Average Error: 43.4 → 20.5
Time: 23.1s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.9893212200252493593310987307631876319647:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.508652436303369411416630768794228112384 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \left(\left(\left(n - \left(i \cdot 0.5\right) \cdot i\right) \cdot \log 1 + i \cdot \left(i \cdot 0.5 + 1\right)\right) \cdot n\right)\right)\\ \mathbf{elif}\;i \le 8.608246042115979009006453659316669705257 \cdot 10^{235}:\\ \;\;\;\;\frac{\frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - {1}^{3}}{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot 1}}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;i \le 1.844738975002478002763642496553644874398 \cdot 10^{296}:\\ \;\;\;\;\frac{\left(\left(1 + i \cdot 1\right) + n \cdot \log 1\right) - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.9893212200252493593310987307631876319647:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 1.508652436303369411416630768794228112384 \cdot 10^{-9}:\\
\;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \left(\left(\left(n - \left(i \cdot 0.5\right) \cdot i\right) \cdot \log 1 + i \cdot \left(i \cdot 0.5 + 1\right)\right) \cdot n\right)\right)\\

\mathbf{elif}\;i \le 8.608246042115979009006453659316669705257 \cdot 10^{235}:\\
\;\;\;\;\frac{\frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - {1}^{3}}{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot 1}}{\frac{i}{n}} \cdot 100\\

\mathbf{elif}\;i \le 1.844738975002478002763642496553644874398 \cdot 10^{296}:\\
\;\;\;\;\frac{\left(\left(1 + i \cdot 1\right) + n \cdot \log 1\right) - 1}{\frac{i}{n}} \cdot 100\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\

\end{array}
double f(double i, double n) {
        double r118669 = 100.0;
        double r118670 = 1.0;
        double r118671 = i;
        double r118672 = n;
        double r118673 = r118671 / r118672;
        double r118674 = r118670 + r118673;
        double r118675 = pow(r118674, r118672);
        double r118676 = r118675 - r118670;
        double r118677 = r118676 / r118673;
        double r118678 = r118669 * r118677;
        return r118678;
}
↓
double f(double i, double n) {
        double r118679 = i;
        double r118680 = -0.9893212200252494;
        bool r118681 = r118679 <= r118680;
        double r118682 = 100.0;
        double r118683 = n;
        double r118684 = r118679 / r118683;
        double r118685 = 1.0;
        double r118686 = r118684 + r118685;
        double r118687 = 2.0;
        double r118688 = r118683 * r118687;
        double r118689 = pow(r118686, r118688);
        double r118690 = r118685 * r118685;
        double r118691 = r118689 - r118690;
        double r118692 = pow(r118686, r118683);
        double r118693 = r118685 + r118692;
        double r118694 = r118691 / r118693;
        double r118695 = r118694 / r118684;
        double r118696 = r118682 * r118695;
        double r118697 = 1.5086524363033694e-09;
        bool r118698 = r118679 <= r118697;
        double r118699 = 1.0;
        double r118700 = r118699 / r118679;
        double r118701 = 0.5;
        double r118702 = r118679 * r118701;
        double r118703 = r118702 * r118679;
        double r118704 = r118683 - r118703;
        double r118705 = log(r118685);
        double r118706 = r118704 * r118705;
        double r118707 = r118702 + r118685;
        double r118708 = r118679 * r118707;
        double r118709 = r118706 + r118708;
        double r118710 = r118709 * r118683;
        double r118711 = r118700 * r118710;
        double r118712 = r118682 * r118711;
        double r118713 = 8.608246042115979e+235;
        bool r118714 = r118679 <= r118713;
        double r118715 = 3.0;
        double r118716 = pow(r118692, r118715);
        double r118717 = pow(r118685, r118715);
        double r118718 = r118716 - r118717;
        double r118719 = r118693 * r118685;
        double r118720 = r118689 + r118719;
        double r118721 = r118718 / r118720;
        double r118722 = r118721 / r118684;
        double r118723 = r118722 * r118682;
        double r118724 = 1.844738975002478e+296;
        bool r118725 = r118679 <= r118724;
        double r118726 = r118679 * r118685;
        double r118727 = r118699 + r118726;
        double r118728 = r118683 * r118705;
        double r118729 = r118727 + r118728;
        double r118730 = r118729 - r118685;
        double r118731 = r118730 / r118684;
        double r118732 = r118731 * r118682;
        double r118733 = r118725 ? r118732 : r118696;
        double r118734 = r118714 ? r118723 : r118733;
        double r118735 = r118698 ? r118712 : r118734;
        double r118736 = r118681 ? r118696 : r118735;
        return r118736;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.4
Target43.3
Herbie20.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -0.9893212200252494 or 1.844738975002478e+296 < i

    1. Initial program 27.9

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

    3. Applied flip--27.9

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

    4. Simplified27.9

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

    5. Simplified27.9

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

    if -0.9893212200252494 < i < 1.5086524363033694e-09

    1. Initial program 51.2

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

    2. Taylor expanded around 0 32.6

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

    3. Simplified32.6

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

    5. Applied div-inv32.7

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

    6. Applied *-un-lft-identity32.7

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

    7. Applied times-frac15.1

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

    8. Simplified15.1

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

    if 1.5086524363033694e-09 < i < 8.608246042115979e+235

    1. Initial program 33.3

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

    3. Applied flip3--33.3

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

    4. Simplified33.3

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

    5. Simplified33.3

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

    if 8.608246042115979e+235 < i < 1.844738975002478e+296

    1. Initial program 32.1

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

    2. Taylor expanded around 0 34.0

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

    3. Simplified34.0

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(\left(1 \cdot i + 1\right) + n \cdot \log 1\right)} - 1}{\frac{i}{n}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification20.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.9893212200252493593310987307631876319647:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.508652436303369411416630768794228112384 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \left(\left(\left(n - \left(i \cdot 0.5\right) \cdot i\right) \cdot \log 1 + i \cdot \left(i \cdot 0.5 + 1\right)\right) \cdot n\right)\right)\\ \mathbf{elif}\;i \le 8.608246042115979009006453659316669705257 \cdot 10^{235}:\\ \;\;\;\;\frac{\frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - {1}^{3}}{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot 1}}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;i \le 1.844738975002478002763642496553644874398 \cdot 10^{296}:\\ \;\;\;\;\frac{\left(\left(1 + i \cdot 1\right) + n \cdot \log 1\right) - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(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))))