<p>In this paper we describe the solutions of the functional equation <Equation ID="Equ29"> <EquationSource Format="TEX">\(\begin{aligned} F\Big (\frac{x+y}{2}\Big )+f_1(x)+f_2(y)= G \big (g_1(x)+g_2(y)) \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>F</mi> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mn>2</mn> </mfrac> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mo>+</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>G</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <msub> <mi>g</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msub> <mi>g</mi> <mn>2</mn> </msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>defined on an open subinterval of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( {\mathbb {R}} \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation>. Improving previous results we assume differentiability on each involved function, eliminate a former condition on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( g'_1 \)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>g</mi> <mn>1</mn> <mo>′</mo> </msubsup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\( g'_2 \, \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>g</mi> <mn>2</mn> <mo>′</mo> </msubsup> <mspace width="0.166667em" /> </mrow> </math></EquationSource> </InlineEquation>, moreover we determine a brand new family of solutions. We also present a particular member of this class as an example. In order to achieve this, we strengthen known results about certain auxiliary functional equations as well.</p>