Average Error: 43.0 → 18.5
Time: 25.0s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.4112290909641321823286830294819083064795:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 116.7795525870314179428532952442765235901:\\ \;\;\;\;\frac{100}{\frac{i}{\left(i \cdot \left(1 + 0.5 \cdot i\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)\right) \cdot n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{i}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.4112290909641321823286830294819083064795:\\
\;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 116.7795525870314179428532952442765235901:\\
\;\;\;\;\frac{100}{\frac{i}{\left(i \cdot \left(1 + 0.5 \cdot i\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)\right) \cdot n}}\\

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

\end{array}
double f(double i, double n) {
        double r126359 = 100.0;
        double r126360 = 1.0;
        double r126361 = i;
        double r126362 = n;
        double r126363 = r126361 / r126362;
        double r126364 = r126360 + r126363;
        double r126365 = pow(r126364, r126362);
        double r126366 = r126365 - r126360;
        double r126367 = r126366 / r126363;
        double r126368 = r126359 * r126367;
        return r126368;
}
↓
double f(double i, double n) {
        double r126369 = i;
        double r126370 = -0.4112290909641322;
        bool r126371 = r126369 <= r126370;
        double r126372 = 100.0;
        double r126373 = n;
        double r126374 = r126369 / r126373;
        double r126375 = pow(r126374, r126373);
        double r126376 = 1.0;
        double r126377 = r126375 - r126376;
        double r126378 = r126377 / r126374;
        double r126379 = r126372 * r126378;
        double r126380 = 116.77955258703142;
        bool r126381 = r126369 <= r126380;
        double r126382 = 0.5;
        double r126383 = r126382 * r126369;
        double r126384 = r126376 + r126383;
        double r126385 = r126369 * r126384;
        double r126386 = log(r126376);
        double r126387 = 2.0;
        double r126388 = pow(r126369, r126387);
        double r126389 = r126382 * r126388;
        double r126390 = r126373 - r126389;
        double r126391 = r126386 * r126390;
        double r126392 = r126385 + r126391;
        double r126393 = r126392 * r126373;
        double r126394 = r126369 / r126393;
        double r126395 = r126372 / r126394;
        double r126396 = r126373 * r126377;
        double r126397 = r126396 / r126369;
        double r126398 = r126372 * r126397;
        double r126399 = r126381 ? r126395 : r126398;
        double r126400 = r126371 ? r126379 : r126399;
        return r126400;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.0
Target42.6
Herbie18.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -0.4112290909641322

    1. Initial program 27.5

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

    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. Simplified17.6

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

    if -0.4112290909641322 < i < 116.77955258703142

    1. Initial program 50.8

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

    3. Applied associate-*r/50.8

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

    5. Applied sub-neg50.8

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

    6. Applied distribute-lft-in50.8

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

    8. Applied distribute-lft-out50.8

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

    9. Applied associate-/l*50.8

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

    10. Simplified50.5

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

    11. Taylor expanded around 0 16.3

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

    12. Simplified16.3

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

    if 116.77955258703142 < i

    1. Initial program 31.3

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

    2. Taylor expanded around inf 29.2

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

    3. Simplified31.3

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

  4. Final simplification18.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.4112290909641321823286830294819083064795:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 116.7795525870314179428532952442765235901:\\ \;\;\;\;\frac{100}{\frac{i}{\left(i \cdot \left(1 + 0.5 \cdot i\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)\right) \cdot n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{i}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :herbie-target
  (* 100 (/ (- (exp (* n (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) 1) (/ i n)))

  (* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))))