Average Error: 42.8 → 19.0
Time: 3.1m
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.2694860566002935:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.0240148652191775 \cdot 10^{-13}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(\frac{50}{3} \cdot i + 50\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot 100\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.2694860566002935:\\
\;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 1.0240148652191775 \cdot 10^{-13}:\\
\;\;\;\;n \cdot \left(100 + i \cdot \left(\frac{50}{3} \cdot i + 50\right)\right)\\

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

\end{array}
double f(double i, double n) {
        double r39001342 = 100.0;
        double r39001343 = 1.0;
        double r39001344 = i;
        double r39001345 = n;
        double r39001346 = r39001344 / r39001345;
        double r39001347 = r39001343 + r39001346;
        double r39001348 = pow(r39001347, r39001345);
        double r39001349 = r39001348 - r39001343;
        double r39001350 = r39001349 / r39001346;
        double r39001351 = r39001342 * r39001350;
        return r39001351;
}
↓
double f(double i, double n) {
        double r39001352 = i;
        double r39001353 = -0.2694860566002935;
        bool r39001354 = r39001352 <= r39001353;
        double r39001355 = 100.0;
        double r39001356 = n;
        double r39001357 = r39001352 / r39001356;
        double r39001358 = pow(r39001357, r39001356);
        double r39001359 = -1.0;
        double r39001360 = r39001358 + r39001359;
        double r39001361 = r39001360 / r39001357;
        double r39001362 = r39001355 * r39001361;
        double r39001363 = 1.0240148652191775e-13;
        bool r39001364 = r39001352 <= r39001363;
        double r39001365 = 16.666666666666668;
        double r39001366 = r39001365 * r39001352;
        double r39001367 = 50.0;
        double r39001368 = r39001366 + r39001367;
        double r39001369 = r39001352 * r39001368;
        double r39001370 = r39001355 + r39001369;
        double r39001371 = r39001356 * r39001370;
        double r39001372 = 1.0;
        double r39001373 = r39001372 + r39001357;
        double r39001374 = pow(r39001373, r39001356);
        double r39001375 = r39001374 - r39001372;
        double r39001376 = r39001375 / r39001352;
        double r39001377 = r39001376 * r39001355;
        double r39001378 = r39001356 * r39001377;
        double r39001379 = r39001364 ? r39001371 : r39001378;
        double r39001380 = r39001354 ? r39001362 : r39001379;
        return r39001380;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.8
Target42.6
Herbie19.0
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -0.2694860566002935

    1. Initial program 29.1

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

    2. Taylor expanded around inf 62.9

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

    3. Simplified18.4

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

    if -0.2694860566002935 < i < 1.0240148652191775e-13

    1. Initial program 50.4

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

    2. Taylor expanded around 0 32.0

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

    3. Simplified32.0

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

    4. Taylor expanded around 0 16.1

      \[\leadsto \color{blue}{\frac{50}{3} \cdot \left({i}^{2} \cdot n\right) + \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)}\]

    5. Simplified16.1

      \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot \left(\frac{50}{3} \cdot i + 50\right)\right)}\]

    6. Taylor expanded around -inf 16.1

      \[\leadsto n \cdot \left(100 + \color{blue}{\left(50 \cdot i + \frac{50}{3} \cdot {i}^{2}\right)}\right)\]

    7. Simplified16.1

      \[\leadsto n \cdot \left(100 + \color{blue}{\left(\frac{50}{3} \cdot i + 50\right) \cdot i}\right)\]

    if 1.0240148652191775e-13 < i

    1. Initial program 31.8

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

    3. Applied associate-/r/31.8

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

    4. Applied associate-*r*31.8

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

  4. Final simplification19.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.2694860566002935:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.0240148652191775 \cdot 10^{-13}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(\frac{50}{3} \cdot i + 50\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot 100\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(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))))