Compound Interest

Average Error: 42.7 → 31.4

Time: 28.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -2.56140166535216897 \cdot 10^{135}:\\ \;\;\;\;\frac{100 \cdot \frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.3992561866449662 \cdot 10^{-10}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.51591290926460688 \cdot 10^{-160}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 6.02622510223326963 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} - \left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{\frac{\frac{\sqrt[3]{i} \cdot \sqrt[3]{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}}{100}}}{\frac{\sqrt[3]{i}}{\sqrt[3]{n}}}\\ \mathbf{elif}\;i \le 8532543483832934860000:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 7.39555913873958208 \cdot 10^{219}:\\ \;\;\;\;\frac{\frac{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{i}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r230979 = 100.0;
        double r230980 = 1.0;
        double r230981 = i;
        double r230982 = n;
        double r230983 = r230981 / r230982;
        double r230984 = r230980 + r230983;
        double r230985 = pow(r230984, r230982);
        double r230986 = r230985 - r230980;
        double r230987 = r230986 / r230983;
        double r230988 = r230979 * r230987;
        return r230988;
}

↓

double f(double i, double n) {
        double r230989 = i;
        double r230990 = -2.561401665352169e+135;
        bool r230991 = r230989 <= r230990;
        double r230992 = 100.0;
        double r230993 = 1.0;
        double r230994 = n;
        double r230995 = r230989 / r230994;
        double r230996 = r230993 + r230995;
        double r230997 = 2.0;
        double r230998 = r230997 * r230994;
        double r230999 = pow(r230996, r230998);
        double r231000 = r230993 * r230993;
        double r231001 = r230999 - r231000;
        double r231002 = exp(r231001);
        double r231003 = log(r231002);
        double r231004 = pow(r230996, r230994);
        double r231005 = r231004 + r230993;
        double r231006 = r231003 / r231005;
        double r231007 = r230992 * r231006;
        double r231008 = r231007 / r230995;
        double r231009 = -1.3992561866449662e-10;
        bool r231010 = r230989 <= r231009;
        double r231011 = pow(r230995, r230994);
        double r231012 = r231011 - r230993;
        double r231013 = r230992 * r231012;
        double r231014 = r231013 / r230995;
        double r231015 = 2.515912909264607e-160;
        bool r231016 = r230989 <= r231015;
        double r231017 = r230993 * r230989;
        double r231018 = 0.5;
        double r231019 = pow(r230989, r230997);
        double r231020 = r231018 * r231019;
        double r231021 = log(r230993);
        double r231022 = r231021 * r230994;
        double r231023 = r231020 + r231022;
        double r231024 = r231017 + r231023;
        double r231025 = r231019 * r231021;
        double r231026 = r231018 * r231025;
        double r231027 = r231024 - r231026;
        double r231028 = r231027 / r230995;
        double r231029 = r230992 * r231028;
        double r231030 = 6.02622510223327e-125;
        bool r231031 = r230989 <= r231030;
        double r231032 = r230997 * r230998;
        double r231033 = pow(r230996, r231032);
        double r231034 = r231000 * r231000;
        double r231035 = r231033 - r231034;
        double r231036 = r231035 / r231005;
        double r231037 = r230999 + r231000;
        double r231038 = r231036 / r231037;
        double r231039 = cbrt(r230989);
        double r231040 = r231039 * r231039;
        double r231041 = cbrt(r230994);
        double r231042 = r231041 * r231041;
        double r231043 = r231040 / r231042;
        double r231044 = r231043 / r230992;
        double r231045 = r231038 / r231044;
        double r231046 = r231039 / r231041;
        double r231047 = r231045 / r231046;
        double r231048 = 8.532543483832935e+21;
        bool r231049 = r230989 <= r231048;
        double r231050 = 7.395559138739582e+219;
        bool r231051 = r230989 <= r231050;
        double r231052 = -r231000;
        double r231053 = r230999 + r231052;
        double r231054 = r231053 / r231005;
        double r231055 = r230992 * r231054;
        double r231056 = r231055 / r230989;
        double r231057 = 1.0;
        double r231058 = r231057 / r230994;
        double r231059 = r231056 / r231058;
        double r231060 = r231022 + r231057;
        double r231061 = r231017 + r231060;
        double r231062 = r231061 - r230993;
        double r231063 = r231062 / r230995;
        double r231064 = r230992 * r231063;
        double r231065 = r231051 ? r231059 : r231064;
        double r231066 = r231049 ? r231029 : r231065;
        double r231067 = r231031 ? r231047 : r231066;
        double r231068 = r231016 ? r231029 : r231067;
        double r231069 = r231010 ? r231014 : r231068;
        double r231070 = r230991 ? r231008 : r231069;
        return r231070;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.7
Target	42.6
Herbie	31.4

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

Derivation

1. Split input into 6 regimes

2. if i < -2.561401665352169e+135

   1. Initial program 15.4

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

   3. Applied associate-*r/ 15.4

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

   5. Applied flip-- 15.4

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

   6. Simplified 15.4

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

   8. Applied add-log-exp 15.4

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

   9. Applied neg-log 15.4

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

   10. Applied add-log-exp 15.4

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

   11. Applied sum-log 15.4

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

   12. Simplified 15.4

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

   if -2.561401665352169e+135 < i < -1.3992561866449662e-10

   1. Initial program 41.3

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

   3. Applied associate-*r/ 41.3

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

   4. Taylor expanded around inf 64.0

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

   5. Simplified 27.6

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

   if -1.3992561866449662e-10 < i < 2.515912909264607e-160 or 6.02622510223327e-125 < i < 8.532543483832935e+21

   1. Initial program 49.9

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

   2. Taylor expanded around 0 33.7

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

   if 2.515912909264607e-160 < i < 6.02622510223327e-125

   1. Initial program 50.4

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

   3. Applied associate-*r/ 50.4

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

   5. Applied flip-- 50.4

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

   6. Simplified 50.4

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

   8. Applied flip-+ 50.4

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

   9. Simplified 50.4

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

   10. Simplified 50.4

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

   12. Applied add-cube-cbrt 50.4

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

   13. Applied add-cube-cbrt 50.4

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

   14. Applied times-frac 50.4

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

   15. Applied associate-/r* 50.1

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

   16. Simplified 50.1

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

   if 8.532543483832935e+21 < i < 7.395559138739582e+219

   1. Initial program 32.1

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

   3. Applied associate-*r/ 32.0

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

   5. Applied flip-- 32.0

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

   6. Simplified 32.0

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

   8. Applied div-inv 32.0

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

   9. Applied associate-/r* 32.0

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

   if 7.395559138739582e+219 < i

   1. Initial program 30.6

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

   2. Taylor expanded around 0 34.7

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right)} - 1}{\frac{i}{n}}\]
3. Recombined 6 regimes into one program.

4. Final simplification 31.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.56140166535216897 \cdot 10^{135}:\\ \;\;\;\;\frac{100 \cdot \frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.3992561866449662 \cdot 10^{-10}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.51591290926460688 \cdot 10^{-160}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 6.02622510223326963 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} - \left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{\frac{\frac{\sqrt[3]{i} \cdot \sqrt[3]{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}}{100}}}{\frac{\sqrt[3]{i}}{\sqrt[3]{n}}}\\ \mathbf{elif}\;i \le 8532543483832934860000:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 7.39555913873958208 \cdot 10^{219}:\\ \;\;\;\;\frac{\frac{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{i}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))))