diff --git a/tests/unit/test-cutils.c b/tests/unit/test-cutils.c
index 0a589567461..1db411489f0 100644
--- a/tests/unit/test-cutils.c
+++ b/tests/unit/test-cutils.c
@@ -3408,19 +3408,18 @@ static void test_qemu_strtosz_float(void)
     /* An empty fraction tail is tolerated */
     do_strtosz("1.k", 0, 1024, 3);

-    /* FIXME An empty fraction head should be tolerated */
-    do_strtosz(" .5k", -EINVAL /* FIXME 0 */, 0 /* FIXME 512 */,
-               0 /* FIXME 4 */);
+    /* An empty fraction head is tolerated */
+    do_strtosz(" .5k", 0, 512, 4);

     /* For convenience, we permit values that are not byte-exact */
     do_strtosz("12.345M", 0, (uint64_t) (12.345 * MiB + 0.5), 7);

-    /* FIXME Fraction tail should round correctly */
+    /* Fraction tail can round up */
     do_strtosz("1.9999k", 0, 2048, 7);
     do_strtosz("1.9999999999999999999999999999999999999999999999999999k", 0,
-               1024 /* FIXME 2048 */, 55);
+               2048, 55);

-    /* FIXME ERANGE underflow in the fraction tail should not matter for 'k' */
+    /* ERANGE underflow in the fraction tail does not matter for 'k' */
     do_strtosz("1."
                "00000000000000000000000000000000000000000000000000"
                "00000000000000000000000000000000000000000000000000"
@@ -3429,7 +3428,7 @@ static void test_qemu_strtosz_float(void)
                "00000000000000000000000000000000000000000000000000"
                "00000000000000000000000000000000000000000000000000"
                "00000000000000000000000000000000000000000000000000"
-               "1k", 0, 1 /* FIXME 1024 */, 354);
+               "1k", 0, 1024, 354);
 }

 static void test_qemu_strtosz_invalid(void)
@@ -3453,10 +3452,9 @@ static void test_qemu_strtosz_invalid(void)
     do_strtosz("1.1B", -EINVAL, 0, 0);
     do_strtosz("1.1", -EINVAL, 0, 0);

-    /* FIXME underflow in the fraction tail should matter for 'B' */
+    /* 'B' cannot have any nonzero fraction, even with rounding or underflow */
     do_strtosz("1.00001B", -EINVAL, 0, 0);
-    do_strtosz("1.00000000000000000001B", 0 /* FIXME -EINVAL */,
-               1 /* FIXME 0 */, 23 /* FIXME 0 */);
+    do_strtosz("1.00000000000000000001B", -EINVAL, 0, 0);
     do_strtosz("1."
                "00000000000000000000000000000000000000000000000000"
                "00000000000000000000000000000000000000000000000000"
@@ -3465,8 +3463,7 @@ static void test_qemu_strtosz_invalid(void)
                "00000000000000000000000000000000000000000000000000"
                "00000000000000000000000000000000000000000000000000"
                "00000000000000000000000000000000000000000000000000"
-               "1B", 0 /* FIXME -EINVAL */, 1 /* FIXME 0 */,
-               354 /* FIXME 0 */);
+               "1B", -EINVAL, 0, 0);

     /* No hex fractions */
     do_strtosz("0x1.8k", -EINVAL, 0, 0);
@@ -3512,28 +3509,20 @@ static void test_qemu_strtosz_trailing(void)
     do_strtosz_full("123-45", qemu_strtosz, 0, 123, 3, -EINVAL, 0);
     do_strtosz_full(" 123 - 45", qemu_strtosz, 0, 123, 4, -EINVAL, 0);

-    /* FIXME should stop parse after 'e'. No floating point exponents */
-    do_strtosz_full("1.5e1k", qemu_strtosz, -EINVAL /* FIXME 0 */,
-                    0 /* FIXME EiB * 1.5 */, 0 /* FIXME 4 */,
-                    -EINVAL, 0);
-    do_strtosz_full("1.5E+0k", qemu_strtosz, -EINVAL /* FIXME 0 */,
-                    0 /* FIXME EiB * 1.5 */, 0 /* FIXME 4 */,
-                    -EINVAL, 0);
-
-    /* FIXME overflow in fraction is still buggy */
-    do_strtosz_full("1.5E999", qemu_strtosz, 0, 1 /* FIXME EiB * 1.5 */,
-                    2 /* FIXME 4 */, -EINVAL, 0);
+    /* Parse stops at 'e', which is not a floating point exponent */
+    do_strtosz_full("1.5e1k", qemu_strtosz, 0, EiB * 1.5, 4, -EINVAL, 0);
+    do_strtosz_full("1.5E+0k", qemu_strtosz, 0, EiB * 1.5, 4, -EINVAL, 0);
+    do_strtosz_full("1.5E999", qemu_strtosz, 0, EiB * 1.5, 4, -EINVAL, 0);
 }

 static void test_qemu_strtosz_erange(void)
 {
-    /* FIXME negative values fit better as ERANGE */
+    /* no negative values */
     do_strtosz(" -0", -ERANGE, 0, 3);
     do_strtosz("-1", -ERANGE, 0, 2);
     do_strtosz_full("-2M", qemu_strtosz, -ERANGE, 0, 2, -EINVAL, 0);
-    do_strtosz(" -.0", -EINVAL /* FIXME -ERANGE */, 0, 0 /* FIXME 4 */);
-    do_strtosz_full("-.1k", qemu_strtosz, -EINVAL /* FIXME -ERANGE */, 0,
-                    0 /* FIXME 3 */, -EINVAL, 0);
+    do_strtosz(" -.0", -ERANGE, 0, 4);
+    do_strtosz_full("-.1k", qemu_strtosz, -ERANGE, 0, 3, -EINVAL, 0);
     do_strtosz_full(" -."
                     "00000000000000000000000000000000000000000000000000"
                     "00000000000000000000000000000000000000000000000000"
@@ -3542,17 +3531,16 @@ static void test_qemu_strtosz_erange(void)
                     "00000000000000000000000000000000000000000000000000"
                     "00000000000000000000000000000000000000000000000000"
                     "00000000000000000000000000000000000000000000000000"
-                    "1M", qemu_strtosz, -EINVAL /* FIXME -ERANGE */, 0,
-                    0 /* FIXME 354 */, -EINVAL, 0);
+                    "1M", qemu_strtosz, -ERANGE, 0, 354, -EINVAL, 0);

     /* 2^64; see strtosz_simple for 2^64-1 */
     do_strtosz("18446744073709551616", -ERANGE, 0, 20);

     do_strtosz("20E", -ERANGE, 0, 3);

-    /* FIXME Fraction tail can cause ERANGE overflow */
+    /* Fraction tail can cause ERANGE overflow */
     do_strtosz("15.9999999999999999999999999999999999999999999999999999E",
-               0 /* FIXME -ERANGE */, 15ULL * EiB /* FIXME 0 */, 56);
+               -ERANGE, 0, 56);

     /* EINVAL has priority over ERANGE */
     do_strtosz_full("100000Pjunk", qemu_strtosz, -ERANGE, 0, 7, -EINVAL, 0);
diff --git a/util/cutils.c b/util/cutils.c
index bde2da59bdd..25373198adc 100644
--- a/util/cutils.c
+++ b/util/cutils.c
@@ -194,15 +194,19 @@ 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
- * - negative values, including -0
- * - floating point exponents, such as 1e3
+ * - hex with scaling suffix, such as 0x20M or 0x1p3 (both fail with
+ *   -EINVAL), while 0x1b is 27 (not 1 with byte scale)
+ * - 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")
+ * - negative values, including -0 (fail with -ERANGE)
+ * - floating point exponents, such as 1e3 (parsed as 1e with trailing
+ *   garbage "3") or 0x1p3 (rejected as hex with scaling suffix)
+ * - non-finite values, such as inf or NaN (fail with -EINVAL)
  *
  * 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
@@ -221,17 +225,17 @@ 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 = parse_uint(nptr, &endptr, 10, &val);
-    if (retval) {
+    if (retval == -ERANGE || !nptr) {
         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) {
@@ -242,27 +246,69 @@ 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;
+            if (signbit(fraction)) {
+                retval = -ERANGE;
+                goto out;
+            }
+        }
+
+        /* 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 {
+            /* We want non-zero valf for any non-zero fraction */
             valf = (uint64_t)(fraction * 0x1p64);
+            if (valf == 0 && fraction > 0.0) {
+                valf = 1;
+            }
         }
     }
+    if (retval) {
+        goto out;
+    }
     c = *endptr;
     mul = suffix_mul(c, unit);
     if (mul > 0) {
