Average Error: 42.6 → 21.6
Time: 40.3s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.77715864264781547 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.88216887028242915 \cdot 10^{-18}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\frac{\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\sqrt[3]{\sqrt[3]{i} \cdot \sqrt[3]{i}}} \cdot \frac{\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\sqrt[3]{\sqrt[3]{i}}}}}\\ \mathbf{elif}\;i \le 2.1606840039657686 \cdot 10^{162}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)}}{i}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -1.77715864264781547 \cdot 10^{-6}:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 2.88216887028242915 \cdot 10^{-18}:\\
\;\;\;\;100 \cdot \frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\frac{\sqrt[3]{\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)} \cdot \sqrt[3]{\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)}}{\sqrt[3]{\sqrt[3]{i} \cdot \sqrt[3]{i}}} \cdot \frac{\sqrt[3]{\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)}}{\sqrt[3]{\sqrt[3]{i}}}}}\\

\mathbf{elif}\;i \le 2.1606840039657686 \cdot 10^{162}:\\
\;\;\;\;100 \cdot \frac{\frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)}}{i}}{\frac{1}{n}}\\

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

\end{array}
double f(double i, double n) {
        double r227977 = 100.0;
        double r227978 = 1.0;
        double r227979 = i;
        double r227980 = n;
        double r227981 = r227979 / r227980;
        double r227982 = r227978 + r227981;
        double r227983 = pow(r227982, r227980);
        double r227984 = r227983 - r227978;
        double r227985 = r227984 / r227981;
        double r227986 = r227977 * r227985;
        return r227986;
}
↓
double f(double i, double n) {
        double r227987 = i;
        double r227988 = -1.7771586426478155e-06;
        bool r227989 = r227987 <= r227988;
        double r227990 = 100.0;
        double r227991 = 1.0;
        double r227992 = n;
        double r227993 = r227987 / r227992;
        double r227994 = r227991 + r227993;
        double r227995 = 2.0;
        double r227996 = r227995 * r227992;
        double r227997 = pow(r227994, r227996);
        double r227998 = r227991 * r227991;
        double r227999 = r227997 - r227998;
        double r228000 = pow(r227994, r227992);
        double r228001 = r228000 + r227991;
        double r228002 = r227999 / r228001;
        double r228003 = r228002 / r227993;
        double r228004 = r227990 * r228003;
        double r228005 = 2.882168870282429e-18;
        bool r228006 = r227987 <= r228005;
        double r228007 = 1.0;
        double r228008 = cbrt(r227987);
        double r228009 = r228008 * r228008;
        double r228010 = r228007 / r228009;
        double r228011 = r228007 / r227992;
        double r228012 = r227991 * r227987;
        double r228013 = 0.5;
        double r228014 = pow(r227987, r227995);
        double r228015 = r228013 * r228014;
        double r228016 = log(r227991);
        double r228017 = r228016 * r227992;
        double r228018 = r228015 + r228017;
        double r228019 = r228012 + r228018;
        double r228020 = r228014 * r228016;
        double r228021 = r228013 * r228020;
        double r228022 = r228019 - r228021;
        double r228023 = cbrt(r228022);
        double r228024 = r228023 * r228023;
        double r228025 = cbrt(r228009);
        double r228026 = r228024 / r228025;
        double r228027 = cbrt(r228008);
        double r228028 = r228023 / r228027;
        double r228029 = r228026 * r228028;
        double r228030 = r228011 / r228029;
        double r228031 = r228010 / r228030;
        double r228032 = r227990 * r228031;
        double r228033 = 2.1606840039657686e+162;
        bool r228034 = r227987 <= r228033;
        double r228035 = 3.0;
        double r228036 = pow(r228000, r228035);
        double r228037 = pow(r227991, r228035);
        double r228038 = r228036 - r228037;
        double r228039 = r227991 * r228001;
        double r228040 = r227997 + r228039;
        double r228041 = r228038 / r228040;
        double r228042 = r228041 / r227987;
        double r228043 = r228042 / r228011;
        double r228044 = r227990 * r228043;
        double r228045 = r228017 + r228007;
        double r228046 = r228012 + r228045;
        double r228047 = r228046 - r227991;
        double r228048 = r228047 / r227993;
        double r228049 = r227990 * r228048;
        double r228050 = r228034 ? r228044 : r228049;
        double r228051 = r228006 ? r228032 : r228050;
        double r228052 = r227989 ? r228004 : r228051;
        return r228052;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.6
Target42.5
Herbie21.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -1.7771586426478155e-06

    1. Initial program 27.7

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

    3. Applied flip--27.7

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

    4. Simplified27.7

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

    if -1.7771586426478155e-06 < i < 2.882168870282429e-18

    1. Initial program 50.5

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

    3. Applied div-inv50.5

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

    4. Applied associate-/r*50.2

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

    5. Taylor expanded around 0 16.7

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

    7. Applied add-cube-cbrt17.6

      \[\leadsto 100 \cdot \frac{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\color{blue}{\left(\sqrt[3]{i} \cdot \sqrt[3]{i}\right) \cdot \sqrt[3]{i}}}}{\frac{1}{n}}\]

    8. Applied *-un-lft-identity17.6

      \[\leadsto 100 \cdot \frac{\frac{\color{blue}{1 \cdot \left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}}{\left(\sqrt[3]{i} \cdot \sqrt[3]{i}\right) \cdot \sqrt[3]{i}}}{\frac{1}{n}}\]

    9. Applied times-frac17.6

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\sqrt[3]{i}}}}{\frac{1}{n}}\]

    10. Applied associate-/l*16.1

      \[\leadsto 100 \cdot \color{blue}{\frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\sqrt[3]{i}}}}}\]
    11. Using strategy rm

    12. Applied add-cube-cbrt16.2

      \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{i} \cdot \sqrt[3]{i}\right) \cdot \sqrt[3]{i}}}}}}\]

    13. Applied cbrt-prod16.3

      \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\color{blue}{\sqrt[3]{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \sqrt[3]{\sqrt[3]{i}}}}}}\]

    14. Applied add-cube-cbrt15.8

      \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\frac{\color{blue}{\left(\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}\right) \cdot \sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}}{\sqrt[3]{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \sqrt[3]{\sqrt[3]{i}}}}}\]

    15. Applied times-frac15.8

      \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\color{blue}{\frac{\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\sqrt[3]{\sqrt[3]{i} \cdot \sqrt[3]{i}}} \cdot \frac{\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\sqrt[3]{\sqrt[3]{i}}}}}}\]

    if 2.882168870282429e-18 < i < 2.1606840039657686e+162

    1. Initial program 36.1

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

    3. Applied div-inv36.1

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

    4. Applied associate-/r*36.1

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

    6. Applied flip3--36.1

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

    7. Simplified36.1

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

    if 2.1606840039657686e+162 < i

    1. Initial program 31.9

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

    2. Taylor expanded around 0 35.5

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

  4. Final simplification21.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.77715864264781547 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.88216887028242915 \cdot 10^{-18}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\frac{\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\sqrt[3]{\sqrt[3]{i} \cdot \sqrt[3]{i}}} \cdot \frac{\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\sqrt[3]{\sqrt[3]{i}}}}}\\ \mathbf{elif}\;i \le 2.1606840039657686 \cdot 10^{162}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)}}{i}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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))))