Average Error: 47.0 → 15.0
Time: 11.3s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq -0.000889700083721587:\\ \;\;\;\;100 \cdot \frac{-1 + {\left(\frac{i}{n}\right)}^{n}}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 289668601567762.06:\\ \;\;\;\;n \cdot \left(\left(100 + \left(i \cdot i\right) \cdot 16.666666666666668\right) + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq -0.000889700083721587:\\
\;\;\;\;100 \cdot \frac{-1 + {\left(\frac{i}{n}\right)}^{n}}{\frac{i}{n}}\\

\mathbf{elif}\;i \leq 289668601567762.06:\\
\;\;\;\;n \cdot \left(\left(100 + \left(i \cdot i\right) \cdot 16.666666666666668\right) + i \cdot 50\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 -0.000889700083721587)
   (* 100.0 (/ (+ -1.0 (pow (/ i n) n)) (/ i n)))
   (if (<= i 289668601567762.06)
     (* n (+ (+ 100.0 (* (* i i) 16.666666666666668)) (* i 50.0)))
     0.0)))
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 <= -0.000889700083721587)) {
		tmp = ((double) (100.0 * (((double) (-1.0 + ((double) pow((i / n), n)))) / (i / n))));
	} else {
		double tmp_1;
		if ((i <= 289668601567762.06)) {
			tmp_1 = ((double) (n * ((double) (((double) (100.0 + ((double) (((double) (i * i)) * 16.666666666666668)))) + ((double) (i * 50.0))))));
		} else {
			tmp_1 = 0.0;
		}
		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

Original47.0
Target47.1
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 < -8.8970008372158695e-4

    1. Initial program 27.7

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

    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. Simplified19.0

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

    if -8.8970008372158695e-4 < i < 289668601567762.062

    1. Initial program 57.4

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

    2. Taylor expanded around 0 27.5

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

    3. Simplified27.5

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

    4. Taylor expanded around 0 10.2

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

    5. Simplified10.2

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

    6. Taylor expanded around 0 10.2

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

    7. Simplified10.2

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

    if 289668601567762.062 < i

    1. Initial program 31.6

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

    2. Taylor expanded around 0 30.9

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

  4. Final simplification15.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.000889700083721587:\\ \;\;\;\;100 \cdot \frac{-1 + {\left(\frac{i}{n}\right)}^{n}}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 289668601567762.06:\\ \;\;\;\;n \cdot \left(\left(100 + \left(i \cdot i\right) \cdot 16.666666666666668\right) + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(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))))