Average Error: 43.0 → 21.5
Time: 16.5s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -5.170598455267974816273456052167524301145 \cdot 10^{128}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -1.17451235129368526597014478538898566314 \cdot 10^{115}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le -103630197189143872:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le 5.964212362723236143156593386256578372635 \cdot 10^{-309}:\\ \;\;\;\;\left(100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{i}\right) \cdot n\\ \mathbf{elif}\;n \le 5.061964505964020145803022024124640179055 \cdot 10^{-200}:\\ \;\;\;\;\frac{100 \cdot e^{\left(\log \left(\frac{1}{n}\right) - \log \left(\frac{1}{i}\right)\right) \cdot n} - 100}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -5.170598455267974816273456052167524301145 \cdot 10^{128}:\\
\;\;\;\;\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\\

\mathbf{elif}\;n \le -1.17451235129368526597014478538898566314 \cdot 10^{115}:\\
\;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;n \le -103630197189143872:\\
\;\;\;\;\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\\

\mathbf{elif}\;n \le 5.964212362723236143156593386256578372635 \cdot 10^{-309}:\\
\;\;\;\;\left(100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{i}\right) \cdot n\\

\mathbf{elif}\;n \le 5.061964505964020145803022024124640179055 \cdot 10^{-200}:\\
\;\;\;\;\frac{100 \cdot e^{\left(\log \left(\frac{1}{n}\right) - \log \left(\frac{1}{i}\right)\right) \cdot n} - 100}{i} \cdot n\\

\mathbf{else}:\\
\;\;\;\;\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\\

\end{array}
double f(double i, double n) {
        double r153316 = 100.0;
        double r153317 = 1.0;
        double r153318 = i;
        double r153319 = n;
        double r153320 = r153318 / r153319;
        double r153321 = r153317 + r153320;
        double r153322 = pow(r153321, r153319);
        double r153323 = r153322 - r153317;
        double r153324 = r153323 / r153320;
        double r153325 = r153316 * r153324;
        return r153325;
}
↓
double f(double i, double n) {
        double r153326 = n;
        double r153327 = -5.170598455267975e+128;
        bool r153328 = r153326 <= r153327;
        double r153329 = 100.0;
        double r153330 = 1.0;
        double r153331 = i;
        double r153332 = r153330 * r153331;
        double r153333 = 0.5;
        double r153334 = 2.0;
        double r153335 = pow(r153331, r153334);
        double r153336 = r153333 * r153335;
        double r153337 = log(r153330);
        double r153338 = r153337 * r153326;
        double r153339 = r153336 + r153338;
        double r153340 = r153332 + r153339;
        double r153341 = r153335 * r153337;
        double r153342 = r153333 * r153341;
        double r153343 = r153340 - r153342;
        double r153344 = r153343 / r153331;
        double r153345 = r153329 * r153344;
        double r153346 = r153345 * r153326;
        double r153347 = -1.1745123512936853e+115;
        bool r153348 = r153326 <= r153347;
        double r153349 = r153331 / r153326;
        double r153350 = r153330 + r153349;
        double r153351 = pow(r153350, r153326);
        double r153352 = r153351 / r153349;
        double r153353 = r153330 / r153349;
        double r153354 = r153352 - r153353;
        double r153355 = r153329 * r153354;
        double r153356 = -1.0363019718914387e+17;
        bool r153357 = r153326 <= r153356;
        double r153358 = 5.964212362723236e-309;
        bool r153359 = r153326 <= r153358;
        double r153360 = r153334 * r153326;
        double r153361 = pow(r153350, r153360);
        double r153362 = r153330 * r153330;
        double r153363 = -r153362;
        double r153364 = r153361 + r153363;
        double r153365 = r153351 + r153330;
        double r153366 = r153364 / r153365;
        double r153367 = r153366 / r153331;
        double r153368 = r153329 * r153367;
        double r153369 = r153368 * r153326;
        double r153370 = 5.06196450596402e-200;
        bool r153371 = r153326 <= r153370;
        double r153372 = 1.0;
        double r153373 = r153372 / r153326;
        double r153374 = log(r153373);
        double r153375 = r153372 / r153331;
        double r153376 = log(r153375);
        double r153377 = r153374 - r153376;
        double r153378 = r153377 * r153326;
        double r153379 = exp(r153378);
        double r153380 = r153329 * r153379;
        double r153381 = r153380 - r153329;
        double r153382 = r153381 / r153331;
        double r153383 = r153382 * r153326;
        double r153384 = r153371 ? r153383 : r153346;
        double r153385 = r153359 ? r153369 : r153384;
        double r153386 = r153357 ? r153346 : r153385;
        double r153387 = r153348 ? r153355 : r153386;
        double r153388 = r153328 ? r153346 : r153387;
        return r153388;
}

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
Target42.5
Herbie21.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if n < -5.170598455267975e+128 or -1.1745123512936853e+115 < n < -1.0363019718914387e+17 or 5.06196450596402e-200 < n

    1. Initial program 52.7

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

    3. Applied associate-/r/52.3

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

    4. Applied associate-*r*52.4

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

    5. Taylor expanded around 0 23.3

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

    if -5.170598455267975e+128 < n < -1.1745123512936853e+115

    1. Initial program 39.6

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

    3. Applied div-sub39.7

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

    if -1.0363019718914387e+17 < n < 5.964212362723236e-309

    1. Initial program 17.0

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

    3. Applied associate-/r/17.6

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

    4. Applied associate-*r*17.6

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

    6. Applied flip--17.6

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

    7. Simplified17.6

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

    if 5.964212362723236e-309 < n < 5.06196450596402e-200

    1. Initial program 38.7

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

    3. Applied associate-/r/38.7

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

    4. Applied associate-*r*38.7

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

    5. Taylor expanded around inf 14.6

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

  4. Final simplification21.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -5.170598455267974816273456052167524301145 \cdot 10^{128}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -1.17451235129368526597014478538898566314 \cdot 10^{115}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le -103630197189143872:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le 5.964212362723236143156593386256578372635 \cdot 10^{-309}:\\ \;\;\;\;\left(100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{i}\right) \cdot n\\ \mathbf{elif}\;n \le 5.061964505964020145803022024124640179055 \cdot 10^{-200}:\\ \;\;\;\;\frac{100 \cdot e^{\left(\log \left(\frac{1}{n}\right) - \log \left(\frac{1}{i}\right)\right) \cdot n} - 100}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(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))))