Average Error: 43.0 → 21.3
Time: 33.2s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -34996776512777996.0:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 4.4771129901808285 \cdot 10^{-246}:\\ \;\;\;\;\left(\sqrt[3]{\left(n \cdot \left(i \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)\right) + n\right) \cdot 100} \cdot \sqrt[3]{\left(n \cdot \left(i \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)\right) + n\right) \cdot 100}\right) \cdot \sqrt[3]{\left(n \cdot \left(i \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)\right) + n\right) \cdot 100}\\ \mathbf{elif}\;i \le 0.11480648704519449:\\ \;\;\;\;100 \cdot \left(\left(\left(\frac{1}{2} \cdot \left(i \cdot i\right) + \left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right)\right) \cdot n\right) \cdot \frac{1}{i}\right)\\ \mathbf{elif}\;i \le 1.8387456869663292 \cdot 10^{+244}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -34996776512777996.0:\\
\;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;i \le 4.4771129901808285 \cdot 10^{-246}:\\
\;\;\;\;\left(\sqrt[3]{\left(n \cdot \left(i \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)\right) + n\right) \cdot 100} \cdot \sqrt[3]{\left(n \cdot \left(i \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)\right) + n\right) \cdot 100}\right) \cdot \sqrt[3]{\left(n \cdot \left(i \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)\right) + n\right) \cdot 100}\\

\mathbf{elif}\;i \le 0.11480648704519449:\\
\;\;\;\;100 \cdot \left(\left(\left(\frac{1}{2} \cdot \left(i \cdot i\right) + \left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right)\right) \cdot n\right) \cdot \frac{1}{i}\right)\\

\mathbf{elif}\;i \le 1.8387456869663292 \cdot 10^{+244}:\\
\;\;\;\;0\\

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

\end{array}
double f(double i, double n) {
        double r4424260 = 100.0;
        double r4424261 = 1.0;
        double r4424262 = i;
        double r4424263 = n;
        double r4424264 = r4424262 / r4424263;
        double r4424265 = r4424261 + r4424264;
        double r4424266 = pow(r4424265, r4424263);
        double r4424267 = r4424266 - r4424261;
        double r4424268 = r4424267 / r4424264;
        double r4424269 = r4424260 * r4424268;
        return r4424269;
}
↓
double f(double i, double n) {
        double r4424270 = i;
        double r4424271 = -34996776512777996.0;
        bool r4424272 = r4424270 <= r4424271;
        double r4424273 = 100.0;
        double r4424274 = n;
        double r4424275 = r4424270 / r4424274;
        double r4424276 = 1.0;
        double r4424277 = r4424275 + r4424276;
        double r4424278 = pow(r4424277, r4424274);
        double r4424279 = r4424278 / r4424275;
        double r4424280 = r4424276 / r4424275;
        double r4424281 = r4424279 - r4424280;
        double r4424282 = r4424273 * r4424281;
        double r4424283 = 4.4771129901808285e-246;
        bool r4424284 = r4424270 <= r4424283;
        double r4424285 = 0.16666666666666666;
        double r4424286 = r4424270 * r4424285;
        double r4424287 = 0.5;
        double r4424288 = r4424286 + r4424287;
        double r4424289 = r4424270 * r4424288;
        double r4424290 = r4424274 * r4424289;
        double r4424291 = r4424290 + r4424274;
        double r4424292 = r4424291 * r4424273;
        double r4424293 = cbrt(r4424292);
        double r4424294 = r4424293 * r4424293;
        double r4424295 = r4424294 * r4424293;
        double r4424296 = 0.11480648704519449;
        bool r4424297 = r4424270 <= r4424296;
        double r4424298 = r4424270 * r4424270;
        double r4424299 = r4424287 * r4424298;
        double r4424300 = r4424298 * r4424270;
        double r4424301 = r4424285 * r4424300;
        double r4424302 = r4424270 + r4424301;
        double r4424303 = r4424299 + r4424302;
        double r4424304 = r4424303 * r4424274;
        double r4424305 = r4424276 / r4424270;
        double r4424306 = r4424304 * r4424305;
        double r4424307 = r4424273 * r4424306;
        double r4424308 = 1.8387456869663292e+244;
        bool r4424309 = r4424270 <= r4424308;
        double r4424310 = 0.0;
        double r4424311 = r4424278 - r4424276;
        double r4424312 = r4424311 / r4424270;
        double r4424313 = r4424273 * r4424312;
        double r4424314 = r4424274 * r4424313;
        double r4424315 = r4424309 ? r4424310 : r4424314;
        double r4424316 = r4424297 ? r4424307 : r4424315;
        double r4424317 = r4424284 ? r4424295 : r4424316;
        double r4424318 = r4424272 ? r4424282 : r4424317;
        return r4424318;
}

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
Target42.2
Herbie21.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 5 regimes

  2. if i < -34996776512777996.0

    1. Initial program 27.0

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

    3. Applied div-sub27.1

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

    if -34996776512777996.0 < i < 4.4771129901808285e-246

    1. Initial program 50.1

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

    2. Taylor expanded around 0 35.5

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

    3. Simplified35.5

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

    5. Applied associate-/r/17.6

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

    6. Taylor expanded around 0 17.6

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

    7. Simplified17.6

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

    9. Applied add-cube-cbrt18.5

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

    if 4.4771129901808285e-246 < i < 0.11480648704519449

    1. Initial program 50.7

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

    2. Taylor expanded around 0 29.9

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

    3. Simplified29.9

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

    5. Applied div-inv29.9

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

    6. Applied *-un-lft-identity29.9

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

    7. Applied *-un-lft-identity29.9

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

    8. Applied distribute-lft-out29.9

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

    9. Applied times-frac15.6

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

    10. Simplified15.5

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

    if 0.11480648704519449 < i < 1.8387456869663292e+244

    1. Initial program 33.7

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

    2. Taylor expanded around 0 30.0

      \[\leadsto \color{blue}{0}\]

    if 1.8387456869663292e+244 < i

    1. Initial program 33.3

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

    3. Applied associate-/r/33.3

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

    4. Applied associate-*r*33.3

      \[\leadsto \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification21.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -34996776512777996.0:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 4.4771129901808285 \cdot 10^{-246}:\\ \;\;\;\;\left(\sqrt[3]{\left(n \cdot \left(i \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)\right) + n\right) \cdot 100} \cdot \sqrt[3]{\left(n \cdot \left(i \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)\right) + n\right) \cdot 100}\right) \cdot \sqrt[3]{\left(n \cdot \left(i \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)\right) + n\right) \cdot 100}\\ \mathbf{elif}\;i \le 0.11480648704519449:\\ \;\;\;\;100 \cdot \left(\left(\left(\frac{1}{2} \cdot \left(i \cdot i\right) + \left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right)\right) \cdot n\right) \cdot \frac{1}{i}\right)\\ \mathbf{elif}\;i \le 1.8387456869663292 \cdot 10^{+244}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 
(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))))