Average Error: 42.0 → 21.3
Time: 37.6s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.9324505473046543 \cdot 10^{-07}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 4.5457902375689345:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i + \left(i \cdot \left(\left(\left(\left(\sqrt[3]{\frac{1}{6}}\right)\right) \cdot \left(i \cdot i\right)\right) \cdot \left(\sqrt[3]{\frac{1}{6}} \cdot \sqrt[3]{\frac{1}{6}}\right)\right) + \left(i \cdot i\right) \cdot \frac{1}{2}\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{1}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -1.9324505473046543 \cdot 10^{-07}:\\
\;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;i \le 4.5457902375689345:\\
\;\;\;\;100 \cdot \left(n \cdot \frac{i + \left(i \cdot \left(\left(\left(\left(\sqrt[3]{\frac{1}{6}}\right)\right) \cdot \left(i \cdot i\right)\right) \cdot \left(\sqrt[3]{\frac{1}{6}} \cdot \sqrt[3]{\frac{1}{6}}\right)\right) + \left(i \cdot i\right) \cdot \frac{1}{2}\right)}{i}\right)\\

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

\end{array}
double f(double i, double n) {
        double r5562651 = 100.0;
        double r5562652 = 1.0;
        double r5562653 = i;
        double r5562654 = n;
        double r5562655 = r5562653 / r5562654;
        double r5562656 = r5562652 + r5562655;
        double r5562657 = pow(r5562656, r5562654);
        double r5562658 = r5562657 - r5562652;
        double r5562659 = r5562658 / r5562655;
        double r5562660 = r5562651 * r5562659;
        return r5562660;
}
↓
double f(double i, double n) {
        double r5562661 = i;
        double r5562662 = -1.9324505473046543e-07;
        bool r5562663 = r5562661 <= r5562662;
        double r5562664 = 100.0;
        double r5562665 = n;
        double r5562666 = r5562661 / r5562665;
        double r5562667 = 1.0;
        double r5562668 = r5562666 + r5562667;
        double r5562669 = pow(r5562668, r5562665);
        double r5562670 = r5562669 / r5562666;
        double r5562671 = r5562667 / r5562666;
        double r5562672 = r5562670 - r5562671;
        double r5562673 = r5562664 * r5562672;
        double r5562674 = 4.5457902375689345;
        bool r5562675 = r5562661 <= r5562674;
        double r5562676 = 0.16666666666666666;
        double r5562677 = cbrt(r5562676);
        double r5562678 = /* ERROR: no posit support in C */;
        double r5562679 = /* ERROR: no posit support in C */;
        double r5562680 = r5562661 * r5562661;
        double r5562681 = r5562679 * r5562680;
        double r5562682 = r5562677 * r5562677;
        double r5562683 = r5562681 * r5562682;
        double r5562684 = r5562661 * r5562683;
        double r5562685 = 0.5;
        double r5562686 = r5562680 * r5562685;
        double r5562687 = r5562684 + r5562686;
        double r5562688 = r5562661 + r5562687;
        double r5562689 = r5562688 / r5562661;
        double r5562690 = r5562665 * r5562689;
        double r5562691 = r5562664 * r5562690;
        double r5562692 = r5562664 / r5562661;
        double r5562693 = r5562669 - r5562667;
        double r5562694 = r5562667 / r5562665;
        double r5562695 = r5562693 / r5562694;
        double r5562696 = r5562692 * r5562695;
        double r5562697 = r5562675 ? r5562691 : r5562696;
        double r5562698 = r5562663 ? r5562673 : r5562697;
        return r5562698;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.0
Target42.3
Herbie21.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -1.9324505473046543e-07

    1. Initial program 27.4

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

    3. Applied div-sub27.4

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

    if -1.9324505473046543e-07 < i < 4.5457902375689345

    1. Initial program 50.1

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

    2. Taylor expanded around 0 33.0

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(\frac{1}{2} \cdot {i}^{2} + \frac{1}{6} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]

    3. Simplified33.0

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \left(\frac{1}{6} \cdot \left(i \cdot i\right)\right) \cdot i\right) + i}}{\frac{i}{n}}\]
    4. Using strategy rm

    5. Applied associate-/r/16.9

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \left(\frac{1}{6} \cdot \left(i \cdot i\right)\right) \cdot i\right) + i}{i} \cdot n\right)}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt16.9

      \[\leadsto 100 \cdot \left(\frac{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \left(\color{blue}{\left(\left(\sqrt[3]{\frac{1}{6}} \cdot \sqrt[3]{\frac{1}{6}}\right) \cdot \sqrt[3]{\frac{1}{6}}\right)} \cdot \left(i \cdot i\right)\right) \cdot i\right) + i}{i} \cdot n\right)\]

    8. Applied associate-*l*16.9

      \[\leadsto 100 \cdot \left(\frac{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \color{blue}{\left(\left(\sqrt[3]{\frac{1}{6}} \cdot \sqrt[3]{\frac{1}{6}}\right) \cdot \left(\sqrt[3]{\frac{1}{6}} \cdot \left(i \cdot i\right)\right)\right)} \cdot i\right) + i}{i} \cdot n\right)\]
    9. Using strategy rm

    10. Applied insert-posit1616.9

      \[\leadsto 100 \cdot \left(\frac{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \left(\left(\sqrt[3]{\frac{1}{6}} \cdot \sqrt[3]{\frac{1}{6}}\right) \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\frac{1}{6}}\right)\right)} \cdot \left(i \cdot i\right)\right)\right) \cdot i\right) + i}{i} \cdot n\right)\]

    if 4.5457902375689345 < i

    1. Initial program 31.0

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

    3. Applied div-inv31.1

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

    4. Applied *-un-lft-identity31.1

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

    5. Applied times-frac31.1

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

    6. Applied associate-*r*31.1

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

    7. Simplified31.0

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

  4. Final simplification21.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.9324505473046543 \cdot 10^{-07}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 4.5457902375689345:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i + \left(i \cdot \left(\left(\left(\left(\sqrt[3]{\frac{1}{6}}\right)\right) \cdot \left(i \cdot i\right)\right) \cdot \left(\sqrt[3]{\frac{1}{6}} \cdot \sqrt[3]{\frac{1}{6}}\right)\right) + \left(i \cdot i\right) \cdot \frac{1}{2}\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{1}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (i n)
  :name "Compound Interest"

  :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))))