Average Error: 42.8 → 21.0
Time: 13.9s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.14141811154607586:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.6939372834177335 \cdot 10^{-290}:\\ \;\;\;\;\frac{100 \cdot \left(\left(\left(1 \cdot i + 0.5 \cdot {i}^{2}\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)\right) \cdot n\right)}{i}\\ \mathbf{elif}\;i \le 8.5870164069987757 \cdot 10^{-230}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;i \le 6.0935215346955289 \cdot 10^{-4}:\\ \;\;\;\;\frac{100 \cdot \left(\left(\left(1 \cdot i + 0.5 \cdot {i}^{2}\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)\right) \cdot n\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.14141811154607586:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 2.6939372834177335 \cdot 10^{-290}:\\
\;\;\;\;\frac{100 \cdot \left(\left(\left(1 \cdot i + 0.5 \cdot {i}^{2}\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)\right) \cdot n\right)}{i}\\

\mathbf{elif}\;i \le 8.5870164069987757 \cdot 10^{-230}:\\
\;\;\;\;\left(100 \cdot \frac{\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)}{i}\right) \cdot n\\

\mathbf{elif}\;i \le 6.0935215346955289 \cdot 10^{-4}:\\
\;\;\;\;\frac{100 \cdot \left(\left(\left(1 \cdot i + 0.5 \cdot {i}^{2}\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)\right) \cdot n\right)}{i}\\

\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 (100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n)));
}
↓
double code(double i, double n) {
	double temp;
	if ((i <= -0.14141811154607586)) {
		temp = (100.0 * (((pow((1.0 + (i / n)), (2.0 * n)) + -(1.0 * 1.0)) / (pow((1.0 + (i / n)), n) + 1.0)) / (i / n)));
	} else {
		double temp_1;
		if ((i <= 2.6939372834177335e-290)) {
			temp_1 = ((100.0 * ((((1.0 * i) + (0.5 * pow(i, 2.0))) + (log(1.0) * (n - (0.5 * pow(i, 2.0))))) * n)) / i);
		} else {
			double temp_2;
			if ((i <= 8.587016406998776e-230)) {
				temp_2 = ((100.0 * ((((1.0 * i) + ((0.5 * pow(i, 2.0)) + (log(1.0) * n))) - (0.5 * (pow(i, 2.0) * log(1.0)))) / i)) * n);
			} else {
				double temp_3;
				if ((i <= 0.0006093521534695529)) {
					temp_3 = ((100.0 * ((((1.0 * i) + (0.5 * pow(i, 2.0))) + (log(1.0) * (n - (0.5 * pow(i, 2.0))))) * n)) / i);
				} else {
					temp_3 = ((100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / i)) * n);
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.8
Target42.5
Herbie21.0
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -0.14141811154607586

    1. Initial program 28.6

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

    3. Applied flip--28.6

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

    4. Simplified28.6

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

    if -0.14141811154607586 < i < 2.6939372834177335e-290 or 8.587016406998776e-230 < i < 0.0006093521534695529

    1. Initial program 50.8

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

    2. Taylor expanded around 0 33.7

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

    4. Applied associate-/r/17.2

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

    5. Applied associate-*r*17.2

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

    7. Applied associate-*r/17.3

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

    8. Applied associate-*l/15.7

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

    10. Applied associate-*l*15.7

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

    11. Simplified15.7

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

    if 2.6939372834177335e-290 < i < 8.587016406998776e-230

    1. Initial program 47.7

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

    2. Taylor expanded around 0 40.8

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

    4. Applied associate-/r/16.4

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

    5. Applied associate-*r*16.4

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

    if 0.0006093521534695529 < i

    1. Initial program 31.9

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

    3. Applied associate-/r/31.8

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

    4. Applied associate-*r*31.9

      \[\leadsto \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification21.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.14141811154607586:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.6939372834177335 \cdot 10^{-290}:\\ \;\;\;\;\frac{100 \cdot \left(\left(\left(1 \cdot i + 0.5 \cdot {i}^{2}\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)\right) \cdot n\right)}{i}\\ \mathbf{elif}\;i \le 8.5870164069987757 \cdot 10^{-230}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;i \le 6.0935215346955289 \cdot 10^{-4}:\\ \;\;\;\;\frac{100 \cdot \left(\left(\left(1 \cdot i + 0.5 \cdot {i}^{2}\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)\right) \cdot n\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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))))