Compound Interest

Average Error: 42.5 → 33.3

Time: 1.8m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le \frac{-3004645614962051}{2251799813685248}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}}{\frac{i}{n} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)} - \frac{\frac{1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le \frac{-7883338139594251}{1.352433999707303030661989849386282512688 \cdot 10^{272}}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\frac{1}{2} \cdot {i}^{2} + \log 1 \cdot n\right)\right) - \frac{1}{2} \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le \frac{2454475895511641}{6.575169876935466884051237353600160831938 \cdot 10^{209}}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\frac{1}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{1}}{i} \cdot \frac{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{1}{n}}\right)\\ \mathbf{elif}\;i \le \frac{5629288812994619}{65536}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\frac{1}{2} \cdot {i}^{2} + \log 1 \cdot n\right)\right) - \frac{1}{2} \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}}{\frac{i}{n} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)} - \frac{\frac{1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\right)\\ \end{array}\]

C

double f(double i, double n) {
        double r534374 = 100.0;
        double r534375 = 1.0;
        double r534376 = i;
        double r534377 = n;
        double r534378 = r534376 / r534377;
        double r534379 = r534375 + r534378;
        double r534380 = pow(r534379, r534377);
        double r534381 = r534380 - r534375;
        double r534382 = r534381 / r534378;
        double r534383 = r534374 * r534382;
        return r534383;
}

↓

double f(double i, double n) {
        double r534384 = i;
        double r534385 = -3004645614962051.0;
        double r534386 = 2251799813685248.0;
        double r534387 = r534385 / r534386;
        bool r534388 = r534384 <= r534387;
        double r534389 = 100.0;
        double r534390 = 1.0;
        double r534391 = n;
        double r534392 = r534384 / r534391;
        double r534393 = r534390 + r534392;
        double r534394 = 2.0;
        double r534395 = r534394 * r534391;
        double r534396 = pow(r534393, r534395);
        double r534397 = pow(r534393, r534391);
        double r534398 = r534397 + r534390;
        double r534399 = r534392 * r534398;
        double r534400 = r534396 / r534399;
        double r534401 = r534390 * r534390;
        double r534402 = r534401 / r534398;
        double r534403 = r534402 / r534392;
        double r534404 = r534400 - r534403;
        double r534405 = r534389 * r534404;
        double r534406 = -7883338139594251.0;
        double r534407 = 1.352433999707303e+272;
        double r534408 = r534406 / r534407;
        bool r534409 = r534384 <= r534408;
        double r534410 = r534390 * r534384;
        double r534411 = 2.0;
        double r534412 = r534390 / r534411;
        double r534413 = pow(r534384, r534394);
        double r534414 = r534412 * r534413;
        double r534415 = log(r534390);
        double r534416 = r534415 * r534391;
        double r534417 = r534414 + r534416;
        double r534418 = r534410 + r534417;
        double r534419 = r534413 * r534415;
        double r534420 = r534412 * r534419;
        double r534421 = r534418 - r534420;
        double r534422 = r534421 / r534392;
        double r534423 = r534389 * r534422;
        double r534424 = 2454475895511641.0;
        double r534425 = 6.575169876935467e+209;
        double r534426 = r534424 / r534425;
        bool r534427 = r534384 <= r534426;
        double r534428 = 1.0;
        double r534429 = r534396 + r534401;
        double r534430 = cbrt(r534429);
        double r534431 = r534430 * r534430;
        double r534432 = r534428 / r534431;
        double r534433 = r534432 / r534428;
        double r534434 = r534433 / r534384;
        double r534435 = r534394 * r534395;
        double r534436 = pow(r534393, r534435);
        double r534437 = r534401 * r534401;
        double r534438 = -r534437;
        double r534439 = r534436 + r534438;
        double r534440 = r534439 / r534430;
        double r534441 = r534440 / r534398;
        double r534442 = r534428 / r534391;
        double r534443 = r534441 / r534442;
        double r534444 = r534434 * r534443;
        double r534445 = r534389 * r534444;
        double r534446 = 5629288812994619.0;
        double r534447 = 65536.0;
        double r534448 = r534446 / r534447;
        bool r534449 = r534384 <= r534448;
        double r534450 = r534449 ? r534423 : r534405;
        double r534451 = r534427 ? r534445 : r534450;
        double r534452 = r534409 ? r534423 : r534451;
        double r534453 = r534388 ? r534405 : r534452;
        return r534453;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.5
Target	42.4
Herbie	33.3

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

Derivation

1. Split input into 3 regimes

2. if i < -1.3343306970279527 or 85896130569.3759 < i

   1. Initial program 28.3

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

   3. Applied flip-- 28.3

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

   4. Simplified 28.3

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

   6. Applied unsub-neg 28.3

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

   7. Applied div-sub 28.3

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

   8. Applied div-sub 28.3

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

   9. Simplified 28.3

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

   if -1.3343306970279527 < i < -5.829000262711808e-257 or 3.7329467397055525e-195 < i < 85896130569.3759

   1. Initial program 50.9

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

   2. Taylor expanded around 0 31.4

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

   3. Simplified 31.4

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

   if -5.829000262711808e-257 < i < 3.7329467397055525e-195

   1. Initial program 48.8

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

   3. Applied flip-- 48.8

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

   4. Simplified 48.8

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

   6. Applied flip-+ 48.8

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

   7. Simplified 48.8

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

   8. Simplified 48.8

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

   10. Applied div-inv 48.8

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

   11. Applied *-un-lft-identity 48.8

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

   12. Applied add-cube-cbrt 48.8

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

   13. Applied *-un-lft-identity 48.8

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

   14. Applied times-frac 48.8

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

   15. Applied times-frac 48.8

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

   16. Applied times-frac 48.2

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

4. Final simplification 33.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le \frac{-3004645614962051}{2251799813685248}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}}{\frac{i}{n} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)} - \frac{\frac{1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le \frac{-7883338139594251}{1.352433999707303030661989849386282512688 \cdot 10^{272}}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\frac{1}{2} \cdot {i}^{2} + \log 1 \cdot n\right)\right) - \frac{1}{2} \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le \frac{2454475895511641}{6.575169876935466884051237353600160831938 \cdot 10^{209}}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\frac{1}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{1}}{i} \cdot \frac{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{1}{n}}\right)\\ \mathbf{elif}\;i \le \frac{5629288812994619}{65536}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\frac{1}{2} \cdot {i}^{2} + \log 1 \cdot n\right)\right) - \frac{1}{2} \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}}{\frac{i}{n} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)} - \frac{\frac{1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))))