Compound Interest

Average Error: 43.1 → 30.1

Time: 28.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -42606.74875587941642152145504951477050781:\\ \;\;\;\;\left(100 \cdot \left(n \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)\right)\right) \cdot \frac{1}{i}\\ \mathbf{elif}\;i \le 3.332988420430884258276291203877273900307 \cdot 10^{-27}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 9.944860624286458495633005962801713322847 \cdot 10^{141}:\\ \;\;\;\;\left(\left(\left(\frac{1}{6} \cdot \frac{{\left(\log i\right)}^{3}}{\frac{i}{100 \cdot {n}^{4}}} + \frac{1}{2} \cdot \left(\frac{100 \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right)}{i} + \frac{{\left(\log i\right)}^{2}}{\frac{i}{100 \cdot {n}^{3}}}\right)\right) + \left(\frac{\left({n}^{2} \cdot 100\right) \cdot \log i}{i} + \frac{\frac{1}{2} \cdot \left(\left(100 \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)\right) \cdot \log i\right)}{i}\right)\right) - \left(\frac{1}{2} \cdot \frac{{\left(\log i\right)}^{2}}{\frac{i}{\left(100 \cdot {n}^{4}\right) \cdot \log n}} + \frac{\frac{1}{6} \cdot \left(\left({n}^{4} \cdot {\left(\log n\right)}^{3}\right) \cdot 100\right)}{i}\right)\right) - \left(\frac{\left(\left({n}^{3} \cdot \log n\right) \cdot 100\right) \cdot \log i}{i} + \frac{100 \cdot \left({n}^{2} \cdot \log n\right)}{i}\right)\\ \mathbf{elif}\;i \le 2.568245662391043108180991997152758915531 \cdot 10^{231}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}}{i} \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} - \sqrt{1}\right) \cdot n\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r112465 = 100.0;
        double r112466 = 1.0;
        double r112467 = i;
        double r112468 = n;
        double r112469 = r112467 / r112468;
        double r112470 = r112466 + r112469;
        double r112471 = pow(r112470, r112468);
        double r112472 = r112471 - r112466;
        double r112473 = r112472 / r112469;
        double r112474 = r112465 * r112473;
        return r112474;
}

↓

double f(double i, double n) {
        double r112475 = i;
        double r112476 = -42606.74875587942;
        bool r112477 = r112475 <= r112476;
        double r112478 = 100.0;
        double r112479 = n;
        double r112480 = r112475 / r112479;
        double r112481 = pow(r112480, r112479);
        double r112482 = 1.0;
        double r112483 = r112481 - r112482;
        double r112484 = r112479 * r112483;
        double r112485 = r112478 * r112484;
        double r112486 = 1.0;
        double r112487 = r112486 / r112475;
        double r112488 = r112485 * r112487;
        double r112489 = 3.332988420430884e-27;
        bool r112490 = r112475 <= r112489;
        double r112491 = r112482 * r112475;
        double r112492 = 0.5;
        double r112493 = 2.0;
        double r112494 = pow(r112475, r112493);
        double r112495 = r112492 * r112494;
        double r112496 = log(r112482);
        double r112497 = r112496 * r112479;
        double r112498 = r112495 + r112497;
        double r112499 = r112491 + r112498;
        double r112500 = r112494 * r112496;
        double r112501 = r112492 * r112500;
        double r112502 = r112499 - r112501;
        double r112503 = r112502 / r112480;
        double r112504 = r112478 * r112503;
        double r112505 = 9.944860624286458e+141;
        bool r112506 = r112475 <= r112505;
        double r112507 = 0.16666666666666666;
        double r112508 = log(r112475);
        double r112509 = 3.0;
        double r112510 = pow(r112508, r112509);
        double r112511 = 4.0;
        double r112512 = pow(r112479, r112511);
        double r112513 = r112478 * r112512;
        double r112514 = r112475 / r112513;
        double r112515 = r112510 / r112514;
        double r112516 = r112507 * r112515;
        double r112517 = 0.5;
        double r112518 = pow(r112479, r112509);
        double r112519 = log(r112479);
        double r112520 = pow(r112519, r112493);
        double r112521 = r112518 * r112520;
        double r112522 = r112478 * r112521;
        double r112523 = r112522 / r112475;
        double r112524 = pow(r112508, r112493);
        double r112525 = r112478 * r112518;
        double r112526 = r112475 / r112525;
        double r112527 = r112524 / r112526;
        double r112528 = r112523 + r112527;
        double r112529 = r112517 * r112528;
        double r112530 = r112516 + r112529;
        double r112531 = pow(r112479, r112493);
        double r112532 = r112531 * r112478;
        double r112533 = r112532 * r112508;
        double r112534 = r112533 / r112475;
        double r112535 = r112512 * r112520;
        double r112536 = r112478 * r112535;
        double r112537 = r112536 * r112508;
        double r112538 = r112517 * r112537;
        double r112539 = r112538 / r112475;
        double r112540 = r112534 + r112539;
        double r112541 = r112530 + r112540;
        double r112542 = r112513 * r112519;
        double r112543 = r112475 / r112542;
        double r112544 = r112524 / r112543;
        double r112545 = r112517 * r112544;
        double r112546 = pow(r112519, r112509);
        double r112547 = r112512 * r112546;
        double r112548 = r112547 * r112478;
        double r112549 = r112507 * r112548;
        double r112550 = r112549 / r112475;
        double r112551 = r112545 + r112550;
        double r112552 = r112541 - r112551;
        double r112553 = r112518 * r112519;
        double r112554 = r112553 * r112478;
        double r112555 = r112554 * r112508;
        double r112556 = r112555 / r112475;
        double r112557 = r112531 * r112519;
        double r112558 = r112478 * r112557;
        double r112559 = r112558 / r112475;
        double r112560 = r112556 + r112559;
        double r112561 = r112552 - r112560;
        double r112562 = 2.568245662391043e+231;
        bool r112563 = r112475 <= r112562;
        double r112564 = r112482 + r112480;
        double r112565 = r112479 / r112493;
        double r112566 = pow(r112564, r112565);
        double r112567 = sqrt(r112482);
        double r112568 = r112566 + r112567;
        double r112569 = r112568 / r112475;
        double r112570 = r112566 - r112567;
        double r112571 = r112570 * r112479;
        double r112572 = r112569 * r112571;
        double r112573 = r112478 * r112572;
        double r112574 = r112497 + r112486;
        double r112575 = r112491 + r112574;
        double r112576 = r112575 - r112482;
        double r112577 = r112576 / r112480;
        double r112578 = r112478 * r112577;
        double r112579 = r112563 ? r112573 : r112578;
        double r112580 = r112506 ? r112561 : r112579;
        double r112581 = r112490 ? r112504 : r112580;
        double r112582 = r112477 ? r112488 : r112581;
        return r112582;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.1
Target	42.8
Herbie	30.1

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

Derivation

1. Split input into 5 regimes

2. if i < -42606.74875587942

   1. Initial program 27.8

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

   2. Taylor expanded around inf 64.0

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

   3. Simplified 18.8

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

   5. Applied div-inv 18.8

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

   6. Applied associate-*r* 19.0

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

   if -42606.74875587942 < i < 3.332988420430884e-27

   1. Initial program 50.7

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

   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 3.332988420430884e-27 < i < 9.944860624286458e+141

   1. Initial program 38.7

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

   2. Taylor expanded around inf 37.2

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

   3. Simplified 39.2

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

   5. Applied add-sqr-sqrt 39.2

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

   6. Applied associate-*l* 39.2

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

   7. Taylor expanded around 0 23.2

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

   8. Simplified 23.2

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

   if 9.944860624286458e+141 < i < 2.568245662391043e+231

   1. Initial program 32.0

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

   3. Applied div-inv 32.0

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

   4. Applied add-sqr-sqrt 32.0

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

   5. Applied sqr-pow 32.1

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

   6. Applied difference-of-squares 32.0

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

   7. Applied times-frac 32.0

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

   8. Simplified 32.0

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

   if 2.568245662391043e+231 < i

   1. Initial program 29.1

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

   2. Taylor expanded around 0 36.1

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right)} - 1}{\frac{i}{n}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 30.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -42606.74875587941642152145504951477050781:\\ \;\;\;\;\left(100 \cdot \left(n \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)\right)\right) \cdot \frac{1}{i}\\ \mathbf{elif}\;i \le 3.332988420430884258276291203877273900307 \cdot 10^{-27}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 9.944860624286458495633005962801713322847 \cdot 10^{141}:\\ \;\;\;\;\left(\left(\left(\frac{1}{6} \cdot \frac{{\left(\log i\right)}^{3}}{\frac{i}{100 \cdot {n}^{4}}} + \frac{1}{2} \cdot \left(\frac{100 \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right)}{i} + \frac{{\left(\log i\right)}^{2}}{\frac{i}{100 \cdot {n}^{3}}}\right)\right) + \left(\frac{\left({n}^{2} \cdot 100\right) \cdot \log i}{i} + \frac{\frac{1}{2} \cdot \left(\left(100 \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)\right) \cdot \log i\right)}{i}\right)\right) - \left(\frac{1}{2} \cdot \frac{{\left(\log i\right)}^{2}}{\frac{i}{\left(100 \cdot {n}^{4}\right) \cdot \log n}} + \frac{\frac{1}{6} \cdot \left(\left({n}^{4} \cdot {\left(\log n\right)}^{3}\right) \cdot 100\right)}{i}\right)\right) - \left(\frac{\left(\left({n}^{3} \cdot \log n\right) \cdot 100\right) \cdot \log i}{i} + \frac{100 \cdot \left({n}^{2} \cdot \log n\right)}{i}\right)\\ \mathbf{elif}\;i \le 2.568245662391043108180991997152758915531 \cdot 10^{231}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}}{i} \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} - \sqrt{1}\right) \cdot n\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))))