Average Error: 42.4 → 21.4
Time: 19.2s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.49613502984700486:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 2.7217117903300314 \cdot 10^{-9}:\\ \;\;\;\;\left(100 \cdot \left(\mathsf{fma}\left(0.5, i, \frac{\log 1 \cdot n}{i} + 1\right) - 0.5 \cdot \left(i \cdot \log 1\right)\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, {\left(\sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, -1\right)}{i}\right) \cdot n\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -1.49613502984700486:\\
\;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;i \le 2.7217117903300314 \cdot 10^{-9}:\\
\;\;\;\;\left(100 \cdot \left(\mathsf{fma}\left(0.5, i, \frac{\log 1 \cdot n}{i} + 1\right) - 0.5 \cdot \left(i \cdot \log 1\right)\right)\right) \cdot n\\

\mathbf{else}:\\
\;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, {\left(\sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, -1\right)}{i}\right) \cdot n\\

\end{array}
double f(double i, double n) {
        double r133317 = 100.0;
        double r133318 = 1.0;
        double r133319 = i;
        double r133320 = n;
        double r133321 = r133319 / r133320;
        double r133322 = r133318 + r133321;
        double r133323 = pow(r133322, r133320);
        double r133324 = r133323 - r133318;
        double r133325 = r133324 / r133321;
        double r133326 = r133317 * r133325;
        return r133326;
}
↓
double f(double i, double n) {
        double r133327 = i;
        double r133328 = -1.4961350298470049;
        bool r133329 = r133327 <= r133328;
        double r133330 = 100.0;
        double r133331 = 1.0;
        double r133332 = n;
        double r133333 = r133327 / r133332;
        double r133334 = r133331 + r133333;
        double r133335 = pow(r133334, r133332);
        double r133336 = r133335 / r133333;
        double r133337 = r133331 / r133333;
        double r133338 = r133336 - r133337;
        double r133339 = r133330 * r133338;
        double r133340 = 2.7217117903300314e-09;
        bool r133341 = r133327 <= r133340;
        double r133342 = 0.5;
        double r133343 = log(r133331);
        double r133344 = r133343 * r133332;
        double r133345 = r133344 / r133327;
        double r133346 = r133345 + r133331;
        double r133347 = fma(r133342, r133327, r133346);
        double r133348 = r133327 * r133343;
        double r133349 = r133342 * r133348;
        double r133350 = r133347 - r133349;
        double r133351 = r133330 * r133350;
        double r133352 = r133351 * r133332;
        double r133353 = cbrt(r133334);
        double r133354 = r133353 * r133353;
        double r133355 = pow(r133354, r133332);
        double r133356 = pow(r133353, r133332);
        double r133357 = -r133331;
        double r133358 = fma(r133355, r133356, r133357);
        double r133359 = r133358 / r133327;
        double r133360 = r133330 * r133359;
        double r133361 = r133360 * r133332;
        double r133362 = r133341 ? r133352 : r133361;
        double r133363 = r133329 ? r133339 : r133362;
        return r133363;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.4
Target42.5
Herbie21.4
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -1.4961350298470049

    1. Initial program 26.7

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

    3. Applied div-sub26.8

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

    if -1.4961350298470049 < i < 2.7217117903300314e-09

    1. Initial program 50.4

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

    3. Applied associate-/r/50.1

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

    4. Applied associate-*r*50.1

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

    5. Taylor expanded around 0 17.1

      \[\leadsto \left(100 \cdot \frac{\color{blue}{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{i}\right) \cdot n\]

    6. Simplified17.1

      \[\leadsto \left(100 \cdot \frac{\color{blue}{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{i}\right) \cdot n\]

    7. Taylor expanded around 0 17.1

      \[\leadsto \left(100 \cdot \color{blue}{\left(\left(1 + \left(0.5 \cdot i + \frac{\log 1 \cdot n}{i}\right)\right) - 0.5 \cdot \left(i \cdot \log 1\right)\right)}\right) \cdot n\]

    8. Simplified17.1

      \[\leadsto \left(100 \cdot \color{blue}{\left(\mathsf{fma}\left(0.5, i, \frac{\log 1 \cdot n}{i} + 1\right) - 0.5 \cdot \left(i \cdot \log 1\right)\right)}\right) \cdot n\]

    if 2.7217117903300314e-09 < i

    1. Initial program 32.0

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

    3. Applied associate-/r/32.0

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

    4. Applied associate-*r*32.0

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

    6. Applied add-cube-cbrt32.0

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

    7. Applied unpow-prod-down32.0

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

    8. Applied fma-neg32.0

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

  4. Final simplification21.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.49613502984700486:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 2.7217117903300314 \cdot 10^{-9}:\\ \;\;\;\;\left(100 \cdot \left(\mathsf{fma}\left(0.5, i, \frac{\log 1 \cdot n}{i} + 1\right) - 0.5 \cdot \left(i \cdot \log 1\right)\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, {\left(\sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, -1\right)}{i}\right) \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :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))))