Average Error: 43.1 → 21.6
Time: 1.1m
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -4.152610626697651 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot i + \left(\left(i \cdot i\right) \cdot \frac{1}{6} + 1\right)}{\frac{1}{n}} \cdot 100\\ \mathbf{elif}\;n \le -3.1597532325517974 \cdot 10^{-85}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}{\frac{1}{n}}\\ \mathbf{elif}\;n \le 2.3277881727215845 \cdot 10^{-107}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \frac{\frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{2} + i\right)}{i}\right) \cdot 100\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -4.152610626697651 \cdot 10^{+69}:\\
\;\;\;\;\frac{\frac{1}{2} \cdot i + \left(\left(i \cdot i\right) \cdot \frac{1}{6} + 1\right)}{\frac{1}{n}} \cdot 100\\

\mathbf{elif}\;n \le -3.1597532325517974 \cdot 10^{-85}:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}{\frac{1}{n}}\\

\mathbf{elif}\;n \le 2.3277881727215845 \cdot 10^{-107}:\\
\;\;\;\;0\\

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

\end{array}
double f(double i, double n) {
        double r5179323 = 100.0;
        double r5179324 = 1.0;
        double r5179325 = i;
        double r5179326 = n;
        double r5179327 = r5179325 / r5179326;
        double r5179328 = r5179324 + r5179327;
        double r5179329 = pow(r5179328, r5179326);
        double r5179330 = r5179329 - r5179324;
        double r5179331 = r5179330 / r5179327;
        double r5179332 = r5179323 * r5179331;
        return r5179332;
}
↓
double f(double i, double n) {
        double r5179333 = n;
        double r5179334 = -4.152610626697651e+69;
        bool r5179335 = r5179333 <= r5179334;
        double r5179336 = 0.5;
        double r5179337 = i;
        double r5179338 = r5179336 * r5179337;
        double r5179339 = r5179337 * r5179337;
        double r5179340 = 0.16666666666666666;
        double r5179341 = r5179339 * r5179340;
        double r5179342 = 1.0;
        double r5179343 = r5179341 + r5179342;
        double r5179344 = r5179338 + r5179343;
        double r5179345 = r5179342 / r5179333;
        double r5179346 = r5179344 / r5179345;
        double r5179347 = 100.0;
        double r5179348 = r5179346 * r5179347;
        double r5179349 = -3.1597532325517974e-85;
        bool r5179350 = r5179333 <= r5179349;
        double r5179351 = r5179337 / r5179333;
        double r5179352 = r5179342 + r5179351;
        double r5179353 = pow(r5179352, r5179333);
        double r5179354 = r5179353 - r5179342;
        double r5179355 = r5179354 / r5179337;
        double r5179356 = r5179355 / r5179345;
        double r5179357 = r5179347 * r5179356;
        double r5179358 = 2.3277881727215845e-107;
        bool r5179359 = r5179333 <= r5179358;
        double r5179360 = 0.0;
        double r5179361 = r5179339 * r5179337;
        double r5179362 = r5179340 * r5179361;
        double r5179363 = r5179339 * r5179336;
        double r5179364 = r5179363 + r5179337;
        double r5179365 = r5179362 + r5179364;
        double r5179366 = r5179365 / r5179337;
        double r5179367 = r5179333 * r5179366;
        double r5179368 = r5179367 * r5179347;
        double r5179369 = r5179359 ? r5179360 : r5179368;
        double r5179370 = r5179350 ? r5179357 : r5179369;
        double r5179371 = r5179335 ? r5179348 : r5179370;
        return r5179371;
}

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
Target42.7
Herbie21.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if n < -4.152610626697651e+69

    1. Initial program 47.7

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

    3. Applied div-inv47.8

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

    4. Applied associate-/r*47.4

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

    5. Taylor expanded around 0 25.8

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

    6. Simplified25.8

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

    7. Taylor expanded around 0 25.6

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

    8. Simplified25.6

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

    if -4.152610626697651e+69 < n < -3.1597532325517974e-85

    1. Initial program 27.7

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

    3. Applied div-inv27.7

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

    4. Applied associate-/r*27.9

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

    if -3.1597532325517974e-85 < n < 2.3277881727215845e-107

    1. Initial program 28.6

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

    2. Taylor expanded around 0 20.8

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

    if 2.3277881727215845e-107 < n

    1. Initial program 59.2

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

    3. Applied div-inv59.2

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

    4. Applied associate-/r*60.0

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

    5. Taylor expanded around 0 16.8

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

    6. Simplified16.8

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

    8. Applied div-inv16.8

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

    9. Simplified16.7

      \[\leadsto 100 \cdot \left(\frac{\left(i + \left(i \cdot i\right) \cdot \frac{1}{2}\right) + \frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}{i} \cdot \color{blue}{n}\right)\]
  3. Recombined 4 regimes into one program.

  4. Final simplification21.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -4.152610626697651 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot i + \left(\left(i \cdot i\right) \cdot \frac{1}{6} + 1\right)}{\frac{1}{n}} \cdot 100\\ \mathbf{elif}\;n \le -3.1597532325517974 \cdot 10^{-85}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}{\frac{1}{n}}\\ \mathbf{elif}\;n \le 2.3277881727215845 \cdot 10^{-107}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \frac{\frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{2} + i\right)}{i}\right) \cdot 100\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(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))))