The functional inequality \(|f(x+ y)- f(x) f(y) |\leq \varepsilon\) where \(f: S\to \mathbb{C}\), \(S\), semigroup, \(\mathbb{C}\) the complex numbers known as stability of the exponential function has been studied by many authors. In this paper the authors study the modified form \[ \biggl|{{f(x+ y)} \over {f(x) f(y)}} -1\biggr|\leq \varepsilon \tag{1} \] where \(f: S\to \mathbb{C}\setminus\{0\}\) can prove among others the following theorem: Let \((S, +)\) be a cancellative Abelian semigroup, and let \(\varepsilon\in [0, 1)\) be a given number. Assume that \(f: S\to \mathbb{C} \setminus \{0\}\) satisfies (1). Then there exists a unique function \(g: S\to \mathbb{C} \setminus \{0\}\) such that \(g(x+ y)= g(x) g(y)\), \(x,y\in S\), \[ \begin{aligned} \biggl|{{f(x)} \over {g(x)}} -1 \biggr|&\leq \sqrt {1+ {1\over {(1- \varepsilon)^2}} -2 \sqrt {{{1+ \varepsilon} \over {1- \varepsilon}}}}, \qquad x\in S,\\ \text{and} \biggl|{{g(x)} \over {f(x)}} -1\biggr|&\leq \sqrt {1+ {1\over {(1- \varepsilon)^2}} -2 \sqrt {{{1+ \varepsilon} \over {1- \varepsilon}}}}, \qquad x\in S. \end{aligned} \] {}.