Average Error: 42.9 → 20.5
Time: 14.2s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.5511580558700998055954300980374682694674:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 6.118499728754747261555536370589189972005 \cdot 10^{-197}:\\ \;\;\;\;100 \cdot \left(\left(\left(\sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}}\right) \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}}\right) \cdot n\right)\\ \mathbf{elif}\;i \le 8648182.72403052262961864471435546875:\\ \;\;\;\;\frac{100 \cdot \left(\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.5511580558700998055954300980374682694674:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 6.118499728754747261555536370589189972005 \cdot 10^{-197}:\\
\;\;\;\;100 \cdot \left(\left(\left(\sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}}\right) \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}}\right) \cdot n\right)\\

\mathbf{elif}\;i \le 8648182.72403052262961864471435546875:\\
\;\;\;\;\frac{100 \cdot \left(\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}{i}\\

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

\end{array}
double f(double i, double n) {
        double r167016 = 100.0;
        double r167017 = 1.0;
        double r167018 = i;
        double r167019 = n;
        double r167020 = r167018 / r167019;
        double r167021 = r167017 + r167020;
        double r167022 = pow(r167021, r167019);
        double r167023 = r167022 - r167017;
        double r167024 = r167023 / r167020;
        double r167025 = r167016 * r167024;
        return r167025;
}
↓
double f(double i, double n) {
        double r167026 = i;
        double r167027 = -0.5511580558700998;
        bool r167028 = r167026 <= r167027;
        double r167029 = 100.0;
        double r167030 = 1.0;
        double r167031 = n;
        double r167032 = r167026 / r167031;
        double r167033 = r167030 + r167032;
        double r167034 = 2.0;
        double r167035 = r167034 * r167031;
        double r167036 = pow(r167033, r167035);
        double r167037 = r167030 * r167030;
        double r167038 = -r167037;
        double r167039 = r167036 + r167038;
        double r167040 = pow(r167033, r167031);
        double r167041 = r167040 + r167030;
        double r167042 = r167039 / r167041;
        double r167043 = r167042 / r167032;
        double r167044 = r167029 * r167043;
        double r167045 = 6.118499728754747e-197;
        bool r167046 = r167026 <= r167045;
        double r167047 = 0.5;
        double r167048 = pow(r167026, r167034);
        double r167049 = log(r167030);
        double r167050 = r167049 * r167031;
        double r167051 = fma(r167047, r167048, r167050);
        double r167052 = r167048 * r167049;
        double r167053 = r167047 * r167052;
        double r167054 = r167051 - r167053;
        double r167055 = fma(r167026, r167030, r167054);
        double r167056 = r167055 / r167026;
        double r167057 = cbrt(r167056);
        double r167058 = r167057 * r167057;
        double r167059 = r167058 * r167057;
        double r167060 = r167059 * r167031;
        double r167061 = r167029 * r167060;
        double r167062 = 8648182.724030523;
        bool r167063 = r167026 <= r167062;
        double r167064 = r167055 * r167031;
        double r167065 = r167029 * r167064;
        double r167066 = r167065 / r167026;
        double r167067 = r167063 ? r167066 : r167044;
        double r167068 = r167046 ? r167061 : r167067;
        double r167069 = r167028 ? r167044 : r167068;
        return r167069;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.9
Target43.1
Herbie20.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -0.5511580558700998 or 8648182.724030523 < i

    1. Initial program 28.3

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

    3. Applied flip--28.4

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

    4. Simplified28.4

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

    if -0.5511580558700998 < i < 6.118499728754747e-197

    1. Initial program 50.7

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

    2. Taylor expanded around 0 35.9

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

    3. Simplified35.9

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

    5. Applied associate-/r/16.3

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

    7. Applied add-cube-cbrt16.3

      \[\leadsto 100 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}}\right) \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}}\right)} \cdot n\right)\]

    if 6.118499728754747e-197 < i < 8648182.724030523

    1. Initial program 51.0

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

    2. Taylor expanded around 0 29.9

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

    3. Simplified29.9

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

    5. Applied associate-/r/19.0

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

    7. Applied associate-*l/16.1

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

    8. Applied associate-*r/16.2

      \[\leadsto \color{blue}{\frac{100 \cdot \left(\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}{i}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification20.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.5511580558700998055954300980374682694674:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 6.118499728754747261555536370589189972005 \cdot 10^{-197}:\\ \;\;\;\;100 \cdot \left(\left(\left(\sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}}\right) \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}}\right) \cdot n\right)\\ \mathbf{elif}\;i \le 8648182.72403052262961864471435546875:\\ \;\;\;\;\frac{100 \cdot \left(\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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))))