Average Error: 47.6 → 16.8
Time: 17.0s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq -3.827469972222611 \cdot 10^{-31}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} + -1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2003.3137174584608:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50 + n \cdot \left(100 + \left(i \cdot i\right) \cdot 16.666666666666668\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq -3.827469972222611 \cdot 10^{-31}:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} + -1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\

\mathbf{elif}\;i \leq 2003.3137174584608:\\
\;\;\;\;\left(i \cdot n\right) \cdot 50 + n \cdot \left(100 + \left(i \cdot i\right) \cdot 16.666666666666668\right)\\

\mathbf{else}:\\
\;\;\;\;0\\

\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 -3.827469972222611e-31)
   (*
    100.0
    (/
     (/
      (+ (pow (+ (/ i n) 1.0) (* n 2.0)) -1.0)
      (+ 1.0 (pow (+ (/ i n) 1.0) n)))
     (/ i n)))
   (if (<= i 2003.3137174584608)
     (+ (* (* i n) 50.0) (* n (+ 100.0 (* (* i i) 16.666666666666668))))
     0.0)))
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 <= -3.827469972222611e-31) {
		tmp = 100.0 * (((pow(((i / n) + 1.0), (n * 2.0)) + -1.0) / (1.0 + pow(((i / n) + 1.0), n))) / (i / n));
	} else if (i <= 2003.3137174584608) {
		tmp = ((i * n) * 50.0) + (n * (100.0 + ((i * i) * 16.666666666666668)));
	} else {
		tmp = 0.0;
	}
	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
Target47.5
Herbie16.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -3.82748e-31

    1. Initial program 29.6

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

    3. Applied flip--_binary64_252629.6

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

    4. Simplified29.6

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

    5. Simplified29.6

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

    if -3.82748e-31 < i < 2003.31

    1. Initial program 58.5

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

    2. Taylor expanded around 0 26.3

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

    3. Simplified26.3

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

    4. Taylor expanded around 0 8.8

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

    5. Simplified8.8

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

    if 2003.31 < i

    1. Initial program 30.7

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

    2. Taylor expanded around 0 30.2

      \[\leadsto \color{blue}{0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification16.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.827469972222611 \cdot 10^{-31}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} + -1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2003.3137174584608:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50 + n \cdot \left(100 + \left(i \cdot i\right) \cdot 16.666666666666668\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(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))))