Average Error: 42.8 → 18.1
Time: 40.1s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -249306.0470798354:\\ \;\;\;\;100 \cdot \left(\left(\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}\right)\right) - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 1.4921312295942013:\\ \;\;\;\;n \cdot \left(i \cdot 50\right) + \left(100 + \left(i \cdot i\right) \cdot \frac{50}{3}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \left(n \cdot {\left(\frac{i}{n} + 1\right)}^{n} - n\right)\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -249306.0470798354:\\
\;\;\;\;100 \cdot \left(\left(\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}\right)\right) - \frac{1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;i \le 1.4921312295942013:\\
\;\;\;\;n \cdot \left(i \cdot 50\right) + \left(100 + \left(i \cdot i\right) \cdot \frac{50}{3}\right) \cdot n\\

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

\end{array}
double f(double i, double n) {
        double r5014388 = 100.0;
        double r5014389 = 1.0;
        double r5014390 = i;
        double r5014391 = n;
        double r5014392 = r5014390 / r5014391;
        double r5014393 = r5014389 + r5014392;
        double r5014394 = pow(r5014393, r5014391);
        double r5014395 = r5014394 - r5014389;
        double r5014396 = r5014395 / r5014392;
        double r5014397 = r5014388 * r5014396;
        return r5014397;
}
↓
double f(double i, double n) {
        double r5014398 = i;
        double r5014399 = -249306.0470798354;
        bool r5014400 = r5014398 <= r5014399;
        double r5014401 = 100.0;
        double r5014402 = n;
        double r5014403 = r5014398 / r5014402;
        double r5014404 = 1.0;
        double r5014405 = r5014403 + r5014404;
        double r5014406 = pow(r5014405, r5014402);
        double r5014407 = r5014406 / r5014403;
        double r5014408 = /* ERROR: no posit support in C */;
        double r5014409 = /* ERROR: no posit support in C */;
        double r5014410 = r5014404 / r5014403;
        double r5014411 = r5014409 - r5014410;
        double r5014412 = r5014401 * r5014411;
        double r5014413 = 1.4921312295942013;
        bool r5014414 = r5014398 <= r5014413;
        double r5014415 = 50.0;
        double r5014416 = r5014398 * r5014415;
        double r5014417 = r5014402 * r5014416;
        double r5014418 = r5014398 * r5014398;
        double r5014419 = 16.666666666666668;
        double r5014420 = r5014418 * r5014419;
        double r5014421 = r5014401 + r5014420;
        double r5014422 = r5014421 * r5014402;
        double r5014423 = r5014417 + r5014422;
        double r5014424 = r5014404 / r5014398;
        double r5014425 = r5014402 * r5014406;
        double r5014426 = r5014425 - r5014402;
        double r5014427 = r5014424 * r5014426;
        double r5014428 = r5014401 * r5014427;
        double r5014429 = r5014414 ? r5014423 : r5014428;
        double r5014430 = r5014400 ? r5014412 : r5014429;
        return r5014430;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.8
Target42.1
Herbie18.1
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -249306.0470798354

    1. Initial program 27.4

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

    3. Applied div-sub27.4

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

    5. Applied insert-posit1611.7

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

    if -249306.0470798354 < i < 1.4921312295942013

    1. Initial program 50.2

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

    3. Applied div-sub50.1

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

    4. Taylor expanded around 0 17.4

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

    5. Simplified17.4

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

    7. Applied distribute-rgt-in17.4

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

    8. Simplified17.4

      \[\leadsto \color{blue}{\left(50 \cdot i\right) \cdot n} + \left(n + n \cdot \left(\left(i \cdot i\right) \cdot \frac{1}{6}\right)\right) \cdot 100\]

    9. Simplified17.4

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

    if 1.4921312295942013 < i

    1. Initial program 32.7

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

    3. Applied div-sub32.7

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

    5. Applied div-inv34.8

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

    6. Applied *-un-lft-identity34.8

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

    7. Applied times-frac36.3

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

    8. Applied div-inv36.4

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

    9. Applied *-un-lft-identity36.4

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

    10. Applied times-frac32.7

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

    11. Applied distribute-lft-out--32.7

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

    12. Simplified32.7

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

  4. Final simplification18.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -249306.0470798354:\\ \;\;\;\;100 \cdot \left(\left(\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}\right)\right) - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 1.4921312295942013:\\ \;\;\;\;n \cdot \left(i \cdot 50\right) + \left(100 + \left(i \cdot i\right) \cdot \frac{50}{3}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \left(n \cdot {\left(\frac{i}{n} + 1\right)}^{n} - n\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(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))))