9 +2848,9 @@ static void test_qemu_strtosz_invalid(void)
     endptr = str;
     res = 0xbaadf00d;
     err = qemu_strtosz(str, &endptr, &res);
-    g_assert_cmpint(err, ==, 0 /* FIXME -EINVAL */);
-    g_assert_cmpuint(res, ==, 1 /* FIXME 0 */);
-    g_assert_true(endptr == str + 354 /* FIXME str */);
+    g_assert_cmpint(err, ==, -EINVAL);
+    g_assert_cmpuint(res, ==, 0);
+    g_assert_true(endptr == str);

     /* No hex fractions */
     str = "0x1.8k";
@@ -3045,14 +3045,14 @@ static void test_qemu_strtosz_trailing(void)
     g_assert_cmpint(err, ==, -EINVAL);
     g_assert_cmpuint(res, ==, 0);

-    /* FIXME should stop parse after 'e'. No floating point exponents */
+    /* Parse stops at 'e', which is not a floating point exponent */
     str = "1.5e1k";
     endptr = NULL;
     res = 0xbaadf00d;
     err = qemu_strtosz(str, &endptr, &res);
-    g_assert_cmpint(err, ==, -EINVAL /* FIXME 0 */);
-    g_assert_cmpuint(res, ==, 0 /* FIXME EiB * 1.5 */);
-    g_assert_true(endptr == str /* FIXME + 4 */);
+    g_assert_cmpint(err, ==, 0);
+    g_assert_cmpuint(res, ==, EiB * 1.5);
+    g_assert_true(endptr == str + 4);

     res = 0xbaadf00d;
     err = qemu_strtosz(str, NULL, &res);
@@ -3063,23 +3063,22 @@ static void test_qemu_strtosz_trailing(void)
     endptr = NULL;
     res = 0xbaadf00d;
     err = qemu_strtosz(str, &endptr, &res);
-    g_assert_cmpint(err, ==, -EINVAL /* FIXME 0 */);
-    g_assert_cmpuint(res, ==, 0 /* FIXME EiB * 1.5 */);
-    g_assert_true(endptr == str /* FIXME + 4 */);
+    g_assert_cmpint(err, ==, 0);
+    g_assert_cmpuint(res, ==, EiB * 1.5);
+    g_assert_true(endptr == str + 4);

     res = 0xbaadf00d;
     err = qemu_strtosz(str, NULL, &res);
     g_assert_cmpint(err, ==, -EINVAL);
     g_assert_cmpuint(res, ==, 0);

-    /* FIXME overflow in fraction is buggy */
     str = "1.5E999";
     endptr = NULL;
     res = 0xbaadf00d;
     err = qemu_strtosz(str, &endptr, &res);
     g_assert_cmpint(err, ==, 0);
-    g_assert_cmpuint(res, ==, 1 /* FIXME EiB * 1.5 */);
-    g_assert(endptr == str + 2 /* FIXME + 4 */);
+    g_assert_cmpuint(res, ==, EiB * 1.5);
+    g_assert_true(endptr == str + 4);

     res = 0xbaadf00d;
     err = qemu_strtosz(str, NULL, &res);
diff --git a/util/cutils.c b/util/cutils.c
index 0e056a27a44..d1dfbc69d16 100644
--- a/util/cutils.c
+++ b/util/cutils.c
@@ -194,14 +194,17 @@ static int64_t suffix_mul(char suffix, int64_t unit)
  * - 12345 - decimal, scale determined by @default_suffix and @unit
  * - 12345{bBkKmMgGtTpPeE} - decimal, scale determined by suffix and @unit
  * - 12345.678{kKmMgGtTpPeE} - decimal, scale determined by suffix, and
- *   fractional portion is truncated to byte
+ *   fractional portion is truncated to byte, either side of . may be empty
  * - 0x7fEE - hexadecimal, unit determined by @default_suffix
  *
  * The following are intentionally not supported
- * - hex with scaling suffix, such as 0x20M
- * - octal, such as 08
- * - fractional hex, such as 0x1.8
- * - floating point exponents, such as 1e3
+ * - hex with scaling suffix, such as 0x20M (0x1b is 27, not 1)
+ * - octal, such as 08 (parsed as decimal instead)
+ * - binary, such as 0b1000 (parsed as 0b with trailing garbage "1000")
+ * - fractional hex, such as 0x1.8 (parsed as 0 with trailing garbage "x1.8")
+ * - floating point exponents, such as 1e3 (parsed as 1e with trailing
+ *   garbage "3") or 0x1p3 (parsed as 1 with trailing garbage "p3")
+ * - non-finite values, such as inf or NaN
  *
  * The end pointer will be returned in *end, if not NULL.  If there is
  * no fraction, the input can be decimal or hexadecimal; if there is a
@@ -220,22 +223,17 @@ static int do_strtosz(const char *nptr, const char **end,
                       uint64_t *result)
 {
     int retval;
-    const char *endptr, *f;
+    const char *endptr;
     unsigned char c;
-    uint64_t val, valf = 0;
+    uint64_t val = 0, valf = 0;
     int64_t mul;

     /* Parse integral portion as decimal. */
     retval = qemu_strtou64(nptr, &endptr, 10, &val);
-    if (retval) {
+    if (retval == -ERANGE || !nptr) {
         goto out;
     }
-    if (memchr(nptr, '-', endptr - nptr) != NULL) {
-        endptr = nptr;
-        retval = -EINVAL;
-        goto out;
-    }
-    if (val == 0 && (*endptr == 'x' || *endptr == 'X')) {
+    if (retval == 0 && val == 0 && (*endptr == 'x' || *endptr == 'X')) {
         /* Input looks like hex; reparse, and insist on no fraction or suffix. */
         retval = qemu_strtou64(nptr, &endptr, 16, &val);
         if (retval) {
@@ -246,27 +244,66 @@ static int do_strtosz(const char *nptr, const char **end,
             retval = -EINVAL;
             goto out;
         }
-    } else if (*endptr == '.') {
+    } else if (*endptr == '.' || (endptr == nptr && strchr(nptr, '.'))) {
         /*
          * Input looks like a fraction.  Make sure even 1.k works
-         * without fractional digits.  If we see an exponent, treat
-         * the entire input as invalid instead.
+         * without fractional digits.  strtod tries to treat 'e' as an
+         * exponent, but we want to treat it as a scaling suffix;
+         * doing this requires modifying a copy of the fraction.
          */
-        double fraction;
+        double fraction = 0.0;

-        f = endptr;
-        retval = qemu_strtod_finite(f, &endptr, &fraction);
-        if (retval) {
+        if (retval == 0 && *endptr == '.' && !isdigit(endptr[1])) {
+            /* If we got here, we parsed at least one digit already. */
             endptr++;
-        } else if (memchr(f, 'e', endptr - f) || memchr(f, 'E', endptr - f)) {
-            endptr = nptr;
-            retval = -EINVAL;
-            goto out;
         } else {
-            /* Extract into a 64-bit fixed-point fraction. */
+            char *e;
+            const char *tail;
+            g_autofree char *copy = g_strdup(endptr);
+
+            e = strchr(copy, 'e');
+            if (e) {
+                *e = '\0';
+            }
+            e = strchr(copy, 'E');
+            if (e) {
+                *e = '\0';
+            }
+            /*
+             * If this is a floating point, we are guaranteed that '.'
+             * appears before any possible digits in copy.  If it is
+             * not a floating point, strtod will fail.  Either way,
+             * there is now no exponent in copy, so if it parses, we
+             * know 0.0 <= abs(result) <= 1.0 (after rounding), and
+             * ERANGE is only possible on underflow which is okay.
+             */
+            retval = qemu_strtod_finite(copy, &tail, &fraction);
+            endptr += tail - copy;
+        }
+
+        /* Extract into a 64-bit fixed-point fraction. */
+        if (fraction == 1.0) {
+            if (val == UINT64_MAX) {
+                retval = -ERANGE;
+                goto out;
+            }
+            val++;
+        } else if (retval == -ERANGE) {
+            /* See comments above about underflow */
+            valf = 1;
+            retval = 0;
+        } else {
             valf = (uint64_t)(fraction * 0x1p64);
         }
     }
+    if (retval) {
+        goto out;
+    }
+    if (memchr(nptr, '-', endptr - nptr) != NULL) {
+        endptr = nptr;
+        retval = -EINVAL;
+        goto out;
+    }
     c = *endptr;
     mul = suffix_mul(c, unit);
     if (mul > 0) {
