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\[\frac{\left(\mathsf{lgamma} \left( 2 \cdot n + 1 \right) - \mathsf{lgamma} \left( n + 2 \right)\right) - \mathsf{lgamma} \left( n + 1 \right)}{\log 10}\]
\frac{\left(\mathsf{lgamma} \left( 2 \cdot n + 1 \right) - \mathsf{lgamma} \left( n + 2 \right)\right) - \mathsf{lgamma} \left( n + 1 \right)}{\log 10}
double f(double n) {
        double r3611820 = 2.0;
        double r3611821 = n;
        double r3611822 = r3611820 * r3611821;
        double r3611823 = 1.0;
        double r3611824 = r3611822 + r3611823;
        double r3611825 = lgamma(r3611824);
        double r3611826 = r3611821 + r3611820;
        double r3611827 = lgamma(r3611826);
        double r3611828 = r3611825 - r3611827;
        double r3611829 = r3611821 + r3611823;
        double r3611830 = lgamma(r3611829);
        double r3611831 = r3611828 - r3611830;
        double r3611832 = 10.0;
        double r3611833 = log(r3611832);
        double r3611834 = r3611831 / r3611833;
        return r3611834;
}

Reproduce

herbie shell --seed 1 
(FPCore (n)
  :name "(lgamma(2n+1)-lgamma(n+2)-lgamma(n+1))/log(10)"
  :precision binary64
  (/ (- (- (lgamma (+ (* 2 n) 1)) (lgamma (+ n 2))) (lgamma (+ n 1))) (log 10)))