Average Error: 47.6 → 14.5
Time: 11.3s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq -4.486060090746444 \cdot 10^{+21}:\\ \;\;\;\;100 \cdot \frac{-1 + {\left(\frac{i}{n}\right)}^{n}}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -3.558185175373979 \cdot 10^{-154}:\\ \;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \left(n \cdot \left({i}^{3} \cdot 0.16666666666666666 + \left(i + \left(i \cdot i\right) \cdot 0.5\right)\right)\right)\right)\\ \mathbf{elif}\;i \leq 250.08617759702352:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50 + 100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{n}{i}\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq -4.486060090746444 \cdot 10^{+21}:\\
\;\;\;\;100 \cdot \frac{-1 + {\left(\frac{i}{n}\right)}^{n}}{\frac{i}{n}}\\

\mathbf{elif}\;i \leq -3.558185175373979 \cdot 10^{-154}:\\
\;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \left(n \cdot \left({i}^{3} \cdot 0.16666666666666666 + \left(i + \left(i \cdot i\right) \cdot 0.5\right)\right)\right)\right)\\

\mathbf{elif}\;i \leq 250.08617759702352:\\
\;\;\;\;\left(i \cdot n\right) \cdot 50 + 100 \cdot n\\

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

\end{array}
(FPCore (i n)
 :precision binary64
 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))
↓
(FPCore (i n)
 :precision binary64
 (if (<= i -4.486060090746444e+21)
   (* 100.0 (/ (+ -1.0 (pow (/ i n) n)) (/ i n)))
   (if (<= i -3.558185175373979e-154)
     (*
      100.0
      (*
       (/ 1.0 i)
       (* n (+ (* (pow i 3.0) 0.16666666666666666) (+ i (* (* i i) 0.5))))))
     (if (<= i 250.08617759702352)
       (+ (* (* i n) 50.0) (* 100.0 n))
       (* 100.0 (- (/ (pow (+ (/ i n) 1.0) n) (/ i n)) (/ n i)))))))
double code(double i, double n) {
	return ((double) (100.0 * (((double) (((double) pow(((double) (1.0 + (i / n))), n)) - 1.0)) / (i / n))));
}
↓
double code(double i, double n) {
	double tmp;
	if ((i <= -4.486060090746444e+21)) {
		tmp = ((double) (100.0 * (((double) (-1.0 + ((double) pow((i / n), n)))) / (i / n))));
	} else {
		double tmp_1;
		if ((i <= -3.558185175373979e-154)) {
			tmp_1 = ((double) (100.0 * ((double) ((1.0 / i) * ((double) (n * ((double) (((double) (((double) pow(i, 3.0)) * 0.16666666666666666)) + ((double) (i + ((double) (((double) (i * i)) * 0.5))))))))))));
		} else {
			double tmp_2;
			if ((i <= 250.08617759702352)) {
				tmp_2 = ((double) (((double) (((double) (i * n)) * 50.0)) + ((double) (100.0 * n))));
			} else {
				tmp_2 = ((double) (100.0 * ((double) ((((double) pow(((double) ((i / n) + 1.0)), n)) / (i / n)) - (n / i)))));
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	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

Original47.6
Target47.4
Herbie14.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -4.4860600907464442e21

    1. Initial program 26.3

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

    2. Taylor expanded around inf 64.0

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

    3. Simplified16.1

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

    if -4.4860600907464442e21 < i < -3.55818517537397883e-154

    1. Initial program 53.5

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

    2. Taylor expanded around 0 27.4

      \[\leadsto 100 \cdot \frac{\color{blue}{0.16666666666666666 \cdot {i}^{3} + \left(0.5 \cdot {i}^{2} + i\right)}}{\frac{i}{n}}\]

    3. Simplified27.4

      \[\leadsto 100 \cdot \frac{\color{blue}{{i}^{3} \cdot 0.16666666666666666 + \left(i + \left(i \cdot i\right) \cdot 0.5\right)}}{\frac{i}{n}}\]
    4. Using strategy rm

    5. Applied div-inv_binary6427.5

      \[\leadsto 100 \cdot \frac{{i}^{3} \cdot 0.16666666666666666 + \left(i + \left(i \cdot i\right) \cdot 0.5\right)}{\color{blue}{i \cdot \frac{1}{n}}}\]

    6. Applied *-un-lft-identity_binary6427.5

      \[\leadsto 100 \cdot \frac{\color{blue}{1 \cdot \left({i}^{3} \cdot 0.16666666666666666 + \left(i + \left(i \cdot i\right) \cdot 0.5\right)\right)}}{i \cdot \frac{1}{n}}\]

    7. Applied times-frac_binary6416.8

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{i} \cdot \frac{{i}^{3} \cdot 0.16666666666666666 + \left(i + \left(i \cdot i\right) \cdot 0.5\right)}{\frac{1}{n}}\right)}\]

    8. Simplified16.7

      \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \color{blue}{\left(\left({i}^{3} \cdot 0.16666666666666666 + \left(i + \left(i \cdot i\right) \cdot 0.5\right)\right) \cdot n\right)}\right)\]

    if -3.55818517537397883e-154 < i < 250.086177597023521

    1. Initial program 59.0

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

    2. Taylor expanded around 0 28.2

      \[\leadsto 100 \cdot \frac{\color{blue}{0.16666666666666666 \cdot {i}^{3} + \left(0.5 \cdot {i}^{2} + i\right)}}{\frac{i}{n}}\]

    3. Simplified28.2

      \[\leadsto 100 \cdot \frac{\color{blue}{{i}^{3} \cdot 0.16666666666666666 + \left(i + \left(i \cdot i\right) \cdot 0.5\right)}}{\frac{i}{n}}\]

    4. Taylor expanded around 0 8.2

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + \left(100 \cdot n + 16.666666666666668 \cdot \left({i}^{2} \cdot n\right)\right)}\]

    5. Simplified8.2

      \[\leadsto \color{blue}{\left(i \cdot n\right) \cdot 50 + n \cdot \left(100 + \left(i \cdot i\right) \cdot 16.666666666666668\right)}\]

    6. Taylor expanded around 0 8.3

      \[\leadsto \left(i \cdot n\right) \cdot 50 + \color{blue}{100 \cdot n}\]

    7. Simplified8.3

      \[\leadsto \left(i \cdot n\right) \cdot 50 + \color{blue}{n \cdot 100}\]

    if 250.086177597023521 < i

    1. Initial program 31.1

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

    3. Applied div-sub_binary6431.1

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

    4. Simplified33.6

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

  4. Final simplification14.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -4.486060090746444 \cdot 10^{+21}:\\ \;\;\;\;100 \cdot \frac{-1 + {\left(\frac{i}{n}\right)}^{n}}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -3.558185175373979 \cdot 10^{-154}:\\ \;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \left(n \cdot \left({i}^{3} \cdot 0.16666666666666666 + \left(i + \left(i \cdot i\right) \cdot 0.5\right)\right)\right)\right)\\ \mathbf{elif}\;i \leq 250.08617759702352:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50 + 100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{n}{i}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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))))