Motor planning is the process of preparing the appropriate motor commands in order to achieve a goal. This process has been largely considered as occurring before movement onset and has been traditionally associated with reaction time. However, in a virtual line bisection task, we observed an overlap between movement planning and execution. In this task performed with a robotic manipulandum, we observed that the participants (N=30) made straight movements when the line was in front of them (near target) but made often curved movements towards a farther target that was located sideways in such a way that they crossed the line perpendicular to it. Unexpectedly, movements to the far targets had shorter reaction times than movements to the near target (mean difference: 32ms, SE: 5ms, max: 104ms). In addition, the curvature of the movement modulated reaction time. A larger increase in movement curvature from the near to the far target was associated with a larger reduction in reaction time. These highly curved movements started with a transport phase during which accuracy demands was not taken into account. We concluded that accuracy demand imposes a reaction time penalty if it is processed before movement onset. This penalty is reduced if the start of the movement can consist of a transport phase and if the movement plan can be refined in function of accuracy demands later in the movement, hence demonstrating an overlap between movement planning and execution.