diff --git a/src/R2_alg.jl b/src/R2_alg.jl
index 91c54543..d23e4b0c 100644
--- a/src/R2_alg.jl
+++ b/src/R2_alg.jl
@@ -69,9 +69,10 @@ function R2(
   f::F,
   ∇f!::G,
   h::H,
-  options::ROSolverOptions,
-  x0::AbstractVector,
-) where {F <: Function, G <: Function, H}
+  options::ROSolverOptions{R},
+  x0::AbstractVector{R},
+  neg_tol::R = -eps(R)^(1/3) ,
+) where {F <: Function, G <: Function, H, R <: Real}
   start_time = time()
   elapsed_time = 0.0
   ϵ = options.ϵ
@@ -83,6 +84,7 @@ function R2(
   η2 = options.η2
   ν = options.ν
   γ = options.γ
+  neg_tol < 0 || error("neg_tol must be negative")
 
   if verbose == 0
     ptf = Inf
@@ -146,13 +148,14 @@ function R2(
     Complex_hist[k] += 1
     mks = mk(s)
     ξ = hk - mks + max(1, abs(hk)) * 10 * eps()
-    ξ > 0 || error("R2: prox-gradient step should produce a decrease but ξ = $(ξ)")
-
-    if sqrt(ξ) < ϵ
+    
+    if neg_tol <= ξ <= ϵ^2
       optimal = true
+      ξ = abs(ξ)
       continue
     end
 
+    ξ > 0 || error("R2: prox-gradient step should produce a decrease but ξ = $(ξ)")
     xkn .= xk .+ s
     fkn = f(xkn)
     hkn = h(xkn)