Compound Interest

Average Error: 42.3 → 22.0

Time: 33.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -8.271685620893127920309681666849172655568 \cdot 10^{-30}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \mathbf{elif}\;i \le -1.07124828629959371703623408587100365571 \cdot 10^{-106}:\\ \;\;\;\;\frac{100 \cdot \left(\left({\left(\log 1 \cdot n + 1 \cdot i\right)}^{3} + {\left(0.5 \cdot \left(i \cdot i - \log 1 \cdot \left(i \cdot i\right)\right)\right)}^{3}\right) \cdot n\right)}{\left(\left(0.5 \cdot \left(i \cdot i - \log 1 \cdot \left(i \cdot i\right)\right)\right) \cdot \left(0.5 \cdot \left(i \cdot i - \log 1 \cdot \left(i \cdot i\right)\right) - \left(\log 1 \cdot n + 1 \cdot i\right)\right) + \left(\log 1 \cdot n + 1 \cdot i\right) \cdot \left(\log 1 \cdot n + 1 \cdot i\right)\right) \cdot i}\\ \mathbf{elif}\;i \le 1.666826687406822778561223565520064088172 \cdot 10^{-69}:\\ \;\;\;\;100 \cdot \left(\frac{\left(\log 1 \cdot n + 1 \cdot i\right) + \log \left(e^{0.5 \cdot \left(i \cdot i - \log 1 \cdot \left(i \cdot i\right)\right)}\right)}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 176217786659469.75:\\ \;\;\;\;\frac{100 \cdot \left(\left({\left(\log 1 \cdot n + 1 \cdot i\right)}^{3} + {\left(0.5 \cdot \left(i \cdot i - \log 1 \cdot \left(i \cdot i\right)\right)\right)}^{3}\right) \cdot n\right)}{\left(\left(0.5 \cdot \left(i \cdot i - \log 1 \cdot \left(i \cdot i\right)\right)\right) \cdot \left(0.5 \cdot \left(i \cdot i - \log 1 \cdot \left(i \cdot i\right)\right) - \left(\log 1 \cdot n + 1 \cdot i\right)\right) + \left(\log 1 \cdot n + 1 \cdot i\right) \cdot \left(\log 1 \cdot n + 1 \cdot i\right)\right) \cdot i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\left({\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot {\left(1 + \frac{i}{n}\right)}^{n} - 1 \cdot \left(1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}{i} \cdot n\right)\\ \end{array}\]

C

double f(double i, double n) {
        double r15785208 = 100.0;
        double r15785209 = 1.0;
        double r15785210 = i;
        double r15785211 = n;
        double r15785212 = r15785210 / r15785211;
        double r15785213 = r15785209 + r15785212;
        double r15785214 = pow(r15785213, r15785211);
        double r15785215 = r15785214 - r15785209;
        double r15785216 = r15785215 / r15785212;
        double r15785217 = r15785208 * r15785216;
        return r15785217;
}

↓

double f(double i, double n) {
        double r15785218 = i;
        double r15785219 = -8.271685620893128e-30;
        bool r15785220 = r15785218 <= r15785219;
        double r15785221 = 100.0;
        double r15785222 = 1.0;
        double r15785223 = n;
        double r15785224 = r15785218 / r15785223;
        double r15785225 = r15785222 + r15785224;
        double r15785226 = pow(r15785225, r15785223);
        double r15785227 = r15785226 - r15785222;
        double r15785228 = r15785227 / r15785218;
        double r15785229 = r15785221 * r15785228;
        double r15785230 = r15785229 * r15785223;
        double r15785231 = -1.0712482862995937e-106;
        bool r15785232 = r15785218 <= r15785231;
        double r15785233 = log(r15785222);
        double r15785234 = r15785233 * r15785223;
        double r15785235 = r15785222 * r15785218;
        double r15785236 = r15785234 + r15785235;
        double r15785237 = 3.0;
        double r15785238 = pow(r15785236, r15785237);
        double r15785239 = 0.5;
        double r15785240 = r15785218 * r15785218;
        double r15785241 = r15785233 * r15785240;
        double r15785242 = r15785240 - r15785241;
        double r15785243 = r15785239 * r15785242;
        double r15785244 = pow(r15785243, r15785237);
        double r15785245 = r15785238 + r15785244;
        double r15785246 = r15785245 * r15785223;
        double r15785247 = r15785221 * r15785246;
        double r15785248 = r15785243 - r15785236;
        double r15785249 = r15785243 * r15785248;
        double r15785250 = r15785236 * r15785236;
        double r15785251 = r15785249 + r15785250;
        double r15785252 = r15785251 * r15785218;
        double r15785253 = r15785247 / r15785252;
        double r15785254 = 1.6668266874068228e-69;
        bool r15785255 = r15785218 <= r15785254;
        double r15785256 = exp(r15785243);
        double r15785257 = log(r15785256);
        double r15785258 = r15785236 + r15785257;
        double r15785259 = r15785258 / r15785218;
        double r15785260 = r15785259 * r15785223;
        double r15785261 = r15785221 * r15785260;
        double r15785262 = 176217786659469.75;
        bool r15785263 = r15785218 <= r15785262;
        double r15785264 = r15785226 * r15785226;
        double r15785265 = r15785264 * r15785226;
        double r15785266 = r15785222 * r15785222;
        double r15785267 = r15785222 * r15785266;
        double r15785268 = r15785265 - r15785267;
        double r15785269 = r15785226 * r15785222;
        double r15785270 = r15785266 + r15785269;
        double r15785271 = r15785264 + r15785270;
        double r15785272 = r15785268 / r15785271;
        double r15785273 = r15785272 / r15785218;
        double r15785274 = r15785273 * r15785223;
        double r15785275 = r15785221 * r15785274;
        double r15785276 = r15785263 ? r15785253 : r15785275;
        double r15785277 = r15785255 ? r15785261 : r15785276;
        double r15785278 = r15785232 ? r15785253 : r15785277;
        double r15785279 = r15785220 ? r15785230 : r15785278;
        return r15785279;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.3
Target	42.8
Herbie	22.0

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

Derivation

1. Split input into 4 regimes

2. if i < -8.271685620893128e-30

   1. Initial program 28.9

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

   3. Applied associate-/r/ 29.4

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

   4. Applied associate-*r* 29.4

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

   if -8.271685620893128e-30 < i < -1.0712482862995937e-106 or 1.6668266874068228e-69 < i < 176217786659469.75

   1. Initial program 50.6

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

   3. Applied associate-/r/ 50.5

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

   4. Taylor expanded around 0 23.1

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

   5. Simplified 23.1

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

   7. Applied associate-*l/ 21.0

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

   8. Applied associate-*r/ 21.2

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

   10. Applied flip3-+ 21.3

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

   11. Applied associate-*l/ 17.8

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

   12. Applied associate-*r/ 17.8

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

   13. Applied associate-/l/ 17.7

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

   14. Simplified 17.7

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

   if -1.0712482862995937e-106 < i < 1.6668266874068228e-69

   1. Initial program 49.8

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

   3. Applied associate-/r/ 49.4

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

   4. Taylor expanded around 0 16.6

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

   5. Simplified 16.6

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

   7. Applied add-log-exp 16.6

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

   if 176217786659469.75 < i

   1. Initial program 31.9

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

   3. Applied associate-/r/ 31.9

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

   5. Applied flip3-- 32.0

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

   6. Simplified 32.0

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

4. Final simplification 22.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -8.271685620893127920309681666849172655568 \cdot 10^{-30}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \mathbf{elif}\;i \le -1.07124828629959371703623408587100365571 \cdot 10^{-106}:\\ \;\;\;\;\frac{100 \cdot \left(\left({\left(\log 1 \cdot n + 1 \cdot i\right)}^{3} + {\left(0.5 \cdot \left(i \cdot i - \log 1 \cdot \left(i \cdot i\right)\right)\right)}^{3}\right) \cdot n\right)}{\left(\left(0.5 \cdot \left(i \cdot i - \log 1 \cdot \left(i \cdot i\right)\right)\right) \cdot \left(0.5 \cdot \left(i \cdot i - \log 1 \cdot \left(i \cdot i\right)\right) - \left(\log 1 \cdot n + 1 \cdot i\right)\right) + \left(\log 1 \cdot n + 1 \cdot i\right) \cdot \left(\log 1 \cdot n + 1 \cdot i\right)\right) \cdot i}\\ \mathbf{elif}\;i \le 1.666826687406822778561223565520064088172 \cdot 10^{-69}:\\ \;\;\;\;100 \cdot \left(\frac{\left(\log 1 \cdot n + 1 \cdot i\right) + \log \left(e^{0.5 \cdot \left(i \cdot i - \log 1 \cdot \left(i \cdot i\right)\right)}\right)}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 176217786659469.75:\\ \;\;\;\;\frac{100 \cdot \left(\left({\left(\log 1 \cdot n + 1 \cdot i\right)}^{3} + {\left(0.5 \cdot \left(i \cdot i - \log 1 \cdot \left(i \cdot i\right)\right)\right)}^{3}\right) \cdot n\right)}{\left(\left(0.5 \cdot \left(i \cdot i - \log 1 \cdot \left(i \cdot i\right)\right)\right) \cdot \left(0.5 \cdot \left(i \cdot i - \log 1 \cdot \left(i \cdot i\right)\right) - \left(\log 1 \cdot n + 1 \cdot i\right)\right) + \left(\log 1 \cdot n + 1 \cdot i\right) \cdot \left(\log 1 \cdot n + 1 \cdot i\right)\right) \cdot i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\left({\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot {\left(1 + \frac{i}{n}\right)}^{n} - 1 \cdot \left(1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}{i} \cdot n\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (i n)
  :name "Compound Interest"

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))