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\[\frac{n - e^{\mathsf{lgamma} \left( n + r \right) - \left(\mathsf{lgamma} \left( n \right) + \mathsf{lgamma} \left( r \right)\right)}}{1 - r}\]

\frac{n - e^{\mathsf{lgamma} \left( n + r \right) - \left(\mathsf{lgamma} \left( n \right) + \mathsf{lgamma} \left( r \right)\right)}}{1 - r}

double code(double n, double r) {
	return ((double) (((double) (n - ((double) exp(((double) (((double) lgamma(((double) (n + r)))) - ((double) (((double) lgamma(n)) + ((double) lgamma(r)))))))))) / ((double) (1.0 - r))));
}

Reproduce

herbie shell --seed 2020152 
(FPCore (n r)
  :name "(/ (- n (exp (- (lgamma (+ n r)) (+ (lgamma n) (lgamma r))))) (- 1 r))"
  :precision binary64
  (/ (- n (exp (- (lgamma (+ n r)) (+ (lgamma n) (lgamma r))))) (- 1.0 r)))