Average Error: 47.9 → 11.7
Time: 13.0s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq -3.311639244018977 \cdot 10^{-06}:\\ \;\;\;\;100 \cdot \frac{e^{i} + -1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 8.431981695178327 \cdot 10^{-12}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;i \leq 1.4026681787903786 \cdot 10^{+231}:\\ \;\;\;\;100 \cdot \frac{\left(0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log n}^{2}\right) + \left(n \cdot \log i + \left(\frac{n \cdot n}{i} + 0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log i}^{2}\right)\right)\right)\right) - \log n \cdot \left(n + n \cdot \left(n \cdot \log i\right)\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{-1 + {\left(\frac{i}{n} + 1\right)}^{n}}{i}\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq -3.311639244018977 \cdot 10^{-06}:\\
\;\;\;\;100 \cdot \frac{e^{i} + -1}{\frac{i}{n}}\\

\mathbf{elif}\;i \leq 8.431981695178327 \cdot 10^{-12}:\\
\;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\

\mathbf{elif}\;i \leq 1.4026681787903786 \cdot 10^{+231}:\\
\;\;\;\;100 \cdot \frac{\left(0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log n}^{2}\right) + \left(n \cdot \log i + \left(\frac{n \cdot n}{i} + 0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log i}^{2}\right)\right)\right)\right) - \log n \cdot \left(n + n \cdot \left(n \cdot \log i\right)\right)}{\frac{i}{n}}\\

\mathbf{else}:\\
\;\;\;\;n \cdot \left(100 \cdot \frac{-1 + {\left(\frac{i}{n} + 1\right)}^{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 -3.311639244018977e-06)
   (* 100.0 (/ (+ (exp i) -1.0) (/ i n)))
   (if (<= i 8.431981695178327e-12)
     (* n (+ 100.0 (* i 50.0)))
     (if (<= i 1.4026681787903786e+231)
       (*
        100.0
        (/
         (-
          (+
           (* 0.5 (* (* n n) (pow (log n) 2.0)))
           (+
            (* n (log i))
            (+ (/ (* n n) i) (* 0.5 (* (* n n) (pow (log i) 2.0))))))
          (* (log n) (+ n (* n (* n (log i))))))
         (/ i n)))
       (* n (* 100.0 (/ (+ -1.0 (pow (+ (/ i n) 1.0) n)) i)))))))
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 tmp;
	if (i <= -3.311639244018977e-06) {
		tmp = 100.0 * ((exp(i) + -1.0) / (i / n));
	} else if (i <= 8.431981695178327e-12) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (i <= 1.4026681787903786e+231) {
		tmp = 100.0 * ((((0.5 * ((n * n) * pow(log(n), 2.0))) + ((n * log(i)) + (((n * n) / i) + (0.5 * ((n * n) * pow(log(i), 2.0)))))) - (log(n) * (n + (n * (n * log(i)))))) / (i / n));
	} else {
		tmp = n * (100.0 * ((-1.0 + pow(((i / n) + 1.0), n)) / i));
	}
	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

Original47.9
Target47.7
Herbie11.7
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -3.3116392440189772e-6

    1. Initial program 29.0

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

    2. Taylor expanded around inf 11.6

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

    3. Simplified11.6

      \[\leadsto 100 \cdot \frac{\color{blue}{e^{i} + -1}}{\frac{i}{n}}\]

    if -3.3116392440189772e-6 < i < 8.43198169517832741e-12

    1. Initial program 58.4

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

    2. Taylor expanded around 0 9.0

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

    3. Simplified9.0

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

    4. Taylor expanded around inf 9.1

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

    if 8.43198169517832741e-12 < i < 1.4026681787903786e231

    1. Initial program 32.6

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

    2. Taylor expanded around 0 20.0

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

    3. Simplified20.0

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

    if 1.4026681787903786e231 < i

    1. Initial program 32.5

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

    3. Applied associate-/r/_binary6432.5

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

    4. Applied associate-*r*_binary6432.5

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

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.311639244018977 \cdot 10^{-06}:\\ \;\;\;\;100 \cdot \frac{e^{i} + -1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 8.431981695178327 \cdot 10^{-12}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;i \leq 1.4026681787903786 \cdot 10^{+231}:\\ \;\;\;\;100 \cdot \frac{\left(0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log n}^{2}\right) + \left(n \cdot \log i + \left(\frac{n \cdot n}{i} + 0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log i}^{2}\right)\right)\right)\right) - \log n \cdot \left(n + n \cdot \left(n \cdot \log i\right)\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{-1 + {\left(\frac{i}{n} + 1\right)}^{n}}{i}\right)\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(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))))