Average Error: 42.5 → 22.6
Time: 13.0s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -3.97244482863566445 \cdot 10^{105}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -8.48364346201296677 \cdot 10^{-263}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 2.91897266268395317 \cdot 10^{-204}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 1.6252547660245352 \cdot 10^{-162}:\\ \;\;\;\;100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\right) \cdot n\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -3.97244482863566445 \cdot 10^{105}:\\
\;\;\;\;\left(100 \cdot \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\\

\mathbf{elif}\;n \le -8.48364346201296677 \cdot 10^{-263}:\\
\;\;\;\;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}\;n \le 2.91897266268395317 \cdot 10^{-204}:\\
\;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\

\mathbf{elif}\;n \le 1.6252547660245352 \cdot 10^{-162}:\\
\;\;\;\;100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{n}{i}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(100 \cdot \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\\

\end{array}
double f(double i, double n) {
        double r139018 = 100.0;
        double r139019 = 1.0;
        double r139020 = i;
        double r139021 = n;
        double r139022 = r139020 / r139021;
        double r139023 = r139019 + r139022;
        double r139024 = pow(r139023, r139021);
        double r139025 = r139024 - r139019;
        double r139026 = r139025 / r139022;
        double r139027 = r139018 * r139026;
        return r139027;
}
↓
double f(double i, double n) {
        double r139028 = n;
        double r139029 = -3.9724448286356644e+105;
        bool r139030 = r139028 <= r139029;
        double r139031 = 100.0;
        double r139032 = i;
        double r139033 = 1.0;
        double r139034 = 0.5;
        double r139035 = 2.0;
        double r139036 = pow(r139032, r139035);
        double r139037 = log(r139033);
        double r139038 = r139037 * r139028;
        double r139039 = fma(r139034, r139036, r139038);
        double r139040 = r139036 * r139037;
        double r139041 = r139034 * r139040;
        double r139042 = r139039 - r139041;
        double r139043 = fma(r139032, r139033, r139042);
        double r139044 = r139043 / r139032;
        double r139045 = r139031 * r139044;
        double r139046 = r139045 * r139028;
        double r139047 = -8.483643462012967e-263;
        bool r139048 = r139028 <= r139047;
        double r139049 = r139032 / r139028;
        double r139050 = r139033 + r139049;
        double r139051 = r139035 * r139028;
        double r139052 = pow(r139050, r139051);
        double r139053 = r139033 * r139033;
        double r139054 = -r139053;
        double r139055 = r139052 + r139054;
        double r139056 = pow(r139050, r139028);
        double r139057 = r139056 + r139033;
        double r139058 = r139055 / r139057;
        double r139059 = r139058 / r139049;
        double r139060 = r139031 * r139059;
        double r139061 = 2.918972662683953e-204;
        bool r139062 = r139028 <= r139061;
        double r139063 = 1.0;
        double r139064 = fma(r139037, r139028, r139063);
        double r139065 = fma(r139033, r139032, r139064);
        double r139066 = r139065 - r139033;
        double r139067 = r139066 / r139049;
        double r139068 = r139031 * r139067;
        double r139069 = 1.6252547660245352e-162;
        bool r139070 = r139028 <= r139069;
        double r139071 = r139056 - r139033;
        double r139072 = r139028 / r139032;
        double r139073 = r139071 * r139072;
        double r139074 = r139031 * r139073;
        double r139075 = r139070 ? r139074 : r139046;
        double r139076 = r139062 ? r139068 : r139075;
        double r139077 = r139048 ? r139060 : r139076;
        double r139078 = r139030 ? r139046 : r139077;
        return r139078;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.5
Target42.4
Herbie22.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if n < -3.9724448286356644e+105 or 1.6252547660245352e-162 < n

    1. Initial program 55.5

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

    2. Taylor expanded around 0 40.0

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

    3. Simplified40.0

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

    5. Applied associate-/r/21.6

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

    6. Applied associate-*r*21.6

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

    if -3.9724448286356644e+105 < n < -8.483643462012967e-263

    1. Initial program 23.8

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

    3. Applied flip--23.8

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

    4. Simplified23.7

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

    if -8.483643462012967e-263 < n < 2.918972662683953e-204

    1. Initial program 28.6

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

    2. Taylor expanded around 0 17.1

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

    3. Simplified17.1

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

    if 2.918972662683953e-204 < n < 1.6252547660245352e-162

    1. Initial program 47.0

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

    3. Applied div-inv47.0

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

    4. Simplified47.0

      \[\leadsto 100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \color{blue}{\frac{n}{i}}\right)\]
  3. Recombined 4 regimes into one program.

  4. Final simplification22.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -3.97244482863566445 \cdot 10^{105}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -8.48364346201296677 \cdot 10^{-263}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 2.91897266268395317 \cdot 10^{-204}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 1.6252547660245352 \cdot 10^{-162}:\\ \;\;\;\;100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\right) \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +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))))