Average Error: 47.3 → 16.6
Time: 13.3s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -7.03094120304917353 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{\mathsf{fma}\left(1, {\left(1 + \frac{i}{n}\right)}^{n} + 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) \cdot i}{n}}\\ \mathbf{elif}\;i \le 0.1391033594988507:\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(-0.5 \cdot i, \log 1, \mathsf{fma}\left(i, 0.5, \mathsf{fma}\left(\frac{\log 1}{i}, n, 1\right)\right)\right) \cdot n\right)\\ \mathbf{elif}\;i \le 8.5994276340506012 \cdot 10^{168}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 3.601044061012584 \cdot 10^{286}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -7.03094120304917353 \cdot 10^{-9}:\\
\;\;\;\;100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{\mathsf{fma}\left(1, {\left(1 + \frac{i}{n}\right)}^{n} + 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) \cdot i}{n}}\\

\mathbf{elif}\;i \le 0.1391033594988507:\\
\;\;\;\;100 \cdot \left(\mathsf{fma}\left(-0.5 \cdot i, \log 1, \mathsf{fma}\left(i, 0.5, \mathsf{fma}\left(\frac{\log 1}{i}, n, 1\right)\right)\right) \cdot n\right)\\

\mathbf{elif}\;i \le 8.5994276340506012 \cdot 10^{168}:\\
\;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 3.601044061012584 \cdot 10^{286}:\\
\;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\

\mathbf{else}:\\
\;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\

\end{array}
double code(double i, double n) {
	return ((double) (100.0 * ((double) (((double) (((double) pow(((double) (1.0 + ((double) (i / n)))), n)) - 1.0)) / ((double) (i / n))))));
}
↓
double code(double i, double n) {
	double VAR;
	if ((i <= -7.0309412030491735e-09)) {
		VAR = ((double) (100.0 * ((double) (((double) (((double) pow(((double) pow(((double) (1.0 + ((double) (i / n)))), n)), 3.0)) - ((double) pow(1.0, 3.0)))) / ((double) (((double) (((double) fma(1.0, ((double) (((double) pow(((double) (1.0 + ((double) (i / n)))), n)) + 1.0)), ((double) pow(((double) (1.0 + ((double) (i / n)))), ((double) (2.0 * n)))))) * i)) / n))))));
	} else {
		double VAR_1;
		if ((i <= 0.13910335949885066)) {
			VAR_1 = ((double) (100.0 * ((double) (((double) fma(((double) -(((double) (0.5 * i)))), ((double) log(1.0)), ((double) fma(i, 0.5, ((double) fma(((double) (((double) log(1.0)) / i)), n, 1.0)))))) * n))));
		} else {
			double VAR_2;
			if ((i <= 8.599427634050601e+168)) {
				VAR_2 = ((double) (((double) (100.0 * ((double) (((double) pow(((double) (1.0 + ((double) (i / n)))), n)) - 1.0)))) / ((double) (i / n))));
			} else {
				double VAR_3;
				if ((i <= 3.601044061012584e+286)) {
					VAR_3 = ((double) (100.0 * ((double) (((double) (((double) fma(1.0, i, ((double) fma(((double) log(1.0)), n, 1.0)))) - 1.0)) / ((double) (i / n))))));
				} else {
					VAR_3 = ((double) (((double) (100.0 * ((double) (((double) (((double) pow(((double) (1.0 + ((double) (i / n)))), n)) - 1.0)) / i)))) * n));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.3
Target47.3
Herbie16.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 5 regimes

  2. if i < -7.0309412030491735e-09

    1. Initial program 28.0

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
    2. Using strategy rm

    3. Applied flip3--28.0

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}{\frac{i}{n}}\]

    4. Applied associate-/l/28.0

      \[\leadsto 100 \cdot \color{blue}{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i}{n} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)\right)}}\]

    5. Simplified28.0

      \[\leadsto 100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\color{blue}{\frac{\mathsf{fma}\left(1, {\left(1 + \frac{i}{n}\right)}^{n} + 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) \cdot i}{n}}}\]

    if -7.0309412030491735e-09 < i < 0.13910335949885066

    1. Initial program 58.2

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

    2. Taylor expanded around 0 25.5

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{i}{n}}\]

    3. Simplified25.5

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}}{\frac{i}{n}}\]
    4. Using strategy rm

    5. Applied associate-/r/8.8

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\right)}\]

    6. Taylor expanded around 0 8.8

      \[\leadsto 100 \cdot \left(\color{blue}{\left(\left(1 + \left(0.5 \cdot i + \frac{\log 1 \cdot n}{i}\right)\right) - 0.5 \cdot \left(i \cdot \log 1\right)\right)} \cdot n\right)\]

    7. Simplified8.8

      \[\leadsto 100 \cdot \left(\color{blue}{\mathsf{fma}\left(-0.5 \cdot i, \log 1, \mathsf{fma}\left(i, 0.5, \mathsf{fma}\left(\frac{\log 1}{i}, n, 1\right)\right)\right)} \cdot n\right)\]

    if 0.13910335949885066 < i < 8.599427634050601e+168

    1. Initial program 30.8

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
    2. Using strategy rm

    3. Applied associate-*r/30.8

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

    if 8.599427634050601e+168 < i < 3.601044061012584e+286

    1. Initial program 32.2

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

    2. Taylor expanded around 0 35.0

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

    3. Simplified35.0

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

    if 3.601044061012584e+286 < i

    1. Initial program 27.7

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
    2. Using strategy rm

    3. Applied associate-/r/27.8

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

    4. Applied associate-*r*27.8

      \[\leadsto \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -7.03094120304917353 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{\mathsf{fma}\left(1, {\left(1 + \frac{i}{n}\right)}^{n} + 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) \cdot i}{n}}\\ \mathbf{elif}\;i \le 0.1391033594988507:\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(-0.5 \cdot i, \log 1, \mathsf{fma}\left(i, 0.5, \mathsf{fma}\left(\frac{\log 1}{i}, n, 1\right)\right)\right) \cdot n\right)\\ \mathbf{elif}\;i \le 8.5994276340506012 \cdot 10^{168}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 3.601044061012584 \cdot 10^{286}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :herbie-target
  (* 100 (/ (- (exp (* n (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) 1) (/ i n)))

  (* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))))