Average Error: 42.8 → 23.5
Time: 22.5s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -6.1428874105226178 \cdot 10^{122}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(1, i, 0.5 \cdot {i}^{2}\right)\right) - 0.5 \cdot \left(\log 1 \cdot {i}^{2}\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -8.23875326441091999 \cdot 10^{106}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -1.8594290872480363 \lor \neg \left(n \le 1.41832978451163636 \cdot 10^{-154}\right):\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(1, i, 0.5 \cdot {i}^{2}\right)\right) - 0.5 \cdot \left(\log 1 \cdot {i}^{2}\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(1, i, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -6.1428874105226178 \cdot 10^{122}:\\
\;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(1, i, 0.5 \cdot {i}^{2}\right)\right) - 0.5 \cdot \left(\log 1 \cdot {i}^{2}\right)}{i}\right) \cdot n\\

\mathbf{elif}\;n \le -8.23875326441091999 \cdot 10^{106}:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\

\mathbf{elif}\;n \le -1.8594290872480363 \lor \neg \left(n \le 1.41832978451163636 \cdot 10^{-154}\right):\\
\;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(1, i, 0.5 \cdot {i}^{2}\right)\right) - 0.5 \cdot \left(\log 1 \cdot {i}^{2}\right)}{i}\right) \cdot n\\

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

\end{array}
double f(double i, double n) {
        double r141723 = 100.0;
        double r141724 = 1.0;
        double r141725 = i;
        double r141726 = n;
        double r141727 = r141725 / r141726;
        double r141728 = r141724 + r141727;
        double r141729 = pow(r141728, r141726);
        double r141730 = r141729 - r141724;
        double r141731 = r141730 / r141727;
        double r141732 = r141723 * r141731;
        return r141732;
}
↓
double f(double i, double n) {
        double r141733 = n;
        double r141734 = -6.142887410522618e+122;
        bool r141735 = r141733 <= r141734;
        double r141736 = 100.0;
        double r141737 = 1.0;
        double r141738 = log(r141737);
        double r141739 = i;
        double r141740 = 0.5;
        double r141741 = 2.0;
        double r141742 = pow(r141739, r141741);
        double r141743 = r141740 * r141742;
        double r141744 = fma(r141737, r141739, r141743);
        double r141745 = fma(r141738, r141733, r141744);
        double r141746 = r141738 * r141742;
        double r141747 = r141740 * r141746;
        double r141748 = r141745 - r141747;
        double r141749 = r141748 / r141739;
        double r141750 = r141736 * r141749;
        double r141751 = r141750 * r141733;
        double r141752 = -8.23875326441092e+106;
        bool r141753 = r141733 <= r141752;
        double r141754 = r141739 / r141733;
        double r141755 = r141737 + r141754;
        double r141756 = r141741 * r141733;
        double r141757 = pow(r141755, r141756);
        double r141758 = r141737 * r141737;
        double r141759 = r141757 - r141758;
        double r141760 = pow(r141755, r141733);
        double r141761 = r141760 + r141737;
        double r141762 = r141759 / r141761;
        double r141763 = r141762 / r141754;
        double r141764 = r141736 * r141763;
        double r141765 = -1.8594290872480363;
        bool r141766 = r141733 <= r141765;
        double r141767 = 1.4183297845116364e-154;
        bool r141768 = r141733 <= r141767;
        double r141769 = !r141768;
        bool r141770 = r141766 || r141769;
        double r141771 = 1.0;
        double r141772 = fma(r141737, r141739, r141771);
        double r141773 = fma(r141738, r141733, r141772);
        double r141774 = r141773 - r141737;
        double r141775 = r141774 / r141754;
        double r141776 = r141736 * r141775;
        double r141777 = r141770 ? r141751 : r141776;
        double r141778 = r141753 ? r141764 : r141777;
        double r141779 = r141735 ? r141751 : r141778;
        return r141779;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.8
Target42.8
Herbie23.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if n < -6.142887410522618e+122 or -8.23875326441092e+106 < n < -1.8594290872480363 or 1.4183297845116364e-154 < n

    1. Initial program 52.8

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

    2. Taylor expanded around 0 39.5

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

    3. Simplified39.5

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

    5. Applied associate-/r/22.5

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

    6. Applied associate-*r*22.5

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

    if -6.142887410522618e+122 < n < -8.23875326441092e+106

    1. Initial program 40.4

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

    3. Applied flip--40.4

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

    4. Simplified40.4

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

    if -1.8594290872480363 < n < 1.4183297845116364e-154

    1. Initial program 23.6

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

    2. Taylor expanded around 0 24.6

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

    3. Simplified24.6

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

  4. Final simplification23.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -6.1428874105226178 \cdot 10^{122}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(1, i, 0.5 \cdot {i}^{2}\right)\right) - 0.5 \cdot \left(\log 1 \cdot {i}^{2}\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -8.23875326441091999 \cdot 10^{106}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -1.8594290872480363 \lor \neg \left(n \le 1.41832978451163636 \cdot 10^{-154}\right):\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(1, i, 0.5 \cdot {i}^{2}\right)\right) - 0.5 \cdot \left(\log 1 \cdot {i}^{2}\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(1, i, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (i n)
  :name "Compound Interest"

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))