Average Error: 48.2 → 14.8
Time: 10.2s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq -6.720774155076935 \cdot 10^{-07}:\\ \;\;\;\;100 \cdot \frac{-1 + {\left(\frac{i}{n}\right)}^{n}}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.7352102327349535 \cdot 10^{-06}:\\ \;\;\;\;\sqrt[3]{\left(i \cdot n\right) \cdot 50} \cdot \left(\sqrt[3]{\left(i \cdot n\right) \cdot 50} \cdot \sqrt[3]{\left(i \cdot n\right) \cdot 50}\right) + 100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(-1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq -6.720774155076935 \cdot 10^{-07}:\\
\;\;\;\;100 \cdot \frac{-1 + {\left(\frac{i}{n}\right)}^{n}}{\frac{i}{n}}\\

\mathbf{elif}\;i \leq 2.7352102327349535 \cdot 10^{-06}:\\
\;\;\;\;\sqrt[3]{\left(i \cdot n\right) \cdot 50} \cdot \left(\sqrt[3]{\left(i \cdot n\right) \cdot 50} \cdot \sqrt[3]{\left(i \cdot n\right) \cdot 50}\right) + 100 \cdot n\\

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

\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 -6.720774155076935e-07)
   (* 100.0 (/ (+ -1.0 (pow (/ i n) n)) (/ i n)))
   (if (<= i 2.7352102327349535e-06)
     (+
      (*
       (cbrt (* (* i n) 50.0))
       (* (cbrt (* (* i n) 50.0)) (cbrt (* (* i n) 50.0))))
      (* 100.0 n))
     (/ (* 100.0 (+ -1.0 (pow (+ (/ i n) 1.0) n))) (/ i n)))))
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 <= -6.720774155076935e-07) {
		tmp = 100.0 * ((-1.0 + pow((i / n), n)) / (i / n));
	} else if (i <= 2.7352102327349535e-06) {
		tmp = (cbrt((i * n) * 50.0) * (cbrt((i * n) * 50.0) * cbrt((i * n) * 50.0))) + (100.0 * n);
	} else {
		tmp = (100.0 * (-1.0 + pow(((i / n) + 1.0), n))) / (i / n);
	}
	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

Original48.2
Target47.6
Herbie14.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -6.72077415507693457e-7

    1. Initial program 29.1

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

    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. Simplified19.4

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

    if -6.72077415507693457e-7 < i < 2.73521023273495348e-6

    1. Initial program 58.3

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

    2. Taylor expanded around 0 25.7

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

    3. Simplified25.7

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

    4. Taylor expanded around 0 8.6

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

    5. Simplified8.6

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

    6. Taylor expanded around 0 8.6

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

    7. Simplified8.6

      \[\leadsto \left(i \cdot n\right) \cdot 50 + \color{blue}{n \cdot 100}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt_binary648.6

      \[\leadsto \color{blue}{\left(\sqrt[3]{\left(i \cdot n\right) \cdot 50} \cdot \sqrt[3]{\left(i \cdot n\right) \cdot 50}\right) \cdot \sqrt[3]{\left(i \cdot n\right) \cdot 50}} + n \cdot 100\]

    if 2.73521023273495348e-6 < i

    1. Initial program 34.5

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

    3. Applied associate-*r/_binary6434.5

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

    4. Simplified34.5

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

  4. Final simplification14.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -6.720774155076935 \cdot 10^{-07}:\\ \;\;\;\;100 \cdot \frac{-1 + {\left(\frac{i}{n}\right)}^{n}}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.7352102327349535 \cdot 10^{-06}:\\ \;\;\;\;\sqrt[3]{\left(i \cdot n\right) \cdot 50} \cdot \left(\sqrt[3]{\left(i \cdot n\right) \cdot 50} \cdot \sqrt[3]{\left(i \cdot n\right) \cdot 50}\right) + 100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(-1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020224 
(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))))