Compound Interest

Average Error: 43.0 → 21.6

Time: 13.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -0.00389737092593035782:\\ \;\;\;\;100 \cdot \left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i} \cdot \left(\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot n\right)\right)\\ \mathbf{elif}\;i \le 1.5425208996765195 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) \cdot n - 0.5 \cdot \left({i}^{2} \cdot \left(\log 1 \cdot n\right)\right)}{i}\\ \mathbf{elif}\;i \le 1.22437183208912241 \cdot 10^{53}:\\ \;\;\;\;\frac{100}{\frac{\sqrt[3]{i} \cdot \sqrt[3]{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{\sqrt[3]{i}}{\sqrt[3]{n}}}\\ \mathbf{elif}\;i \le 5.6632912188108867 \cdot 10^{265}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r123047 = 100.0;
        double r123048 = 1.0;
        double r123049 = i;
        double r123050 = n;
        double r123051 = r123049 / r123050;
        double r123052 = r123048 + r123051;
        double r123053 = pow(r123052, r123050);
        double r123054 = r123053 - r123048;
        double r123055 = r123054 / r123051;
        double r123056 = r123047 * r123055;
        return r123056;
}

↓

double f(double i, double n) {
        double r123057 = i;
        double r123058 = -0.003897370925930358;
        bool r123059 = r123057 <= r123058;
        double r123060 = 100.0;
        double r123061 = 1.0;
        double r123062 = n;
        double r123063 = r123057 / r123062;
        double r123064 = r123061 + r123063;
        double r123065 = pow(r123064, r123062);
        double r123066 = r123065 - r123061;
        double r123067 = cbrt(r123066);
        double r123068 = r123067 * r123067;
        double r123069 = r123068 / r123057;
        double r123070 = r123067 * r123062;
        double r123071 = r123069 * r123070;
        double r123072 = r123060 * r123071;
        double r123073 = 1.5425208996765195e-07;
        bool r123074 = r123057 <= r123073;
        double r123075 = r123061 * r123057;
        double r123076 = 0.5;
        double r123077 = 2.0;
        double r123078 = pow(r123057, r123077);
        double r123079 = r123076 * r123078;
        double r123080 = log(r123061);
        double r123081 = r123080 * r123062;
        double r123082 = r123079 + r123081;
        double r123083 = r123075 + r123082;
        double r123084 = r123083 * r123062;
        double r123085 = r123078 * r123081;
        double r123086 = r123076 * r123085;
        double r123087 = r123084 - r123086;
        double r123088 = r123087 / r123057;
        double r123089 = r123060 * r123088;
        double r123090 = 1.2243718320891224e+53;
        bool r123091 = r123057 <= r123090;
        double r123092 = cbrt(r123057);
        double r123093 = r123092 * r123092;
        double r123094 = cbrt(r123062);
        double r123095 = r123094 * r123094;
        double r123096 = r123093 / r123095;
        double r123097 = r123060 / r123096;
        double r123098 = r123092 / r123094;
        double r123099 = r123066 / r123098;
        double r123100 = r123097 * r123099;
        double r123101 = 5.663291218810887e+265;
        bool r123102 = r123057 <= r123101;
        double r123103 = 1.0;
        double r123104 = r123081 + r123103;
        double r123105 = r123075 + r123104;
        double r123106 = r123105 - r123061;
        double r123107 = r123106 / r123063;
        double r123108 = r123060 * r123107;
        double r123109 = r123060 / r123057;
        double r123110 = r123103 / r123062;
        double r123111 = r123066 / r123110;
        double r123112 = r123109 * r123111;
        double r123113 = r123102 ? r123108 : r123112;
        double r123114 = r123091 ? r123100 : r123113;
        double r123115 = r123074 ? r123089 : r123114;
        double r123116 = r123059 ? r123072 : r123115;
        return r123116;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.0
Target	42.4
Herbie	21.6

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

Derivation

1. Split input into 5 regimes

2. if i < -0.003897370925930358

   1. Initial program 28.4

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

   3. Applied div-inv 28.4

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

   4. Applied add-cube-cbrt 28.4

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

   5. Applied times-frac 28.8

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

   6. Simplified 28.8

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

   if -0.003897370925930358 < i < 1.5425208996765195e-07

   1. Initial program 50.8

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

   2. Taylor expanded around 0 34.1

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

   4. Applied div-inv 34.2

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

   5. Applied *-un-lft-identity 34.2

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

   6. Applied times-frac 15.7

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

   7. Applied associate-*r* 16.1

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

   8. Simplified 16.1

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

   10. Applied div-sub 16.1

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

   11. Simplified 16.0

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

   12. Simplified 16.0

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

   14. Applied div-inv 16.1

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

   15. Applied associate-*l* 15.7

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

   16. Simplified 15.5

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

   if 1.5425208996765195e-07 < i < 1.2243718320891224e+53

   1. Initial program 34.7

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

   3. Applied add-cube-cbrt 34.7

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

   4. Applied add-cube-cbrt 34.8

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

   5. Applied times-frac 34.8

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

   6. Applied *-un-lft-identity 34.8

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

   7. Applied times-frac 34.8

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

   8. Applied associate-*r* 34.8

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

   9. Simplified 34.8

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

   if 1.2243718320891224e+53 < i < 5.663291218810887e+265

   1. Initial program 31.8

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

   2. Taylor expanded around 0 39.6

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

   if 5.663291218810887e+265 < i

   1. Initial program 30.0

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

   3. Applied div-inv 30.1

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

   4. Applied *-un-lft-identity 30.1

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

   5. Applied times-frac 30.1

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

   6. Applied associate-*r* 30.1

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

   7. Simplified 30.1

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

4. Final simplification 21.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.00389737092593035782:\\ \;\;\;\;100 \cdot \left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i} \cdot \left(\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot n\right)\right)\\ \mathbf{elif}\;i \le 1.5425208996765195 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) \cdot n - 0.5 \cdot \left({i}^{2} \cdot \left(\log 1 \cdot n\right)\right)}{i}\\ \mathbf{elif}\;i \le 1.22437183208912241 \cdot 10^{53}:\\ \;\;\;\;\frac{100}{\frac{\sqrt[3]{i} \cdot \sqrt[3]{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{\sqrt[3]{i}}{\sqrt[3]{n}}}\\ \mathbf{elif}\;i \le 5.6632912188108867 \cdot 10^{265}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(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))))