Average Error: 47.7 → 11.4
Time: 12.0s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
↓
\[\begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -1.8422714442376505 \cdot 10^{-192}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5.21080278599253 \cdot 10^{-310}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.1323542808442225 \cdot 10^{-164}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(\log i - \log n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 0.0023543790910373224:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\
\mathbf{if}\;n \leq -1.8422714442376505 \cdot 10^{-192}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;n \leq -5.21080278599253 \cdot 10^{-310}:\\
\;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)}{\frac{i}{n}}\\

\mathbf{elif}\;n \leq 1.1323542808442225 \cdot 10^{-164}:\\
\;\;\;\;100 \cdot \frac{n \cdot \left(\log i - \log n\right)}{\frac{i}{n}}\\

\mathbf{elif}\;n \leq 0.0023543790910373224:\\
\;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

(FPCore (i n)
 :precision binary64
 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))
↓
(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -1.8422714442376505e-192)
     t_0
     (if (<= n -5.21080278599253e-310)
       (* 100.0 (/ (expm1 (* n (log (+ 1.0 (/ i n))))) (/ i n)))
       (if (<= n 1.1323542808442225e-164)
         (* 100.0 (/ (* n (- (log i) (log n))) (/ i n)))
         (if (<= n 0.0023543790910373224) (* 100.0 (/ i (/ i n))) t_0))))))
double code(double i, double n) {
	return 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}
↓
double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -1.8422714442376505e-192) {
		tmp = t_0;
	} else if (n <= -5.21080278599253e-310) {
		tmp = 100.0 * (expm1(n * log(1.0 + (i / n))) / (i / n));
	} else if (n <= 1.1323542808442225e-164) {
		tmp = 100.0 * ((n * (log(i) - log(n))) / (i / n));
	} else if (n <= 0.0023543790910373224) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.7
Target47.3
Herbie11.4
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \]

Derivation

  1. Split input into 4 regimes

  2. if n < -1.84227144423765048e-192 or 0.00235437909103732241 < n

    1. Initial program 50.6

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

    2. Taylor expanded in n around inf 46.9

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]

    3. Simplified11.7

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]

    4. Applied associate-/l*_binary649.0

      \[\leadsto 100 \cdot \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

    if -1.84227144423765048e-192 < n < -5.21080278599253e-310

    1. Initial program 16.0

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

    2. Applied pow-to-exp_binary6416.0

      \[\leadsto 100 \cdot \frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{\frac{i}{n}} \]

    3. Applied expm1-def_binary649.7

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)}}{\frac{i}{n}} \]

    if -5.21080278599253e-310 < n < 1.1323542808442225e-164

    1. Initial program 41.5

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

    2. Taylor expanded in n around 0 14.1

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(\log i - \log n\right) \cdot n}}{\frac{i}{n}} \]

    if 1.1323542808442225e-164 < n < 0.00235437909103732241

    1. Initial program 54.9

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

    2. Taylor expanded in i around 0 23.0

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.8422714442376505 \cdot 10^{-192}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq -5.21080278599253 \cdot 10^{-310}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.1323542808442225 \cdot 10^{-164}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(\log i - \log n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 0.0023543790910373224:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))