Average Error: 48.0 → 17.0
Time: 10.2s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.0077728093341000708:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 2.51907991628111937 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \log 1, {i}^{2}, \mathsf{fma}\left(i, 1, \log 1 \cdot n\right)\right)}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 3.50959367729407485 \cdot 10^{238}:\\ \;\;\;\;\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;i \le 2.37383627367145743 \cdot 10^{285}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{\frac{i}{n}}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.0077728093341000708:\\
\;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;i \le 2.51907991628111937 \cdot 10^{-6}:\\
\;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \log 1, {i}^{2}, \mathsf{fma}\left(i, 1, \log 1 \cdot n\right)\right)}{i} \cdot n\right)\\

\mathbf{elif}\;i \le 3.50959367729407485 \cdot 10^{238}:\\
\;\;\;\;\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100\right) \cdot \frac{n}{i}\\

\mathbf{elif}\;i \le 2.37383627367145743 \cdot 10^{285}:\\
\;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\

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

\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 VAR;
	if ((i <= -0.007772809334100071)) {
		VAR = (100.0 * ((pow((1.0 + (i / n)), n) / (i / n)) - (1.0 / (i / n))));
	} else {
		double VAR_1;
		if ((i <= 2.5190799162811194e-06)) {
			VAR_1 = (100.0 * ((fma((0.5 - (0.5 * log(1.0))), pow(i, 2.0), fma(i, 1.0, (log(1.0) * n))) / i) * n));
		} else {
			double VAR_2;
			if ((i <= 3.509593677294075e+238)) {
				VAR_2 = (((pow((1.0 + (i / n)), n) - 1.0) * 100.0) * (n / i));
			} else {
				double VAR_3;
				if ((i <= 2.3738362736714574e+285)) {
					VAR_3 = (100.0 * ((fma(1.0, i, fma(log(1.0), n, 1.0)) - 1.0) / (i / n)));
				} else {
					VAR_3 = (100.0 / ((i / n) / (pow((1.0 + (i / n)), n) - 1.0)));
				}
				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

Original48.0
Target47.1
Herbie17.0
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 5 regimes

  2. if i < -0.007772809334100071

    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 div-sub28.6

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

    if -0.007772809334100071 < i < 2.5190799162811194e-06

    1. Initial program 58.0

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

    2. Taylor expanded around 0 26.0

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

    3. Simplified26.0

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

    5. Applied associate-/r/9.2

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

    6. Simplified9.2

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

    if 2.5190799162811194e-06 < i < 3.509593677294075e+238

    1. Initial program 34.1

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

    3. Applied clear-num34.1

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

    4. Applied associate-/r/34.1

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

    5. Applied associate-*r*34.1

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

    6. Simplified34.1

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

    if 3.509593677294075e+238 < i < 2.3738362736714574e+285

    1. Initial program 32.8

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

    2. Taylor expanded around 0 31.6

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

    3. Simplified31.6

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

    if 2.3738362736714574e+285 < i

    1. Initial program 34.6

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

    3. Applied clear-num34.6

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

    4. Applied un-div-inv34.6

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.0077728093341000708:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 2.51907991628111937 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \log 1, {i}^{2}, \mathsf{fma}\left(i, 1, \log 1 \cdot n\right)\right)}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 3.50959367729407485 \cdot 10^{238}:\\ \;\;\;\;\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;i \le 2.37383627367145743 \cdot 10^{285}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{\frac{i}{n}}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +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))))