Compound Interest

Average Error: 43.0 → 21.6

Time: 14.2s

Precision: 64

Math

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

↓

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

C

double f(double i, double n) {
        double r162040 = 100.0;
        double r162041 = 1.0;
        double r162042 = i;
        double r162043 = n;
        double r162044 = r162042 / r162043;
        double r162045 = r162041 + r162044;
        double r162046 = pow(r162045, r162043);
        double r162047 = r162046 - r162041;
        double r162048 = r162047 / r162044;
        double r162049 = r162040 * r162048;
        return r162049;
}

↓

double f(double i, double n) {
        double r162050 = i;
        double r162051 = -0.003897370925930358;
        bool r162052 = r162050 <= r162051;
        double r162053 = 100.0;
        double r162054 = 1.0;
        double r162055 = n;
        double r162056 = r162050 / r162055;
        double r162057 = r162054 + r162056;
        double r162058 = pow(r162057, r162055);
        double r162059 = sqrt(r162058);
        double r162060 = sqrt(r162054);
        double r162061 = r162059 + r162060;
        double r162062 = r162061 / r162050;
        double r162063 = r162059 - r162060;
        double r162064 = r162063 * r162055;
        double r162065 = r162062 * r162064;
        double r162066 = r162053 * r162065;
        double r162067 = 1.5425208996765195e-07;
        bool r162068 = r162050 <= r162067;
        double r162069 = 0.5;
        double r162070 = 2.0;
        double r162071 = pow(r162050, r162070);
        double r162072 = log(r162054);
        double r162073 = r162072 * r162055;
        double r162074 = fma(r162069, r162071, r162073);
        double r162075 = r162071 * r162072;
        double r162076 = r162069 * r162075;
        double r162077 = r162074 - r162076;
        double r162078 = fma(r162050, r162054, r162077);
        double r162079 = r162078 * r162055;
        double r162080 = r162079 / r162050;
        double r162081 = r162053 * r162080;
        double r162082 = 1.2243718320891224e+53;
        bool r162083 = r162050 <= r162082;
        double r162084 = cbrt(r162050);
        double r162085 = r162084 * r162084;
        double r162086 = cbrt(r162055);
        double r162087 = r162086 * r162086;
        double r162088 = r162085 / r162087;
        double r162089 = r162053 / r162088;
        double r162090 = r162058 - r162054;
        double r162091 = r162084 / r162086;
        double r162092 = r162090 / r162091;
        double r162093 = r162089 * r162092;
        double r162094 = 5.663291218810887e+265;
        bool r162095 = r162050 <= r162094;
        double r162096 = 1.0;
        double r162097 = fma(r162072, r162055, r162096);
        double r162098 = fma(r162054, r162050, r162097);
        double r162099 = r162098 - r162054;
        double r162100 = r162099 / r162056;
        double r162101 = r162053 * r162100;
        double r162102 = r162053 / r162050;
        double r162103 = r162096 / r162055;
        double r162104 = r162090 / r162103;
        double r162105 = r162102 * r162104;
        double r162106 = r162095 ? r162101 : r162105;
        double r162107 = r162083 ? r162093 : r162106;
        double r162108 = r162068 ? r162081 : r162107;
        double r162109 = r162052 ? r162066 : r162108;
        return r162109;
}

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-sqr-sqrt 28.4

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

   5. Applied add-sqr-sqrt 28.4

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

   6. Applied difference-of-squares 28.4

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

   7. Applied times-frac 28.8

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

   8. Simplified 28.8

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

   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. Simplified 34.1

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

   5. Applied div-inv 34.2

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

   6. Applied *-un-lft-identity 34.2

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

   7. Applied times-frac 15.7

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

   8. Applied associate-*r* 16.1

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

   9. Simplified 16.1

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

   11. Applied *-un-lft-identity 16.1

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

   12. Applied add-sqr-sqrt 16.1

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

   13. Applied times-frac 16.1

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

   14. Applied *-un-lft-identity 16.1

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

   15. Applied times-frac 16.1

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

   16. Simplified 16.1

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

   17. Simplified 16.0

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

   19. Applied div-inv 16.1

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

   20. Applied associate-*l* 15.7

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

   21. Simplified 15.5

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

   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

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

   3. Simplified 39.6

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

   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

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

Reproduce

herbie shell --seed 2020033 +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))))