Average Error: 42.8 → 19.1
Time: 53.7s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -5.163360055787723405136702348322010579229 \cdot 10^{-11}:\\ \;\;\;\;\frac{{\left(\frac{1}{n} \cdot i\right)}^{n} \cdot 100 - 100}{i} \cdot n\\ \mathbf{elif}\;i \le 6.305809243265488292923253978118289242438 \cdot 10^{-10}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \left(\left(\log 1 \cdot \left(n - i \cdot \left(i \cdot 0.5\right)\right) + i \cdot \left(i \cdot 0.5\right)\right) + 1 \cdot i\right)\right)}{i}\\ \mathbf{elif}\;i \le 2.565130334196551364557031790272495063197 \cdot 10^{190}:\\ \;\;\;\;\frac{{\left(\frac{1}{n} \cdot i\right)}^{n} \cdot 100 - 100}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(1 + n \cdot \log 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -5.163360055787723405136702348322010579229 \cdot 10^{-11}:\\
\;\;\;\;\frac{{\left(\frac{1}{n} \cdot i\right)}^{n} \cdot 100 - 100}{i} \cdot n\\

\mathbf{elif}\;i \le 6.305809243265488292923253978118289242438 \cdot 10^{-10}:\\
\;\;\;\;\frac{100 \cdot \left(n \cdot \left(\left(\log 1 \cdot \left(n - i \cdot \left(i \cdot 0.5\right)\right) + i \cdot \left(i \cdot 0.5\right)\right) + 1 \cdot i\right)\right)}{i}\\

\mathbf{elif}\;i \le 2.565130334196551364557031790272495063197 \cdot 10^{190}:\\
\;\;\;\;\frac{{\left(\frac{1}{n} \cdot i\right)}^{n} \cdot 100 - 100}{i} \cdot n\\

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

\end{array}
double f(double i, double n) {
        double r7113982 = 100.0;
        double r7113983 = 1.0;
        double r7113984 = i;
        double r7113985 = n;
        double r7113986 = r7113984 / r7113985;
        double r7113987 = r7113983 + r7113986;
        double r7113988 = pow(r7113987, r7113985);
        double r7113989 = r7113988 - r7113983;
        double r7113990 = r7113989 / r7113986;
        double r7113991 = r7113982 * r7113990;
        return r7113991;
}
↓
double f(double i, double n) {
        double r7113992 = i;
        double r7113993 = -5.1633600557877234e-11;
        bool r7113994 = r7113992 <= r7113993;
        double r7113995 = 1.0;
        double r7113996 = n;
        double r7113997 = r7113995 / r7113996;
        double r7113998 = r7113997 * r7113992;
        double r7113999 = pow(r7113998, r7113996);
        double r7114000 = 100.0;
        double r7114001 = r7113999 * r7114000;
        double r7114002 = r7114001 - r7114000;
        double r7114003 = r7114002 / r7113992;
        double r7114004 = r7114003 * r7113996;
        double r7114005 = 6.305809243265488e-10;
        bool r7114006 = r7113992 <= r7114005;
        double r7114007 = 1.0;
        double r7114008 = log(r7114007);
        double r7114009 = 0.5;
        double r7114010 = r7113992 * r7114009;
        double r7114011 = r7113992 * r7114010;
        double r7114012 = r7113996 - r7114011;
        double r7114013 = r7114008 * r7114012;
        double r7114014 = r7114013 + r7114011;
        double r7114015 = r7114007 * r7113992;
        double r7114016 = r7114014 + r7114015;
        double r7114017 = r7113996 * r7114016;
        double r7114018 = r7114000 * r7114017;
        double r7114019 = r7114018 / r7113992;
        double r7114020 = 2.5651303341965514e+190;
        bool r7114021 = r7113992 <= r7114020;
        double r7114022 = r7113996 * r7114008;
        double r7114023 = r7113995 + r7114022;
        double r7114024 = r7114015 + r7114023;
        double r7114025 = r7114024 - r7114007;
        double r7114026 = r7113992 / r7113996;
        double r7114027 = r7114025 / r7114026;
        double r7114028 = r7114000 * r7114027;
        double r7114029 = r7114021 ? r7114004 : r7114028;
        double r7114030 = r7114006 ? r7114019 : r7114029;
        double r7114031 = r7113994 ? r7114004 : r7114030;
        return r7114031;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.8
Target42.7
Herbie19.1
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -5.1633600557877234e-11 or 6.305809243265488e-10 < i < 2.5651303341965514e+190

    1. Initial program 29.3

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

    3. Applied associate-/r/29.8

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

    4. Applied associate-*r*29.8

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

    5. Taylor expanded around inf 55.4

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

    6. Simplified23.4

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

    if -5.1633600557877234e-11 < i < 6.305809243265488e-10

    1. Initial program 50.7

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

    3. Applied associate-/r/50.4

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

    4. Applied associate-*r*50.4

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

    5. Taylor expanded around 0 16.3

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

    6. Simplified16.3

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

    8. Applied associate-*l*16.3

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

    10. Applied associate-*l/15.3

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

    11. Applied associate-*r/15.5

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

    if 2.5651303341965514e+190 < i

    1. Initial program 33.1

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

    2. Taylor expanded around 0 33.3

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

  4. Final simplification19.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -5.163360055787723405136702348322010579229 \cdot 10^{-11}:\\ \;\;\;\;\frac{{\left(\frac{1}{n} \cdot i\right)}^{n} \cdot 100 - 100}{i} \cdot n\\ \mathbf{elif}\;i \le 6.305809243265488292923253978118289242438 \cdot 10^{-10}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \left(\left(\log 1 \cdot \left(n - i \cdot \left(i \cdot 0.5\right)\right) + i \cdot \left(i \cdot 0.5\right)\right) + 1 \cdot i\right)\right)}{i}\\ \mathbf{elif}\;i \le 2.565130334196551364557031790272495063197 \cdot 10^{190}:\\ \;\;\;\;\frac{{\left(\frac{1}{n} \cdot i\right)}^{n} \cdot 100 - 100}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(1 + n \cdot \log 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(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))))