Compound Interest

Average Error: 43.2 → 31.8

Time: 15.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -6.8977356701701615 \cdot 10^{-9}:\\ \;\;\;\;\left(\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}\right) \cdot 100\right) \cdot \left(\frac{{\left(\frac{i}{n}\right)}^{\left(\frac{n}{2}\right)}}{\frac{i}{n}} - \frac{\sqrt{1}}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 2.5875098730826361 \cdot 10^{-28}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 4.20451188706486744 \cdot 10^{212}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{2} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right) + \left(\left(\left(\frac{1}{6} \cdot \left({\left(\log i\right)}^{3} \cdot {n}^{4}\right) + \log i \cdot {n}^{2}\right) + \frac{1}{2} \cdot \left(\log i \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right) + {\left(\log i\right)}^{2} \cdot {n}^{3}\right)\right) - \left(\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot \left({n}^{4} \cdot \log n\right)\right) + \left({n}^{2} \cdot \log n + \left(\frac{1}{6} \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{3}\right) + \log i \cdot \left({n}^{3} \cdot \log n\right)\right)\right)\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}\right) \cdot 100\right) \cdot \left(\frac{{\left(\frac{i}{n}\right)}^{\left(\frac{n}{2}\right)}}{\frac{i}{n}} - \frac{\sqrt{1}}{\frac{i}{n}}\right)\\ \end{array}\]

C

double f(double i, double n) {
        double r136094 = 100.0;
        double r136095 = 1.0;
        double r136096 = i;
        double r136097 = n;
        double r136098 = r136096 / r136097;
        double r136099 = r136095 + r136098;
        double r136100 = pow(r136099, r136097);
        double r136101 = r136100 - r136095;
        double r136102 = r136101 / r136098;
        double r136103 = r136094 * r136102;
        return r136103;
}

↓

double f(double i, double n) {
        double r136104 = i;
        double r136105 = -6.8977356701701615e-09;
        bool r136106 = r136104 <= r136105;
        double r136107 = 1.0;
        double r136108 = n;
        double r136109 = r136104 / r136108;
        double r136110 = r136107 + r136109;
        double r136111 = 2.0;
        double r136112 = r136108 / r136111;
        double r136113 = pow(r136110, r136112);
        double r136114 = sqrt(r136107);
        double r136115 = r136113 + r136114;
        double r136116 = 100.0;
        double r136117 = r136115 * r136116;
        double r136118 = pow(r136109, r136112);
        double r136119 = r136118 / r136109;
        double r136120 = r136114 / r136109;
        double r136121 = r136119 - r136120;
        double r136122 = r136117 * r136121;
        double r136123 = 2.587509873082636e-28;
        bool r136124 = r136104 <= r136123;
        double r136125 = r136107 * r136104;
        double r136126 = 0.5;
        double r136127 = pow(r136104, r136111);
        double r136128 = r136126 * r136127;
        double r136129 = log(r136107);
        double r136130 = r136129 * r136108;
        double r136131 = r136128 + r136130;
        double r136132 = r136125 + r136131;
        double r136133 = r136127 * r136129;
        double r136134 = r136126 * r136133;
        double r136135 = r136132 - r136134;
        double r136136 = r136135 / r136109;
        double r136137 = r136116 * r136136;
        double r136138 = 4.2045118870648674e+212;
        bool r136139 = r136104 <= r136138;
        double r136140 = 0.5;
        double r136141 = 3.0;
        double r136142 = pow(r136108, r136141);
        double r136143 = log(r136108);
        double r136144 = pow(r136143, r136111);
        double r136145 = r136142 * r136144;
        double r136146 = r136140 * r136145;
        double r136147 = 0.16666666666666666;
        double r136148 = log(r136104);
        double r136149 = pow(r136148, r136141);
        double r136150 = 4.0;
        double r136151 = pow(r136108, r136150);
        double r136152 = r136149 * r136151;
        double r136153 = r136147 * r136152;
        double r136154 = pow(r136108, r136111);
        double r136155 = r136148 * r136154;
        double r136156 = r136153 + r136155;
        double r136157 = r136151 * r136144;
        double r136158 = r136148 * r136157;
        double r136159 = pow(r136148, r136111);
        double r136160 = r136159 * r136142;
        double r136161 = r136158 + r136160;
        double r136162 = r136140 * r136161;
        double r136163 = r136156 + r136162;
        double r136164 = r136151 * r136143;
        double r136165 = r136159 * r136164;
        double r136166 = r136140 * r136165;
        double r136167 = r136154 * r136143;
        double r136168 = pow(r136143, r136141);
        double r136169 = r136151 * r136168;
        double r136170 = r136147 * r136169;
        double r136171 = r136142 * r136143;
        double r136172 = r136148 * r136171;
        double r136173 = r136170 + r136172;
        double r136174 = r136167 + r136173;
        double r136175 = r136166 + r136174;
        double r136176 = r136163 - r136175;
        double r136177 = r136146 + r136176;
        double r136178 = r136177 / r136104;
        double r136179 = r136116 * r136178;
        double r136180 = r136139 ? r136179 : r136122;
        double r136181 = r136124 ? r136137 : r136180;
        double r136182 = r136106 ? r136122 : r136181;
        return r136182;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.2
Target	42.9
Herbie	31.8

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

Derivation

1. Split input into 3 regimes

2. if i < -6.8977356701701615e-09 or 4.2045118870648674e+212 < i

   1. Initial program 29.7

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

   3. Applied *-un-lft-identity 29.7

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

   4. Applied *-un-lft-identity 29.7

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

   5. Applied times-frac 29.7

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

   6. Applied add-sqr-sqrt 29.7

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

   7. Applied sqr-pow 29.7

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

   8. Applied difference-of-squares 29.7

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

   9. Applied times-frac 29.7

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

   10. Applied associate-*r* 29.7

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

   11. Simplified 29.7

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

   12. Taylor expanded around inf 63.5

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

   13. Simplified 29.6

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

   if -6.8977356701701615e-09 < i < 2.587509873082636e-28

   1. Initial program 51.1

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

   2. Taylor expanded around 0 34.2

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

   if 2.587509873082636e-28 < i < 4.2045118870648674e+212

   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-inv 36.1

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

   4. Applied add-sqr-sqrt 36.1

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

   5. Applied sqr-pow 36.2

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

   6. Applied difference-of-squares 36.2

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

   7. Applied times-frac 36.2

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

   8. Simplified 36.1

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

   10. Applied associate-*l/ 36.2

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

   11. Simplified 36.0

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

   12. Taylor expanded around 0 24.3

       \[\leadsto 100 \cdot \frac{\color{blue}{\left(\frac{1}{2} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right) + \left(\frac{1}{6} \cdot \left({\left(\log i\right)}^{3} \cdot {n}^{4}\right) + \left(\log i \cdot {n}^{2} + \left(\frac{1}{2} \cdot \left(\log i \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot {n}^{3}\right)\right)\right)\right)\right) - \left(\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot \left({n}^{4} \cdot \log n\right)\right) + \left({n}^{2} \cdot \log n + \left(\frac{1}{6} \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{3}\right) + \log i \cdot \left({n}^{3} \cdot \log n\right)\right)\right)\right)}}{i}\]

   13. Simplified 24.3

       \[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{2} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right) + \left(\left(\left(\frac{1}{6} \cdot \left({\left(\log i\right)}^{3} \cdot {n}^{4}\right) + \log i \cdot {n}^{2}\right) + \frac{1}{2} \cdot \left(\log i \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right) + {\left(\log i\right)}^{2} \cdot {n}^{3}\right)\right) - \left(\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot \left({n}^{4} \cdot \log n\right)\right) + \left({n}^{2} \cdot \log n + \left(\frac{1}{6} \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{3}\right) + \log i \cdot \left({n}^{3} \cdot \log n\right)\right)\right)\right)\right)}}{i}\]
3. Recombined 3 regimes into one program.

4. Final simplification 31.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -6.8977356701701615 \cdot 10^{-9}:\\ \;\;\;\;\left(\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}\right) \cdot 100\right) \cdot \left(\frac{{\left(\frac{i}{n}\right)}^{\left(\frac{n}{2}\right)}}{\frac{i}{n}} - \frac{\sqrt{1}}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 2.5875098730826361 \cdot 10^{-28}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 4.20451188706486744 \cdot 10^{212}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{2} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right) + \left(\left(\left(\frac{1}{6} \cdot \left({\left(\log i\right)}^{3} \cdot {n}^{4}\right) + \log i \cdot {n}^{2}\right) + \frac{1}{2} \cdot \left(\log i \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right) + {\left(\log i\right)}^{2} \cdot {n}^{3}\right)\right) - \left(\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot \left({n}^{4} \cdot \log n\right)\right) + \left({n}^{2} \cdot \log n + \left(\frac{1}{6} \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{3}\right) + \log i \cdot \left({n}^{3} \cdot \log n\right)\right)\right)\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}\right) \cdot 100\right) \cdot \left(\frac{{\left(\frac{i}{n}\right)}^{\left(\frac{n}{2}\right)}}{\frac{i}{n}} - \frac{\sqrt{1}}{\frac{i}{n}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))))