Average Error: 43.1 → 22.2
Time: 15.7s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -2.23087318151625042 \cdot 10^{136}:\\ \;\;\;\;\left(\left(100 \cdot \sqrt{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}}\right) \cdot \sqrt{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}}\right) \cdot n\\ \mathbf{elif}\;n \le -1.09580041647786345 \cdot 10^{86}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -35142169422648208:\\ \;\;\;\;\left(\left(100 \cdot \sqrt{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}}\right) \cdot \sqrt{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}}\right) \cdot n\\ \mathbf{elif}\;n \le -1.2431085101949388 \cdot 10^{-214}:\\ \;\;\;\;100 \cdot \frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 2.0850330521039454 \cdot 10^{-172}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(100 \cdot \sqrt{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}}\right) \cdot \sqrt{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}}\right) \cdot n\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -2.23087318151625042 \cdot 10^{136}:\\
\;\;\;\;\left(\left(100 \cdot \sqrt{\frac{\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)}{i}}\right) \cdot \sqrt{\frac{\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)}{i}}\right) \cdot n\\

\mathbf{elif}\;n \le -1.09580041647786345 \cdot 10^{86}:\\
\;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\

\mathbf{elif}\;n \le -35142169422648208:\\
\;\;\;\;\left(\left(100 \cdot \sqrt{\frac{\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)}{i}}\right) \cdot \sqrt{\frac{\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)}{i}}\right) \cdot n\\

\mathbf{elif}\;n \le -1.2431085101949388 \cdot 10^{-214}:\\
\;\;\;\;100 \cdot \frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right)}{\frac{i}{n}}\\

\mathbf{elif}\;n \le 2.0850330521039454 \cdot 10^{-172}:\\
\;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\

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

\end{array}
double f(double i, double n) {
        double r137315 = 100.0;
        double r137316 = 1.0;
        double r137317 = i;
        double r137318 = n;
        double r137319 = r137317 / r137318;
        double r137320 = r137316 + r137319;
        double r137321 = pow(r137320, r137318);
        double r137322 = r137321 - r137316;
        double r137323 = r137322 / r137319;
        double r137324 = r137315 * r137323;
        return r137324;
}
↓
double f(double i, double n) {
        double r137325 = n;
        double r137326 = -2.2308731815162504e+136;
        bool r137327 = r137325 <= r137326;
        double r137328 = 100.0;
        double r137329 = 1.0;
        double r137330 = i;
        double r137331 = r137329 * r137330;
        double r137332 = 0.5;
        double r137333 = 2.0;
        double r137334 = pow(r137330, r137333);
        double r137335 = r137332 * r137334;
        double r137336 = log(r137329);
        double r137337 = r137336 * r137325;
        double r137338 = r137335 + r137337;
        double r137339 = r137331 + r137338;
        double r137340 = r137334 * r137336;
        double r137341 = r137332 * r137340;
        double r137342 = r137339 - r137341;
        double r137343 = r137342 / r137330;
        double r137344 = sqrt(r137343);
        double r137345 = r137328 * r137344;
        double r137346 = r137345 * r137344;
        double r137347 = r137346 * r137325;
        double r137348 = -1.0958004164778634e+86;
        bool r137349 = r137325 <= r137348;
        double r137350 = r137330 / r137325;
        double r137351 = r137329 + r137350;
        double r137352 = pow(r137351, r137325);
        double r137353 = r137352 - r137329;
        double r137354 = r137328 * r137353;
        double r137355 = r137354 / r137350;
        double r137356 = -3.514216942264821e+16;
        bool r137357 = r137325 <= r137356;
        double r137358 = -1.2431085101949388e-214;
        bool r137359 = r137325 <= r137358;
        double r137360 = exp(r137353);
        double r137361 = log(r137360);
        double r137362 = r137361 / r137350;
        double r137363 = r137328 * r137362;
        double r137364 = 2.0850330521039454e-172;
        bool r137365 = r137325 <= r137364;
        double r137366 = 1.0;
        double r137367 = r137337 + r137366;
        double r137368 = r137331 + r137367;
        double r137369 = r137368 - r137329;
        double r137370 = r137369 / r137350;
        double r137371 = r137328 * r137370;
        double r137372 = r137365 ? r137371 : r137347;
        double r137373 = r137359 ? r137363 : r137372;
        double r137374 = r137357 ? r137347 : r137373;
        double r137375 = r137349 ? r137355 : r137374;
        double r137376 = r137327 ? r137347 : r137375;
        return r137376;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.1
Target42.9
Herbie22.2
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if n < -2.2308731815162504e+136 or -1.0958004164778634e+86 < n < -3.514216942264821e+16 or 2.0850330521039454e-172 < n

    1. Initial program 54.1

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

    2. Taylor expanded around 0 38.7

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

    4. Applied associate-/r/22.0

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

    5. Applied associate-*r*22.0

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

    7. Applied add-sqr-sqrt22.1

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

    8. Applied associate-*r*22.1

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

    if -2.2308731815162504e+136 < n < -1.0958004164778634e+86

    1. Initial program 35.1

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

    3. Applied associate-*r/35.0

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

    if -3.514216942264821e+16 < n < -1.2431085101949388e-214

    1. Initial program 19.3

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

    3. Applied add-log-exp19.3

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

    4. Applied add-log-exp19.4

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

    5. Applied diff-log19.4

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

    6. Simplified19.4

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

    if -1.2431085101949388e-214 < n < 2.0850330521039454e-172

    1. Initial program 28.7

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

    2. Taylor expanded around 0 21.5

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right)} - 1}{\frac{i}{n}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification22.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.23087318151625042 \cdot 10^{136}:\\ \;\;\;\;\left(\left(100 \cdot \sqrt{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}}\right) \cdot \sqrt{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}}\right) \cdot n\\ \mathbf{elif}\;n \le -1.09580041647786345 \cdot 10^{86}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -35142169422648208:\\ \;\;\;\;\left(\left(100 \cdot \sqrt{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}}\right) \cdot \sqrt{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}}\right) \cdot n\\ \mathbf{elif}\;n \le -1.2431085101949388 \cdot 10^{-214}:\\ \;\;\;\;100 \cdot \frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 2.0850330521039454 \cdot 10^{-172}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(100 \cdot \sqrt{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}}\right) \cdot \sqrt{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}}\right) \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(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))))