Compound Interest

Average Error: 41.1 → 24.6

Time: 17.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1.8403630755190044 \cdot 10^{260}:\\ \;\;\;\;100 \cdot \frac{\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n}{i}\\ \mathbf{elif}\;n \le 1.2344778126423281 \cdot 10^{-267}:\\ \;\;\;\;\frac{100}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot n - 1 \cdot n\right)\\ \mathbf{elif}\;n \le 4.27971623982010377 \cdot 10^{-165}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\\ \end{array}\]

C

double f(double i, double n) {
        double r172346 = 100.0;
        double r172347 = 1.0;
        double r172348 = i;
        double r172349 = n;
        double r172350 = r172348 / r172349;
        double r172351 = r172347 + r172350;
        double r172352 = pow(r172351, r172349);
        double r172353 = r172352 - r172347;
        double r172354 = r172353 / r172350;
        double r172355 = r172346 * r172354;
        return r172355;
}

↓

double f(double i, double n) {
        double r172356 = n;
        double r172357 = -1.8403630755190044e+260;
        bool r172358 = r172356 <= r172357;
        double r172359 = 100.0;
        double r172360 = 1.0;
        double r172361 = i;
        double r172362 = r172360 * r172361;
        double r172363 = 0.5;
        double r172364 = 2.0;
        double r172365 = pow(r172361, r172364);
        double r172366 = r172363 * r172365;
        double r172367 = log(r172360);
        double r172368 = r172367 * r172356;
        double r172369 = r172366 + r172368;
        double r172370 = r172362 + r172369;
        double r172371 = r172365 * r172367;
        double r172372 = r172363 * r172371;
        double r172373 = r172370 - r172372;
        double r172374 = r172373 * r172356;
        double r172375 = r172374 / r172361;
        double r172376 = r172359 * r172375;
        double r172377 = 1.234477812642328e-267;
        bool r172378 = r172356 <= r172377;
        double r172379 = r172359 / r172361;
        double r172380 = r172361 / r172356;
        double r172381 = r172360 + r172380;
        double r172382 = pow(r172381, r172356);
        double r172383 = r172382 * r172356;
        double r172384 = r172360 * r172356;
        double r172385 = r172383 - r172384;
        double r172386 = r172379 * r172385;
        double r172387 = 4.279716239820104e-165;
        bool r172388 = r172356 <= r172387;
        double r172389 = 1.0;
        double r172390 = r172368 + r172389;
        double r172391 = r172362 + r172390;
        double r172392 = r172391 - r172360;
        double r172393 = r172392 / r172380;
        double r172394 = r172359 * r172393;
        double r172395 = r172359 * r172373;
        double r172396 = r172395 / r172361;
        double r172397 = r172396 * r172356;
        double r172398 = r172388 ? r172394 : r172397;
        double r172399 = r172378 ? r172386 : r172398;
        double r172400 = r172358 ? r172376 : r172399;
        return r172400;
}

Error

Bits error versus i

Bits error versus n

Target

Original	41.1
Target	40.9
Herbie	24.6

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

Derivation

1. Split input into 4 regimes

2. if n < -1.8403630755190044e+260

   1. Initial program 56.0

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

   3. Applied div-inv 56.1

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

   4. Applied *-un-lft-identity 56.1

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

   5. Applied times-frac 55.6

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

   6. Applied associate-*r* 55.6

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

   7. Simplified 55.6

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

   8. Taylor expanded around 0 37.0

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

   10. Applied div-inv 37.1

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

   11. Applied associate-*l* 37.0

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

   12. Simplified 36.9

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

   if -1.8403630755190044e+260 < n < 1.234477812642328e-267

   1. Initial program 26.1

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

   3. Applied div-inv 26.1

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

   4. Applied *-un-lft-identity 26.1

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

   5. Applied times-frac 26.3

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

   6. Applied associate-*r* 26.4

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

   7. Simplified 26.4

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

   9. Applied div-sub 26.4

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

   10. Simplified 29.0

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

   11. Simplified 26.4

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

   if 1.234477812642328e-267 < n < 4.279716239820104e-165

   1. Initial program 42.2

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

   2. Taylor expanded around 0 33.4

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

   if 4.279716239820104e-165 < n

   1. Initial program 58.8

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

   3. Applied div-inv 58.8

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

   4. Applied *-un-lft-identity 58.8

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

   5. Applied times-frac 58.5

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

   6. Applied associate-*r* 58.5

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

   7. Simplified 58.5

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

   8. Taylor expanded around 0 23.5

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

   10. Applied associate-/r/ 23.4

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

   11. Applied associate-*r* 19.6

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

   12. Simplified 19.4

       \[\leadsto \color{blue}{\frac{100 \cdot \left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}} \cdot n\]
3. Recombined 4 regimes into one program.

4. Final simplification 24.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.8403630755190044 \cdot 10^{260}:\\ \;\;\;\;100 \cdot \frac{\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n}{i}\\ \mathbf{elif}\;n \le 1.2344778126423281 \cdot 10^{-267}:\\ \;\;\;\;\frac{100}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot n - 1 \cdot n\right)\\ \mathbf{elif}\;n \le 4.27971623982010377 \cdot 10^{-165}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(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))))