Average Error: 43.2 → 19.3
Time: 29.1s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.1198183818045873705315784718550276011229:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 5.738844998942817632325841259444132447243:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;i \le 3.877114676469803572790117232248422698017 \cdot 10^{110}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.1198183818045873705315784718550276011229:\\
\;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 5.738844998942817632325841259444132447243:\\
\;\;\;\;\left(100 \cdot \frac{\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\\

\mathbf{elif}\;i \le 3.877114676469803572790117232248422698017 \cdot 10^{110}:\\
\;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\

\end{array}
double f(double i, double n) {
        double r104151 = 100.0;
        double r104152 = 1.0;
        double r104153 = i;
        double r104154 = n;
        double r104155 = r104153 / r104154;
        double r104156 = r104152 + r104155;
        double r104157 = pow(r104156, r104154);
        double r104158 = r104157 - r104152;
        double r104159 = r104158 / r104155;
        double r104160 = r104151 * r104159;
        return r104160;
}
↓
double f(double i, double n) {
        double r104161 = i;
        double r104162 = -0.11981838180458737;
        bool r104163 = r104161 <= r104162;
        double r104164 = 100.0;
        double r104165 = n;
        double r104166 = r104161 / r104165;
        double r104167 = pow(r104166, r104165);
        double r104168 = 1.0;
        double r104169 = r104167 - r104168;
        double r104170 = r104164 * r104169;
        double r104171 = r104170 / r104166;
        double r104172 = 5.738844998942818;
        bool r104173 = r104161 <= r104172;
        double r104174 = 0.5;
        double r104175 = 2.0;
        double r104176 = pow(r104161, r104175);
        double r104177 = log(r104168);
        double r104178 = r104177 * r104165;
        double r104179 = fma(r104174, r104176, r104178);
        double r104180 = fma(r104168, r104161, r104179);
        double r104181 = r104176 * r104177;
        double r104182 = r104174 * r104181;
        double r104183 = r104180 - r104182;
        double r104184 = r104183 / r104161;
        double r104185 = r104164 * r104184;
        double r104186 = r104185 * r104165;
        double r104187 = 3.8771146764698036e+110;
        bool r104188 = r104161 <= r104187;
        double r104189 = 1.0;
        double r104190 = fma(r104177, r104165, r104189);
        double r104191 = fma(r104168, r104161, r104190);
        double r104192 = r104191 - r104168;
        double r104193 = r104192 / r104166;
        double r104194 = r104164 * r104193;
        double r104195 = r104188 ? r104171 : r104194;
        double r104196 = r104173 ? r104186 : r104195;
        double r104197 = r104163 ? r104171 : r104196;
        return r104197;
}

Error

Bits error versus i

Bits error versus n

Target

Original43.2
Target42.7
Herbie19.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -0.11981838180458737 or 5.738844998942818 < i < 3.8771146764698036e+110

    1. Initial program 28.6

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

    2. Taylor expanded around inf 59.3

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

    3. Simplified20.4

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

    if -0.11981838180458737 < i < 5.738844998942818

    1. Initial program 50.9

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

    2. Taylor expanded around 0 34.6

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

    3. Simplified34.6

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

    5. Applied associate-/r/16.6

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

    6. Applied associate-*r*16.6

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

    if 3.8771146764698036e+110 < i

    1. Initial program 34.7

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

    2. Taylor expanded around 0 34.3

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

    3. Simplified34.3

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

  4. Final simplification19.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.1198183818045873705315784718550276011229:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 5.738844998942817632325841259444132447243:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;i \le 3.877114676469803572790117232248422698017 \cdot 10^{110}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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))))