Average Error: 47.7 → 15.0
Time: 13.3s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq -9.326225423734636 \cdot 10^{-11}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.001881761355889925:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i \cdot 1 + \left(n \cdot \log 1 + \left(i \cdot i\right) \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} - n \cdot \frac{1}{i}\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq -9.326225423734636 \cdot 10^{-11}:\\
\;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\

\mathbf{elif}\;i \leq 0.001881761355889925:\\
\;\;\;\;100 \cdot \left(n \cdot \frac{i \cdot 1 + \left(n \cdot \log 1 + \left(i \cdot i\right) \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}\right)\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} - n \cdot \frac{1}{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 -9.326225423734636e-11)
   (* 100.0 (/ (- (pow (/ i n) n) 1.0) (/ i n)))
   (if (<= i 0.001881761355889925)
     (*
      100.0
      (*
       n
       (/
        (+ (* i 1.0) (+ (* n (log 1.0)) (* (* i i) (- 0.5 (* (log 1.0) 0.5)))))
        i)))
     (* 100.0 (- (* n (/ (pow (+ (/ i n) 1.0) n) i)) (* n (/ 1.0 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 VAR;
	if ((i <= -9.326225423734636e-11)) {
		VAR = ((double) (100.0 * (((double) (((double) pow((i / n), n)) - 1.0)) / (i / n))));
	} else {
		double VAR_1;
		if ((i <= 0.001881761355889925)) {
			VAR_1 = ((double) (100.0 * ((double) (n * (((double) (((double) (i * 1.0)) + ((double) (((double) (n * ((double) log(1.0)))) + ((double) (((double) (i * i)) * ((double) (0.5 - ((double) (((double) log(1.0)) * 0.5)))))))))) / i)))));
		} else {
			VAR_1 = ((double) (100.0 * ((double) (((double) (n * (((double) pow(((double) ((i / n) + 1.0)), n)) / i))) - ((double) (n * (1.0 / i)))))));
		}
		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

Original47.7
Target47.5
Herbie15.0
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -9.3262254237346363e-11

    1. Initial program 28.4

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

    2. Taylor expanded around inf 64.0

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

    3. Simplified19.7

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

    if -9.3262254237346363e-11 < i < 0.00188176135588992498

    1. Initial program 58.1

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

    2. Taylor expanded around 0 26.2

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

    3. Simplified26.2

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

    5. Applied associate-/r/9.5

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

    6. Simplified9.5

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

    if 0.00188176135588992498 < i

    1. Initial program 33.0

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

    3. Applied div-sub33.0

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

    4. Simplified35.8

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

    5. Simplified33.0

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

  4. Final simplification15.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -9.326225423734636 \cdot 10^{-11}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.001881761355889925:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i \cdot 1 + \left(n \cdot \log 1 + \left(i \cdot i\right) \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} - n \cdot \frac{1}{i}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(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))))