Compound Interest

Average Error: 42.5 → 15.1

Time: 3.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -1.967890440787981 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{\frac{i}{n}} \cdot \mathsf{fma}\left(100, \left(e^{\mathsf{log1p}\left(\left(\frac{i}{n}\right)\right) \cdot n}\right), -100\right)\\ \mathbf{elif}\;i \le 0.23936266726166916:\\ \;\;\;\;\frac{\frac{i \cdot \mathsf{fma}\left(i, \left(\mathsf{fma}\left(i, \frac{50}{3}, 50\right)\right), 100\right)}{i}}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 4.0545436608851676 \cdot 10^{+138}:\\ \;\;\;\;\frac{\mathsf{fma}\left(50, \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\log i \cdot \left(n \cdot \left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right)\right)\right), \left(\mathsf{fma}\left(100, \left(\log i \cdot n\right), \left(\left(\log i \cdot \left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot n\right)\right) \cdot 50 + 50 \cdot \left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right)\right)\right)\right)\right)\right)\right) - \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot n\right) \cdot \log n\right), \left(\mathsf{fma}\left(\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot \log i\right)\right), \frac{50}{3}, \left(100 \cdot \left(n \cdot \log n\right)\right)\right)\right)\right) + \left(\mathsf{fma}\left(\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot \log i\right)\right), \frac{100}{3}, \left(\left(\log i \cdot 50\right) \cdot \left(\log n \cdot \left(n \cdot n\right)\right)\right)\right) + \left(\log i \cdot 50\right) \cdot \left(\log n \cdot \left(n \cdot n\right)\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 7.203465032064257 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{1}{\frac{i}{\mathsf{fma}\left(100, \left({\left(\frac{i}{n} + 1\right)}^{n}\right), -100\right)}}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(50, \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\log i \cdot \left(n \cdot \left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right)\right)\right), \left(\mathsf{fma}\left(100, \left(\log i \cdot n\right), \left(\left(\log i \cdot \left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot n\right)\right) \cdot 50 + 50 \cdot \left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right)\right)\right)\right)\right)\right)\right) - \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot n\right) \cdot \log n\right), \left(\mathsf{fma}\left(\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot \log i\right)\right), \frac{50}{3}, \left(100 \cdot \left(n \cdot \log n\right)\right)\right)\right)\right) + \left(\mathsf{fma}\left(\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot \log i\right)\right), \frac{100}{3}, \left(\left(\log i \cdot 50\right) \cdot \left(\log n \cdot \left(n \cdot n\right)\right)\right)\right) + \left(\log i \cdot 50\right) \cdot \left(\log n \cdot \left(n \cdot n\right)\right)\right)\right)}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r31124399 = 100.0;
        double r31124400 = 1.0;
        double r31124401 = i;
        double r31124402 = n;
        double r31124403 = r31124401 / r31124402;
        double r31124404 = r31124400 + r31124403;
        double r31124405 = pow(r31124404, r31124402);
        double r31124406 = r31124405 - r31124400;
        double r31124407 = r31124406 / r31124403;
        double r31124408 = r31124399 * r31124407;
        return r31124408;
}

↓

double f(double i, double n) {
        double r31124409 = i;
        double r31124410 = -1.967890440787981e-12;
        bool r31124411 = r31124409 <= r31124410;
        double r31124412 = 1.0;
        double r31124413 = n;
        double r31124414 = r31124409 / r31124413;
        double r31124415 = r31124412 / r31124414;
        double r31124416 = 100.0;
        double r31124417 = log1p(r31124414);
        double r31124418 = r31124417 * r31124413;
        double r31124419 = exp(r31124418);
        double r31124420 = -100.0;
        double r31124421 = fma(r31124416, r31124419, r31124420);
        double r31124422 = r31124415 * r31124421;
        double r31124423 = 0.23936266726166916;
        bool r31124424 = r31124409 <= r31124423;
        double r31124425 = 16.666666666666668;
        double r31124426 = 50.0;
        double r31124427 = fma(r31124409, r31124425, r31124426);
        double r31124428 = fma(r31124409, r31124427, r31124416);
        double r31124429 = r31124409 * r31124428;
        double r31124430 = r31124429 / r31124409;
        double r31124431 = r31124412 / r31124413;
        double r31124432 = r31124430 / r31124431;
        double r31124433 = 4.0545436608851676e+138;
        bool r31124434 = r31124409 <= r31124433;
        double r31124435 = log(r31124413);
        double r31124436 = r31124413 * r31124435;
        double r31124437 = r31124436 * r31124436;
        double r31124438 = log(r31124409);
        double r31124439 = r31124438 * r31124413;
        double r31124440 = r31124439 * r31124439;
        double r31124441 = r31124413 * r31124440;
        double r31124442 = r31124438 * r31124441;
        double r31124443 = r31124437 * r31124413;
        double r31124444 = r31124438 * r31124443;
        double r31124445 = r31124444 * r31124426;
        double r31124446 = r31124426 * r31124440;
        double r31124447 = r31124445 + r31124446;
        double r31124448 = fma(r31124416, r31124439, r31124447);
        double r31124449 = fma(r31124425, r31124442, r31124448);
        double r31124450 = fma(r31124426, r31124437, r31124449);
        double r31124451 = r31124443 * r31124435;
        double r31124452 = r31124413 * r31124413;
        double r31124453 = r31124436 * r31124452;
        double r31124454 = r31124438 * r31124438;
        double r31124455 = r31124453 * r31124454;
        double r31124456 = r31124416 * r31124436;
        double r31124457 = fma(r31124455, r31124425, r31124456);
        double r31124458 = fma(r31124425, r31124451, r31124457);
        double r31124459 = 33.333333333333336;
        double r31124460 = r31124438 * r31124426;
        double r31124461 = r31124435 * r31124452;
        double r31124462 = r31124460 * r31124461;
        double r31124463 = fma(r31124455, r31124459, r31124462);
        double r31124464 = r31124463 + r31124462;
        double r31124465 = r31124458 + r31124464;
        double r31124466 = r31124450 - r31124465;
        double r31124467 = r31124466 / r31124414;
        double r31124468 = 7.203465032064257e+231;
        bool r31124469 = r31124409 <= r31124468;
        double r31124470 = r31124414 + r31124412;
        double r31124471 = pow(r31124470, r31124413);
        double r31124472 = fma(r31124416, r31124471, r31124420);
        double r31124473 = r31124409 / r31124472;
        double r31124474 = r31124412 / r31124473;
        double r31124475 = r31124474 / r31124431;
        double r31124476 = r31124469 ? r31124475 : r31124467;
        double r31124477 = r31124434 ? r31124467 : r31124476;
        double r31124478 = r31124424 ? r31124432 : r31124477;
        double r31124479 = r31124411 ? r31124422 : r31124478;
        return r31124479;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.5
Target	41.6
Herbie	15.1

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

Derivation

1. Split input into 4 regimes

2. if i < -1.967890440787981e-12

   1. Initial program 29.5

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

   2. Simplified 29.4

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

   4. Applied add-exp-log 29.4

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

   5. Applied pow-exp 29.4

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

   6. Simplified 6.2

      \[\leadsto \frac{\mathsf{fma}\left(100, \left(e^{\color{blue}{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}}\right), -100\right)}{\frac{i}{n}}\]
   7. Using strategy rm

   8. Applied div-inv 6.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right) \cdot \frac{1}{\frac{i}{n}}}\]

   if -1.967890440787981e-12 < i < 0.23936266726166916

   1. Initial program 49.9

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

   2. Simplified 49.9

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

   4. Applied div-inv 49.9

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

   5. Applied associate-/r* 50.1

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

   6. Taylor expanded around 0 17.1

      \[\leadsto \frac{\frac{\color{blue}{100 \cdot i + \left(50 \cdot {i}^{2} + \frac{50}{3} \cdot {i}^{3}\right)}}{i}}{\frac{1}{n}}\]

   7. Simplified 17.1

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(i, \left(\mathsf{fma}\left(i, \frac{50}{3}, 50\right)\right), 100\right) \cdot i}}{i}}{\frac{1}{n}}\]

   if 0.23936266726166916 < i < 4.0545436608851676e+138 or 7.203465032064257e+231 < i

   1. Initial program 33.3

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

   2. Simplified 33.3

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

   4. Applied add-exp-log 49.5

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

   5. Applied pow-exp 49.5

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

   6. Simplified 48.5

      \[\leadsto \frac{\mathsf{fma}\left(100, \left(e^{\color{blue}{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}}\right), -100\right)}{\frac{i}{n}}\]

   7. Taylor expanded around 0 18.0

      \[\leadsto \frac{\color{blue}{\left(50 \cdot \left({n}^{2} \cdot {\left(\log n\right)}^{2}\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot {\left(\log i\right)}^{3}\right) + \left(100 \cdot \left(n \cdot \log i\right) + \left(\frac{100}{3} \cdot \left({n}^{3} \cdot \left({\left(\log n\right)}^{2} \cdot \log i\right)\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot \left(\log i \cdot {\left(\log n\right)}^{2}\right)\right) + 50 \cdot \left({n}^{2} \cdot {\left(\log i\right)}^{2}\right)\right)\right)\right)\right)\right) - \left(\frac{100}{3} \cdot \left({n}^{3} \cdot \left({\left(\log i\right)}^{2} \cdot \log n\right)\right) + \left(50 \cdot \left({n}^{2} \cdot \left(\log n \cdot \log i\right)\right) + \left(50 \cdot \left({n}^{2} \cdot \left(\log i \cdot \log n\right)\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot \left(\log n \cdot {\left(\log i\right)}^{2}\right)\right) + 100 \cdot \left(n \cdot \log n\right)\right)\right)\right)\right)\right)}}{\frac{i}{n}}\]

   8. Simplified 18.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(50, \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\log i \cdot \left(n \cdot \left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right)\right)\right), \left(\mathsf{fma}\left(100, \left(n \cdot \log i\right), \left(\left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right) \cdot 50 + \left(\log i \cdot \left(n \cdot \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right)\right)\right) \cdot 50\right)\right)\right)\right)\right)\right) - \left(\left(\mathsf{fma}\left(\left(\left(\log i \cdot \log i\right) \cdot \left(\left(n \cdot n\right) \cdot \left(n \cdot \log n\right)\right)\right), \frac{100}{3}, \left(\left(\left(n \cdot n\right) \cdot \log n\right) \cdot \left(\log i \cdot 50\right)\right)\right) + \left(\left(n \cdot n\right) \cdot \log n\right) \cdot \left(\log i \cdot 50\right)\right) + \mathsf{fma}\left(\frac{50}{3}, \left(\log n \cdot \left(n \cdot \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right)\right)\right), \left(\mathsf{fma}\left(\left(\left(\log i \cdot \log i\right) \cdot \left(\left(n \cdot n\right) \cdot \left(n \cdot \log n\right)\right)\right), \frac{50}{3}, \left(100 \cdot \left(n \cdot \log n\right)\right)\right)\right)\right)\right)}}{\frac{i}{n}}\]

   if 4.0545436608851676e+138 < i < 7.203465032064257e+231

   1. Initial program 30.5

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

   2. Simplified 30.5

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

   4. Applied div-inv 30.5

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

   5. Applied associate-/r* 30.6

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

   7. Applied *-un-lft-identity 30.6

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

   8. Applied associate-/l* 30.6

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

4. Final simplification 15.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.967890440787981 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{\frac{i}{n}} \cdot \mathsf{fma}\left(100, \left(e^{\mathsf{log1p}\left(\left(\frac{i}{n}\right)\right) \cdot n}\right), -100\right)\\ \mathbf{elif}\;i \le 0.23936266726166916:\\ \;\;\;\;\frac{\frac{i \cdot \mathsf{fma}\left(i, \left(\mathsf{fma}\left(i, \frac{50}{3}, 50\right)\right), 100\right)}{i}}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 4.0545436608851676 \cdot 10^{+138}:\\ \;\;\;\;\frac{\mathsf{fma}\left(50, \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\log i \cdot \left(n \cdot \left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right)\right)\right), \left(\mathsf{fma}\left(100, \left(\log i \cdot n\right), \left(\left(\log i \cdot \left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot n\right)\right) \cdot 50 + 50 \cdot \left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right)\right)\right)\right)\right)\right)\right) - \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot n\right) \cdot \log n\right), \left(\mathsf{fma}\left(\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot \log i\right)\right), \frac{50}{3}, \left(100 \cdot \left(n \cdot \log n\right)\right)\right)\right)\right) + \left(\mathsf{fma}\left(\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot \log i\right)\right), \frac{100}{3}, \left(\left(\log i \cdot 50\right) \cdot \left(\log n \cdot \left(n \cdot n\right)\right)\right)\right) + \left(\log i \cdot 50\right) \cdot \left(\log n \cdot \left(n \cdot n\right)\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 7.203465032064257 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{1}{\frac{i}{\mathsf{fma}\left(100, \left({\left(\frac{i}{n} + 1\right)}^{n}\right), -100\right)}}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(50, \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\log i \cdot \left(n \cdot \left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right)\right)\right), \left(\mathsf{fma}\left(100, \left(\log i \cdot n\right), \left(\left(\log i \cdot \left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot n\right)\right) \cdot 50 + 50 \cdot \left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right)\right)\right)\right)\right)\right)\right) - \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot n\right) \cdot \log n\right), \left(\mathsf{fma}\left(\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot \log i\right)\right), \frac{50}{3}, \left(100 \cdot \left(n \cdot \log n\right)\right)\right)\right)\right) + \left(\mathsf{fma}\left(\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot \log i\right)\right), \frac{100}{3}, \left(\left(\log i \cdot 50\right) \cdot \left(\log n \cdot \left(n \cdot n\right)\right)\right)\right) + \left(\log i \cdot 50\right) \cdot \left(\log n \cdot \left(n \cdot n\right)\right)\right)\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (i n)
  :name "Compound Interest"

  :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))))