Average Error: 42.7 → 17.8
Time: 24.7s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.8795028361709332:\\ \;\;\;\;100 \cdot \left(\left(\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}\right)\right) - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 0.0004866774544189746:\\ \;\;\;\;100 \cdot \left(\left(n + i \cdot \left(n \cdot \frac{1}{2}\right)\right) + n \cdot \left(i \cdot \left(i \cdot \frac{1}{6}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} - \frac{1}{i}\right)\right) \cdot 100\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -1.8795028361709332:\\
\;\;\;\;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 0.0004866774544189746:\\
\;\;\;\;100 \cdot \left(\left(n + i \cdot \left(n \cdot \frac{1}{2}\right)\right) + n \cdot \left(i \cdot \left(i \cdot \frac{1}{6}\right)\right)\right)\\

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

\end{array}
double f(double i, double n) {
        double r6429272 = 100.0;
        double r6429273 = 1.0;
        double r6429274 = i;
        double r6429275 = n;
        double r6429276 = r6429274 / r6429275;
        double r6429277 = r6429273 + r6429276;
        double r6429278 = pow(r6429277, r6429275);
        double r6429279 = r6429278 - r6429273;
        double r6429280 = r6429279 / r6429276;
        double r6429281 = r6429272 * r6429280;
        return r6429281;
}
↓
double f(double i, double n) {
        double r6429282 = i;
        double r6429283 = -1.8795028361709332;
        bool r6429284 = r6429282 <= r6429283;
        double r6429285 = 100.0;
        double r6429286 = n;
        double r6429287 = r6429282 / r6429286;
        double r6429288 = 1.0;
        double r6429289 = r6429287 + r6429288;
        double r6429290 = pow(r6429289, r6429286);
        double r6429291 = r6429290 / r6429287;
        double r6429292 = /* ERROR: no posit support in C */;
        double r6429293 = /* ERROR: no posit support in C */;
        double r6429294 = r6429288 / r6429287;
        double r6429295 = r6429293 - r6429294;
        double r6429296 = r6429285 * r6429295;
        double r6429297 = 0.0004866774544189746;
        bool r6429298 = r6429282 <= r6429297;
        double r6429299 = 0.5;
        double r6429300 = r6429286 * r6429299;
        double r6429301 = r6429282 * r6429300;
        double r6429302 = r6429286 + r6429301;
        double r6429303 = 0.16666666666666666;
        double r6429304 = r6429282 * r6429303;
        double r6429305 = r6429282 * r6429304;
        double r6429306 = r6429286 * r6429305;
        double r6429307 = r6429302 + r6429306;
        double r6429308 = r6429285 * r6429307;
        double r6429309 = r6429290 / r6429282;
        double r6429310 = r6429288 / r6429282;
        double r6429311 = r6429309 - r6429310;
        double r6429312 = r6429286 * r6429311;
        double r6429313 = r6429312 * r6429285;
        double r6429314 = r6429298 ? r6429308 : r6429313;
        double r6429315 = r6429284 ? r6429296 : r6429314;
        return r6429315;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.7
Target42.2
Herbie17.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -1.8795028361709332

    1. Initial program 28.6

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

    3. Applied div-sub28.6

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

    5. Applied insert-posit1612.7

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

    if -1.8795028361709332 < i < 0.0004866774544189746

    1. Initial program 50.1

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
    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 16.7

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

    5. Simplified16.7

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

    if 0.0004866774544189746 < i

    1. Initial program 31.4

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

    3. Applied div-sub31.4

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

    5. Applied associate-/r/34.4

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

    6. Applied associate-/r/31.4

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

    7. Applied distribute-rgt-out--31.4

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.8795028361709332:\\ \;\;\;\;100 \cdot \left(\left(\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}\right)\right) - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 0.0004866774544189746:\\ \;\;\;\;100 \cdot \left(\left(n + i \cdot \left(n \cdot \frac{1}{2}\right)\right) + n \cdot \left(i \cdot \left(i \cdot \frac{1}{6}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} - \frac{1}{i}\right)\right) \cdot 100\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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))))