Average Error: 47.3 → 17.3
Time: 15.3s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.17879906071750874 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 8.74609462986419547 \cdot 10^{-16}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -1.17879906071750874 \cdot 10^{-7}:\\
\;\;\;\;100 \cdot \frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right)}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 8.74609462986419547 \cdot 10^{-16}:\\
\;\;\;\;100 \cdot \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)\\

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

\end{array}
double f(double i, double n) {
        double r155693 = 100.0;
        double r155694 = 1.0;
        double r155695 = i;
        double r155696 = n;
        double r155697 = r155695 / r155696;
        double r155698 = r155694 + r155697;
        double r155699 = pow(r155698, r155696);
        double r155700 = r155699 - r155694;
        double r155701 = r155700 / r155697;
        double r155702 = r155693 * r155701;
        return r155702;
}
↓
double f(double i, double n) {
        double r155703 = i;
        double r155704 = -1.1787990607175087e-07;
        bool r155705 = r155703 <= r155704;
        double r155706 = 100.0;
        double r155707 = 1.0;
        double r155708 = n;
        double r155709 = r155703 / r155708;
        double r155710 = r155707 + r155709;
        double r155711 = pow(r155710, r155708);
        double r155712 = r155711 - r155707;
        double r155713 = exp(r155712);
        double r155714 = log(r155713);
        double r155715 = r155714 / r155709;
        double r155716 = r155706 * r155715;
        double r155717 = 8.746094629864195e-16;
        bool r155718 = r155703 <= r155717;
        double r155719 = 0.5;
        double r155720 = 2.0;
        double r155721 = pow(r155703, r155720);
        double r155722 = log(r155707);
        double r155723 = r155722 * r155708;
        double r155724 = fma(r155719, r155721, r155723);
        double r155725 = r155721 * r155722;
        double r155726 = r155719 * r155725;
        double r155727 = r155724 - r155726;
        double r155728 = fma(r155703, r155707, r155727);
        double r155729 = r155728 / r155703;
        double r155730 = r155729 * r155708;
        double r155731 = r155706 * r155730;
        double r155732 = r155706 / r155703;
        double r155733 = 1.0;
        double r155734 = r155733 / r155708;
        double r155735 = r155712 / r155734;
        double r155736 = r155732 * r155735;
        double r155737 = r155718 ? r155731 : r155736;
        double r155738 = r155705 ? r155716 : r155737;
        return r155738;
}

Error

Bits error versus i

Bits error versus n

Target

Original47.3
Target47.0
Herbie17.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -1.1787990607175087e-07

    1. Initial program 29.1

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

    3. Applied add-log-exp29.1

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

    4. Applied add-log-exp29.1

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

    5. Applied diff-log29.1

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

    6. Simplified29.1

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

    if -1.1787990607175087e-07 < i < 8.746094629864195e-16

    1. Initial program 57.8

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

    2. Taylor expanded around 0 27.1

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

    3. Simplified27.1

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

    5. Applied associate-/r/9.2

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

    if 8.746094629864195e-16 < i

    1. Initial program 32.5

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

    3. Applied div-inv32.6

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

    4. Applied *-un-lft-identity32.6

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

    5. Applied times-frac32.6

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

    6. Applied associate-*r*32.6

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

    7. Simplified32.5

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

  4. Final simplification17.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.17879906071750874 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 8.74609462986419547 \cdot 10^{-16}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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))))