Average Error: 42.7 → 17.8
Time: 28.1s
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 r4868175 = 100.0;
        double r4868176 = 1.0;
        double r4868177 = i;
        double r4868178 = n;
        double r4868179 = r4868177 / r4868178;
        double r4868180 = r4868176 + r4868179;
        double r4868181 = pow(r4868180, r4868178);
        double r4868182 = r4868181 - r4868176;
        double r4868183 = r4868182 / r4868179;
        double r4868184 = r4868175 * r4868183;
        return r4868184;
}
↓
double f(double i, double n) {
        double r4868185 = i;
        double r4868186 = -1.8795028361709332;
        bool r4868187 = r4868185 <= r4868186;
        double r4868188 = 100.0;
        double r4868189 = n;
        double r4868190 = r4868185 / r4868189;
        double r4868191 = 1.0;
        double r4868192 = r4868190 + r4868191;
        double r4868193 = pow(r4868192, r4868189);
        double r4868194 = r4868193 / r4868190;
        double r4868195 = /* ERROR: no posit support in C */;
        double r4868196 = /* ERROR: no posit support in C */;
        double r4868197 = r4868191 / r4868190;
        double r4868198 = r4868196 - r4868197;
        double r4868199 = r4868188 * r4868198;
        double r4868200 = 0.0004866774544189746;
        bool r4868201 = r4868185 <= r4868200;
        double r4868202 = 0.5;
        double r4868203 = r4868189 * r4868202;
        double r4868204 = r4868185 * r4868203;
        double r4868205 = r4868189 + r4868204;
        double r4868206 = 0.16666666666666666;
        double r4868207 = r4868185 * r4868206;
        double r4868208 = r4868185 * r4868207;
        double r4868209 = r4868189 * r4868208;
        double r4868210 = r4868205 + r4868209;
        double r4868211 = r4868188 * r4868210;
        double r4868212 = r4868193 / r4868185;
        double r4868213 = r4868191 / r4868185;
        double r4868214 = r4868212 - r4868213;
        double r4868215 = r4868189 * r4868214;
        double r4868216 = r4868215 * r4868188;
        double r4868217 = r4868201 ? r4868211 : r4868216;
        double r4868218 = r4868187 ? r4868199 : r4868217;
        return r4868218;
}

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))))