Average Error: 47.1 → 16.9
Time: 12.1s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.0390445009989891442:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 22.1533166385413587:\\ \;\;\;\;\mathsf{fma}\left(i, 50, \mathsf{fma}\left(100, \frac{\log 1 \cdot n}{i}, 100\right) - 50 \cdot \left(i \cdot \log 1\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.0390445009989891442:\\
\;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\

\mathbf{elif}\;i \le 22.1533166385413587:\\
\;\;\;\;\mathsf{fma}\left(i, 50, \mathsf{fma}\left(100, \frac{\log 1 \cdot n}{i}, 100\right) - 50 \cdot \left(i \cdot \log 1\right)\right) \cdot n\\

\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 r125602 = 100.0;
        double r125603 = 1.0;
        double r125604 = i;
        double r125605 = n;
        double r125606 = r125604 / r125605;
        double r125607 = r125603 + r125606;
        double r125608 = pow(r125607, r125605);
        double r125609 = r125608 - r125603;
        double r125610 = r125609 / r125606;
        double r125611 = r125602 * r125610;
        return r125611;
}
↓
double f(double i, double n) {
        double r125612 = i;
        double r125613 = -0.039044500998989144;
        bool r125614 = r125612 <= r125613;
        double r125615 = 100.0;
        double r125616 = r125615 / r125612;
        double r125617 = 1.0;
        double r125618 = n;
        double r125619 = r125612 / r125618;
        double r125620 = r125617 + r125619;
        double r125621 = pow(r125620, r125618);
        double r125622 = r125621 - r125617;
        double r125623 = 1.0;
        double r125624 = r125623 / r125618;
        double r125625 = r125622 / r125624;
        double r125626 = r125616 * r125625;
        double r125627 = 22.15331663854136;
        bool r125628 = r125612 <= r125627;
        double r125629 = 50.0;
        double r125630 = log(r125617);
        double r125631 = r125630 * r125618;
        double r125632 = r125631 / r125612;
        double r125633 = fma(r125615, r125632, r125615);
        double r125634 = r125612 * r125630;
        double r125635 = r125629 * r125634;
        double r125636 = r125633 - r125635;
        double r125637 = fma(r125612, r125629, r125636);
        double r125638 = r125637 * r125618;
        double r125639 = 2.0;
        double r125640 = r125639 * r125618;
        double r125641 = pow(r125620, r125640);
        double r125642 = r125617 * r125617;
        double r125643 = -r125642;
        double r125644 = r125641 + r125643;
        double r125645 = r125621 + r125617;
        double r125646 = r125644 / r125645;
        double r125647 = r125646 / r125619;
        double r125648 = r125615 * r125647;
        double r125649 = r125628 ? r125638 : r125648;
        double r125650 = r125614 ? r125626 : r125649;
        return r125650;
}

Error

Bits error versus i

Bits error versus n

Target

Original47.1
Target46.8
Herbie16.9
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -0.039044500998989144

    1. Initial program 27.3

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

    3. Applied div-inv27.3

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

    4. Applied *-un-lft-identity27.3

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

    5. Applied times-frac27.8

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

    6. Applied associate-*r*27.8

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

    7. Simplified27.8

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

    if -0.039044500998989144 < i < 22.15331663854136

    1. Initial program 57.8

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

    2. Taylor expanded around 0 26.8

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

    3. Simplified26.8

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

    5. Applied associate-/r/9.8

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

    6. Applied associate-*r*9.8

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

    7. Taylor expanded around 0 9.8

      \[\leadsto \color{blue}{\left(\left(100 + \left(50 \cdot i + 100 \cdot \frac{\log 1 \cdot n}{i}\right)\right) - 50 \cdot \left(i \cdot \log 1\right)\right)} \cdot n\]

    8. Simplified9.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, 50, \mathsf{fma}\left(100, \frac{\log 1 \cdot n}{i}, 100\right) - 50 \cdot \left(i \cdot \log 1\right)\right)} \cdot n\]

    if 22.15331663854136 < i

    1. Initial program 31.6

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

    3. Applied flip--31.6

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

    4. Simplified31.6

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

  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.0390445009989891442:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 22.1533166385413587:\\ \;\;\;\;\mathsf{fma}\left(i, 50, \mathsf{fma}\left(100, \frac{\log 1 \cdot n}{i}, 100\right) - 50 \cdot \left(i \cdot \log 1\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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))))