Compound Interest

Average Error: 42.7 → 31.4

Time: 29.9s

Precision: 64

Math

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

↓

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

C

double f(double i, double n) {
        double r328890 = 100.0;
        double r328891 = 1.0;
        double r328892 = i;
        double r328893 = n;
        double r328894 = r328892 / r328893;
        double r328895 = r328891 + r328894;
        double r328896 = pow(r328895, r328893);
        double r328897 = r328896 - r328891;
        double r328898 = r328897 / r328894;
        double r328899 = r328890 * r328898;
        return r328899;
}

↓

double f(double i, double n) {
        double r328900 = i;
        double r328901 = -2.561401665352169e+135;
        bool r328902 = r328900 <= r328901;
        double r328903 = 1.0;
        double r328904 = n;
        double r328905 = r328900 / r328904;
        double r328906 = r328903 + r328905;
        double r328907 = 2.0;
        double r328908 = r328907 * r328904;
        double r328909 = pow(r328906, r328908);
        double r328910 = r328903 * r328903;
        double r328911 = r328909 - r328910;
        double r328912 = exp(r328911);
        double r328913 = log(r328912);
        double r328914 = 100.0;
        double r328915 = r328913 * r328914;
        double r328916 = pow(r328906, r328904);
        double r328917 = r328916 + r328903;
        double r328918 = r328915 / r328917;
        double r328919 = r328918 / r328905;
        double r328920 = -1.3992561866449662e-10;
        bool r328921 = r328900 <= r328920;
        double r328922 = pow(r328905, r328904);
        double r328923 = r328922 - r328903;
        double r328924 = r328914 * r328923;
        double r328925 = r328924 / r328905;
        double r328926 = 2.515912909264607e-160;
        bool r328927 = r328900 <= r328926;
        double r328928 = 0.5;
        double r328929 = pow(r328900, r328907);
        double r328930 = log(r328903);
        double r328931 = r328930 * r328904;
        double r328932 = fma(r328928, r328929, r328931);
        double r328933 = r328929 * r328930;
        double r328934 = r328928 * r328933;
        double r328935 = r328932 - r328934;
        double r328936 = fma(r328900, r328903, r328935);
        double r328937 = r328936 / r328905;
        double r328938 = r328914 * r328937;
        double r328939 = 6.02622510223327e-125;
        bool r328940 = r328900 <= r328939;
        double r328941 = 1.0;
        double r328942 = r328941 / r328900;
        double r328943 = r328907 * r328908;
        double r328944 = pow(r328906, r328943);
        double r328945 = -r328910;
        double r328946 = r328945 * r328945;
        double r328947 = -r328946;
        double r328948 = r328944 + r328947;
        double r328949 = r328909 + r328910;
        double r328950 = r328948 / r328949;
        double r328951 = r328950 * r328914;
        double r328952 = r328951 / r328917;
        double r328953 = r328952 * r328904;
        double r328954 = r328942 * r328953;
        double r328955 = 8.532543483832935e+21;
        bool r328956 = r328900 <= r328955;
        double r328957 = 7.395559138739582e+219;
        bool r328958 = r328900 <= r328957;
        double r328959 = fma(r328930, r328904, r328941);
        double r328960 = fma(r328903, r328900, r328959);
        double r328961 = r328960 - r328903;
        double r328962 = r328961 / r328905;
        double r328963 = r328914 * r328962;
        double r328964 = r328958 ? r328954 : r328963;
        double r328965 = r328956 ? r328938 : r328964;
        double r328966 = r328940 ? r328954 : r328965;
        double r328967 = r328927 ? r328938 : r328966;
        double r328968 = r328921 ? r328925 : r328967;
        double r328969 = r328902 ? r328919 : r328968;
        return r328969;
}

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 5 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. Applied associate-*r/ 15.4

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

   7. Simplified 15.4

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

   9. Applied add-log-exp 15.4

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

   10. Applied neg-log 15.4

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

   11. Applied add-log-exp 15.4

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

   12. Applied sum-log 15.4

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

   13. Simplified 15.4

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

   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

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

   3. Simplified 33.7

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

   if 2.515912909264607e-160 < i < 6.02622510223327e-125 or 8.532543483832935e+21 < i < 7.395559138739582e+219

   1. Initial program 38.2

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

   3. Applied associate-*r/ 38.1

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

   5. Applied flip-- 38.1

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

   6. Applied associate-*r/ 38.1

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

   7. Simplified 38.1

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

   9. Applied flip-+ 38.2

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

   10. Simplified 38.1

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

   11. Simplified 38.1

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

   13. Applied div-inv 38.2

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

   14. Applied *-un-lft-identity 38.2

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

   15. Applied times-frac 38.1

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

   16. Simplified 38.1

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

   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. Simplified 34.7

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right)} - 1}{\frac{i}{n}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 31.4

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(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))))