diff --git a/src/arb.h b/src/arb.h
index 2e7a23ab7d..103a67c361 100644
--- a/src/arb.h
+++ b/src/arb.h
@@ -29,6 +29,7 @@ extern "C" {
 #define arb_radref(x) (&(x)->rad)
 
 #define ARB_IS_LAGOM(x) (ARF_IS_LAGOM(arb_midref(x)) && MAG_IS_LAGOM(arb_radref(x)))
+#define ARB_IS_SPECIAL(x) (ARF_IS_SPECIAL(arb_midref(x)) || MAG_IS_SPECIAL(arb_radref(x)))
 
 #define ARB_RND ARF_RND_DOWN
 
diff --git a/src/arb/mul.c b/src/arb/mul.c
index e32538846c..b7e562d3a6 100644
--- a/src/arb/mul.c
+++ b/src/arb/mul.c
@@ -69,6 +69,80 @@ arb_mul_arf(arb_t z, const arb_t x, const arf_t y, slong prec)
     }
 }
 
+#if FLINT_HAVE_ARF_MUL_RND_SLOPPY
+FLINT_STATIC_NOINLINE
+void _arb_mul_special(arb_t z, const arb_t x, const arb_t y)
+{
+    /* FIXME: Treat special cases. */
+}
+
+void
+arb_mul(arb_t z, const arb_t x, const arb_t y, slong prec)
+{
+    if (FLINT_LIKELY(!ARB_IS_SPECIAL(x) && !ARB_IS_SPECIAL(y)))
+    {
+        /* Neither x or y are zero, exact, NaN or infinity. */
+        mag_t zr, xm, ym;
+        int is_lagom;
+
+        is_lagom = (ARB_IS_LAGOM(x) && ARB_IS_LAGOM(y) && ARB_IS_LAGOM(z));
+        mag_init(zr);
+
+        if (is_lagom)
+        {
+            mag_fast_init_set_arf(xm, arb_midref(x));
+            mag_fast_init_set_arf(ym, arb_midref(y));
+
+            mag_fast_mul(zr, xm, arb_radref(y));
+            mag_fast_addmul(zr, ym, arb_radref(x));
+            mag_fast_addmul(zr, arb_radref(x), arb_radref(y));
+        }
+        else
+        {
+            mag_init_set_arf(xm, arb_midref(x));
+            mag_init_set_arf(ym, arb_midref(y));
+
+            mag_mul(zr, xm, arb_radref(y));
+            mag_addmul(zr, ym, arb_radref(x));
+            mag_addmul(zr, arb_radref(x), arb_radref(y));
+        }
+
+        /* NOTE: Should we check if precision is less than what x and y has to
+           offer? */
+        /* NOTE: This is inexact by nature (unless tails are zero). */
+        arf_mul_rnd_sloppy(arb_midref(z), arb_midref(x), arb_midref(y));
+
+        /* FIXME: arf_mul_rnd_sloppy uses rounded inputs in order to only
+         * evaluate an n-by-n high multiplication. This needs to be reflected on
+         * the mag. */
+
+        if (is_lagom)
+        {
+            arf_mag_fast_add_ulp(zr, zr, arb_midref(z), prec);
+            *arb_radref(z) = *zr;
+        }
+        else
+        {
+            arf_mag_add_ulp(arb_radref(z), zr, arb_midref(z), prec);
+            mag_clear(xm);
+            mag_clear(ym);
+            mag_clear(zr);
+        }
+    }
+    else if (arb_is_exact(x) || arb_is_exact(y))
+    {
+        if (arb_is_exact(x))
+            arb_mul_arf(z, y, arb_midref(x), prec);
+        else
+            arb_mul_arf(z, x, arb_midref(y), prec);
+    }
+    else
+    {
+        /* x and y cannot be exact, and some value has to special. */
+        _arb_mul_special(z, x, y);
+    }
+}
+#else
 void
 arb_mul(arb_t z, const arb_t x, const arb_t y, slong prec)
 {
@@ -132,6 +206,7 @@ arb_mul(arb_t z, const arb_t x, const arb_t y, slong prec)
         mag_clear(zr);
     }
 }
+#endif
 
 void
 arb_mul_si(arb_t z, const arb_t x, slong y, slong prec)
diff --git a/src/arf.h b/src/arf.h
index cd88e366a8..7cd92ade67 100644
--- a/src/arf.h
+++ b/src/arf.h
@@ -776,6 +776,11 @@ int arf_mul_rnd_down(arf_ptr z, arf_srcptr x, arf_srcptr y, slong prec);
         ? arf_mul_rnd_down(z, x, y, prec)        \
         : arf_mul_rnd_any(z, x, y, prec, rnd))
 
+#if FLINT_HAVE_MPN_MULHIGH_NORMALISED
+# define FLINT_HAVE_ARF_MUL_RND_SLOPPY 1
+void arf_mul_rnd_sloppy(arf_ptr, arf_srcptr, arf_srcptr);
+#endif
+
 ARF_INLINE int
 arf_neg_mul(arf_t z, const arf_t x, const arf_t y, slong prec, arf_rnd_t rnd)
 {
diff --git a/src/arf/mul_rnd_sloppy.c b/src/arf/mul_rnd_sloppy.c
new file mode 100644
index 0000000000..2a8a439d27
--- /dev/null
+++ b/src/arf/mul_rnd_sloppy.c
@@ -0,0 +1,86 @@
+/*
+    Copyright (C) 2024 Albin Ahlbäck
+
+    This file is part of FLINT.
+
+    FLINT is free software: you can redistribute it and/or modify it under
+    the terms of the GNU Lesser General Public License (LGPL) as published
+    by the Free Software Foundation; either version 3 of the License, or
+    (at your option) any later version.  See <https://www.gnu.org/licenses/>.
+*/
+
+#include "mpn_extras.h"
+#include "arf.h"
+
+#if FLINT_HAVE_MPN_MULHIGH_NORMALISED
+/* NOTE: Assumes no special values */
+void
+arf_mul_rnd_sloppy(arf_ptr z, arf_srcptr x, arf_srcptr y)
+{
+    mp_size_t xn, yn, zn;
+    mp_srcptr xp, yp;
+    mp_ptr zp;
+    int sgnbit;
+    struct mp_limb_pair_t mulret;
+
+    xn = ARF_XSIZE(x);
+    yn = ARF_XSIZE(y);
+    zn = ARF_XSIZE(z);
+    sgnbit = (xn ^ yn) & 1;
+    xn >>= 1;
+    yn >>= 1;
+    zn >>= 1;
+
+    if (yn > xn)
+    {
+        FLINT_SWAP(arf_srcptr, x, y);
+        FLINT_SWAP(mp_size_t, xn, yn);
+    }
+
+    if (yn > 2)
+    {
+        xp = ARF_PTR_D(x);
+        yp = ARF_PTR_D(y);
+    }
+    else
+    {
+        yp = ARF_NOPTR_D(y);
+        if (xn <= 2)
+            xp = ARF_NOPTR_D(x);
+        else
+            xp = ARF_PTR_D(x);
+    }
+
+    xp += xn - yn; /* Make xp have the same length as yp */
+
+    if (zn < yn)
+    {
+        _arf_promote(z, yn);
+        zp = ARF_PTR_D(z);
+    }
+    else if (zn > 2 && yn <= 2)
+    {
+        _arf_demote(z);
+        zp = ARF_NOPTR_D(z);
+    }
+
+    if (!FLINT_HAVE_MULHIGH_NORMALISED_FUNC(yn) && (zp == xp || zp == yp))
+    {
+        /* These cases do not allow aliasing */
+        mp_ptr tmp = flint_malloc(sizeof(mp_limb_t) * yn);
+        mulret = flint_mpn_mulhigh_normalised(tmp, xp, yp, yn);
+        flint_mpn_copyi(zp, tmp, yn);
+        flint_free(tmp);
+    }
+    else
+    {
+        /* Here we actually allow aliasing. */
+        mulret = flint_mpn_mulhigh_normalised(zp, xp, yp, yn);
+    }
+
+    ARF_XSIZE(z) = ARF_MAKE_XSIZE(yn, sgnbit);
+    _fmpz_add2_fast(ARF_EXPREF(z), ARF_EXPREF(x), ARF_EXPREF(y), mulret.m2);
+}
+#else
+typedef int fileisempty;
+#endif
diff --git a/src/mag.h b/src/mag.h
index 0122941051..7fc041ae05 100644
--- a/src/mag.h
+++ b/src/mag.h
@@ -109,6 +109,7 @@ _fmpz_sub2_fast(fmpz_t z, const fmpz_t x, const fmpz_t y, slong c)
 /* Finite and with lagom big exponents. */
 #define MAG_IS_LAGOM(x) (MAG_EXP(x) >= MAG_MIN_LAGOM_EXP && \
                          MAG_EXP(x) <= MAG_MAX_LAGOM_EXP)
+#define MAG_IS_SPECIAL(x) (MAG_MAN(x) == 0)
 
 #define MAG_EXPREF(x) (&(x)->exp)
 #define MAG_EXP(x) ((x)->exp)
@@ -222,7 +223,7 @@ mag_one(mag_t x)
 MAG_INLINE int
 mag_is_special(const mag_t x)
 {
-    return MAG_MAN(x) == 0;
+    return MAG_IS_SPECIAL(x);
 }
 
 MAG_INLINE int
diff --git a/src/mpn_extras.h b/src/mpn_extras.h
index 0df2521597..5ca5f46c55 100644
--- a/src/mpn_extras.h
+++ b/src/mpn_extras.h
@@ -268,6 +268,9 @@ FLINT_DLL extern const flint_mpn_mulhigh_normalised_func_t flint_mpn_mulhigh_nor
 # define FLINT_HAVE_NATIVE_MPN_MULHIGH_BASECASE 1
 # define FLINT_HAVE_NATIVE_MPN_SQRHIGH_BASECASE 1
 
+# define FLINT_HAVE_MPN_MULHIGH 1
+# define FLINT_HAVE_MPN_MULHIGH_NORMALISED 1
+
 /* NOTE: This function only works for n >= 6 */
 mp_limb_t _flint_mpn_mulhigh_basecase(mp_ptr, mp_srcptr, mp_srcptr, mp_size_t);
 
@@ -318,6 +321,8 @@ struct mp_limb_pair_t flint_mpn_mulhigh_normalised(mp_ptr rp, mp_srcptr xp, mp_s
     {
         struct mp_limb_pair_t ret;
 
+        /* FIXME: Currently _flint_mpn_mulhigh allows aliasing above a certain
+         * size. */
         FLINT_ASSERT(rp != xp && rp != yp);
 
         ret.m1 = _flint_mpn_mulhigh(rp, xp, yp, n);