Average Error: 42.5 → 21.9
Time: 50.9s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -1.3875157595860486 \cdot 10^{+126}:\\ \;\;\;\;\left(\left(n \cdot \left(\left(\sqrt[3]{\frac{1}{6}} \cdot i\right) \cdot \left(\sqrt[3]{\frac{1}{6}} \cdot i\right)\right)\right) \cdot \sqrt[3]{\frac{1}{6}} + \left(\left(\frac{1}{2} \cdot i\right) \cdot n + n\right)\right) \cdot 100\\ \mathbf{elif}\;n \le -1.7911077630361746 \cdot 10^{+47}:\\ \;\;\;\;\left(\sqrt[3]{\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}} \cdot \left(\sqrt[3]{\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}} \cdot \sqrt[3]{\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}}\right) - \frac{1}{\frac{i}{n}}\right) \cdot 100\\ \mathbf{elif}\;n \le -9.669625897906558 \cdot 10^{+22}:\\ \;\;\;\;\left(\left(n \cdot \left(\left(\sqrt[3]{\frac{1}{6}} \cdot i\right) \cdot \left(\sqrt[3]{\frac{1}{6}} \cdot i\right)\right)\right) \cdot \sqrt[3]{\frac{1}{6}} + \left(\left(\frac{1}{2} \cdot i\right) \cdot n + n\right)\right) \cdot 100\\ \mathbf{elif}\;n \le -5.9286511822489366 \cdot 10^{-49}:\\ \;\;\;\;100 \cdot \left(\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} - \frac{1}{i}\right) \cdot n\right)\\ \mathbf{elif}\;n \le 9.575664251285104 \cdot 10^{-135}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(n \cdot \left(\left(\sqrt[3]{\frac{1}{6}} \cdot i\right) \cdot \left(\sqrt[3]{\frac{1}{6}} \cdot i\right)\right)\right) \cdot \sqrt[3]{\frac{1}{6}} + \left(\left(\frac{1}{2} \cdot i\right) \cdot n + n\right)\right) \cdot 100\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -1.3875157595860486 \cdot 10^{+126}:\\
\;\;\;\;\left(\left(n \cdot \left(\left(\sqrt[3]{\frac{1}{6}} \cdot i\right) \cdot \left(\sqrt[3]{\frac{1}{6}} \cdot i\right)\right)\right) \cdot \sqrt[3]{\frac{1}{6}} + \left(\left(\frac{1}{2} \cdot i\right) \cdot n + n\right)\right) \cdot 100\\

\mathbf{elif}\;n \le -1.7911077630361746 \cdot 10^{+47}:\\
\;\;\;\;\left(\sqrt[3]{\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}} \cdot \left(\sqrt[3]{\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}} \cdot \sqrt[3]{\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}}\right) - \frac{1}{\frac{i}{n}}\right) \cdot 100\\

\mathbf{elif}\;n \le -9.669625897906558 \cdot 10^{+22}:\\
\;\;\;\;\left(\left(n \cdot \left(\left(\sqrt[3]{\frac{1}{6}} \cdot i\right) \cdot \left(\sqrt[3]{\frac{1}{6}} \cdot i\right)\right)\right) \cdot \sqrt[3]{\frac{1}{6}} + \left(\left(\frac{1}{2} \cdot i\right) \cdot n + n\right)\right) \cdot 100\\

\mathbf{elif}\;n \le -5.9286511822489366 \cdot 10^{-49}:\\
\;\;\;\;100 \cdot \left(\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} - \frac{1}{i}\right) \cdot n\right)\\

\mathbf{elif}\;n \le 9.575664251285104 \cdot 10^{-135}:\\
\;\;\;\;0\\

\mathbf{else}:\\
\;\;\;\;\left(\left(n \cdot \left(\left(\sqrt[3]{\frac{1}{6}} \cdot i\right) \cdot \left(\sqrt[3]{\frac{1}{6}} \cdot i\right)\right)\right) \cdot \sqrt[3]{\frac{1}{6}} + \left(\left(\frac{1}{2} \cdot i\right) \cdot n + n\right)\right) \cdot 100\\

\end{array}
double f(double i, double n) {
        double r4554354 = 100.0;
        double r4554355 = 1.0;
        double r4554356 = i;
        double r4554357 = n;
        double r4554358 = r4554356 / r4554357;
        double r4554359 = r4554355 + r4554358;
        double r4554360 = pow(r4554359, r4554357);
        double r4554361 = r4554360 - r4554355;
        double r4554362 = r4554361 / r4554358;
        double r4554363 = r4554354 * r4554362;
        return r4554363;
}
↓
double f(double i, double n) {
        double r4554364 = n;
        double r4554365 = -1.3875157595860486e+126;
        bool r4554366 = r4554364 <= r4554365;
        double r4554367 = 0.16666666666666666;
        double r4554368 = cbrt(r4554367);
        double r4554369 = i;
        double r4554370 = r4554368 * r4554369;
        double r4554371 = r4554370 * r4554370;
        double r4554372 = r4554364 * r4554371;
        double r4554373 = r4554372 * r4554368;
        double r4554374 = 0.5;
        double r4554375 = r4554374 * r4554369;
        double r4554376 = r4554375 * r4554364;
        double r4554377 = r4554376 + r4554364;
        double r4554378 = r4554373 + r4554377;
        double r4554379 = 100.0;
        double r4554380 = r4554378 * r4554379;
        double r4554381 = -1.7911077630361746e+47;
        bool r4554382 = r4554364 <= r4554381;
        double r4554383 = r4554369 / r4554364;
        double r4554384 = 1.0;
        double r4554385 = r4554383 + r4554384;
        double r4554386 = pow(r4554385, r4554364);
        double r4554387 = r4554386 / r4554383;
        double r4554388 = cbrt(r4554387);
        double r4554389 = r4554388 * r4554388;
        double r4554390 = r4554388 * r4554389;
        double r4554391 = r4554384 / r4554383;
        double r4554392 = r4554390 - r4554391;
        double r4554393 = r4554392 * r4554379;
        double r4554394 = -9.669625897906558e+22;
        bool r4554395 = r4554364 <= r4554394;
        double r4554396 = -5.9286511822489366e-49;
        bool r4554397 = r4554364 <= r4554396;
        double r4554398 = r4554386 / r4554369;
        double r4554399 = r4554384 / r4554369;
        double r4554400 = r4554398 - r4554399;
        double r4554401 = r4554400 * r4554364;
        double r4554402 = r4554379 * r4554401;
        double r4554403 = 9.575664251285104e-135;
        bool r4554404 = r4554364 <= r4554403;
        double r4554405 = 0.0;
        double r4554406 = r4554404 ? r4554405 : r4554380;
        double r4554407 = r4554397 ? r4554402 : r4554406;
        double r4554408 = r4554395 ? r4554380 : r4554407;
        double r4554409 = r4554382 ? r4554393 : r4554408;
        double r4554410 = r4554366 ? r4554380 : r4554409;
        return r4554410;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.5
Target41.8
Herbie21.9
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if n < -1.3875157595860486e+126 or -1.7911077630361746e+47 < n < -9.669625897906558e+22 or 9.575664251285104e-135 < n

    1. Initial program 54.8

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

    3. Applied div-sub54.8

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

    4. Taylor expanded around 0 21.2

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({i}^{2} \cdot n\right) + \left(n + \frac{1}{2} \cdot \left(i \cdot n\right)\right)\right)}\]

    5. Simplified21.2

      \[\leadsto 100 \cdot \color{blue}{\left(\left(\left(i \cdot i\right) \cdot n\right) \cdot \frac{1}{6} + \left(\left(i \cdot \frac{1}{2}\right) \cdot n + n\right)\right)}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt21.2

      \[\leadsto 100 \cdot \left(\left(\left(i \cdot i\right) \cdot n\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{6}} \cdot \sqrt[3]{\frac{1}{6}}\right) \cdot \sqrt[3]{\frac{1}{6}}\right)} + \left(\left(i \cdot \frac{1}{2}\right) \cdot n + n\right)\right)\]

    8. Applied associate-*r*21.2

      \[\leadsto 100 \cdot \left(\color{blue}{\left(\left(\left(i \cdot i\right) \cdot n\right) \cdot \left(\sqrt[3]{\frac{1}{6}} \cdot \sqrt[3]{\frac{1}{6}}\right)\right) \cdot \sqrt[3]{\frac{1}{6}}} + \left(\left(i \cdot \frac{1}{2}\right) \cdot n + n\right)\right)\]

    9. Simplified21.2

      \[\leadsto 100 \cdot \left(\color{blue}{\left(n \cdot \left(\left(\sqrt[3]{\frac{1}{6}} \cdot i\right) \cdot \left(\sqrt[3]{\frac{1}{6}} \cdot i\right)\right)\right)} \cdot \sqrt[3]{\frac{1}{6}} + \left(\left(i \cdot \frac{1}{2}\right) \cdot n + n\right)\right)\]

    if -1.3875157595860486e+126 < n < -1.7911077630361746e+47

    1. Initial program 38.6

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

    3. Applied div-sub38.6

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

    5. Applied add-cube-cbrt38.4

      \[\leadsto 100 \cdot \left(\color{blue}{\left(\sqrt[3]{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}} \cdot \sqrt[3]{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}}\right) \cdot \sqrt[3]{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}}} - \frac{1}{\frac{i}{n}}\right)\]

    if -9.669625897906558e+22 < n < -5.9286511822489366e-49

    1. Initial program 24.5

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

    3. Applied div-sub24.6

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

    5. Applied associate-/r/33.6

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

    6. Applied associate-/r/25.0

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

    7. Applied distribute-rgt-out--25.0

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

    if -5.9286511822489366e-49 < n < 9.575664251285104e-135

    1. Initial program 25.6

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

    3. Applied associate-*r/25.6

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

    4. Taylor expanded around 0 17.7

      \[\leadsto \color{blue}{0}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification21.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.3875157595860486 \cdot 10^{+126}:\\ \;\;\;\;\left(\left(n \cdot \left(\left(\sqrt[3]{\frac{1}{6}} \cdot i\right) \cdot \left(\sqrt[3]{\frac{1}{6}} \cdot i\right)\right)\right) \cdot \sqrt[3]{\frac{1}{6}} + \left(\left(\frac{1}{2} \cdot i\right) \cdot n + n\right)\right) \cdot 100\\ \mathbf{elif}\;n \le -1.7911077630361746 \cdot 10^{+47}:\\ \;\;\;\;\left(\sqrt[3]{\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}} \cdot \left(\sqrt[3]{\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}} \cdot \sqrt[3]{\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}}\right) - \frac{1}{\frac{i}{n}}\right) \cdot 100\\ \mathbf{elif}\;n \le -9.669625897906558 \cdot 10^{+22}:\\ \;\;\;\;\left(\left(n \cdot \left(\left(\sqrt[3]{\frac{1}{6}} \cdot i\right) \cdot \left(\sqrt[3]{\frac{1}{6}} \cdot i\right)\right)\right) \cdot \sqrt[3]{\frac{1}{6}} + \left(\left(\frac{1}{2} \cdot i\right) \cdot n + n\right)\right) \cdot 100\\ \mathbf{elif}\;n \le -5.9286511822489366 \cdot 10^{-49}:\\ \;\;\;\;100 \cdot \left(\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} - \frac{1}{i}\right) \cdot n\right)\\ \mathbf{elif}\;n \le 9.575664251285104 \cdot 10^{-135}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(n \cdot \left(\left(\sqrt[3]{\frac{1}{6}} \cdot i\right) \cdot \left(\sqrt[3]{\frac{1}{6}} \cdot i\right)\right)\right) \cdot \sqrt[3]{\frac{1}{6}} + \left(\left(\frac{1}{2} \cdot i\right) \cdot n + n\right)\right) \cdot 100\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 
(FPCore (i n)
  :name "Compound Interest"

  :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))))