Compound Interest

Average Error: 42.3 → 32.9

Time: 17.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -4.726437827530884962234924984159079031087 \cdot 10^{-4}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}^{3} + {\left(-1 \cdot 1\right)}^{3}}{\left(-1 \cdot 1\right) \cdot \left(\left(-1 \cdot 1\right) - {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) + {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)}}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.964684591502211290172320573095869134171 \cdot 10^{-206}:\\ \;\;\;\;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 -2.882850856787981581879145472439421418808 \cdot 10^{-226}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n}}\right) - 1}}{i}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\\ \mathbf{elif}\;i \le -1.019093496626147386492788288826940719188 \cdot 10^{-241}:\\ \;\;\;\;100 \cdot \frac{\frac{2 \cdot \left(i + {i}^{2}\right) + \left(2 \cdot \left(\log 1 \cdot n\right) - 2 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -3.645341750186080323375566759305922898034 \cdot 10^{-293}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n}}\right) - 1}}{i}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 9.27900448690603418810951552586629986763:\\ \;\;\;\;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{else}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}^{3} + {\left(-1 \cdot 1\right)}^{3}}{\left(-1 \cdot 1\right) \cdot \left(\left(-1 \cdot 1\right) - {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) + {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)}}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r171382 = 100.0;
        double r171383 = 1.0;
        double r171384 = i;
        double r171385 = n;
        double r171386 = r171384 / r171385;
        double r171387 = r171383 + r171386;
        double r171388 = pow(r171387, r171385);
        double r171389 = r171388 - r171383;
        double r171390 = r171389 / r171386;
        double r171391 = r171382 * r171390;
        return r171391;
}

↓

double f(double i, double n) {
        double r171392 = i;
        double r171393 = -0.0004726437827530885;
        bool r171394 = r171392 <= r171393;
        double r171395 = 100.0;
        double r171396 = 1.0;
        double r171397 = n;
        double r171398 = r171392 / r171397;
        double r171399 = r171396 + r171398;
        double r171400 = 2.0;
        double r171401 = r171400 * r171397;
        double r171402 = pow(r171399, r171401);
        double r171403 = 3.0;
        double r171404 = pow(r171402, r171403);
        double r171405 = r171396 * r171396;
        double r171406 = -r171405;
        double r171407 = pow(r171406, r171403);
        double r171408 = r171404 + r171407;
        double r171409 = r171406 - r171402;
        double r171410 = r171406 * r171409;
        double r171411 = r171400 * r171401;
        double r171412 = pow(r171399, r171411);
        double r171413 = r171410 + r171412;
        double r171414 = r171408 / r171413;
        double r171415 = pow(r171399, r171397);
        double r171416 = r171415 + r171396;
        double r171417 = r171414 / r171416;
        double r171418 = r171417 / r171398;
        double r171419 = r171395 * r171418;
        double r171420 = -1.9646845915022113e-206;
        bool r171421 = r171392 <= r171420;
        double r171422 = r171396 * r171392;
        double r171423 = 0.5;
        double r171424 = pow(r171392, r171400);
        double r171425 = r171423 * r171424;
        double r171426 = log(r171396);
        double r171427 = r171426 * r171397;
        double r171428 = r171425 + r171427;
        double r171429 = r171422 + r171428;
        double r171430 = r171424 * r171426;
        double r171431 = r171423 * r171430;
        double r171432 = r171429 - r171431;
        double r171433 = r171432 / r171398;
        double r171434 = r171395 * r171433;
        double r171435 = -2.8828508567879816e-226;
        bool r171436 = r171392 <= r171435;
        double r171437 = r171415 - r171396;
        double r171438 = cbrt(r171437);
        double r171439 = exp(r171415);
        double r171440 = log(r171439);
        double r171441 = r171440 - r171396;
        double r171442 = cbrt(r171441);
        double r171443 = r171438 * r171442;
        double r171444 = r171443 / r171392;
        double r171445 = r171395 * r171444;
        double r171446 = 1.0;
        double r171447 = r171446 / r171397;
        double r171448 = r171438 / r171447;
        double r171449 = r171445 * r171448;
        double r171450 = -1.0190934966261474e-241;
        bool r171451 = r171392 <= r171450;
        double r171452 = 2.0;
        double r171453 = r171392 + r171424;
        double r171454 = r171452 * r171453;
        double r171455 = r171400 * r171427;
        double r171456 = r171452 * r171430;
        double r171457 = r171455 - r171456;
        double r171458 = r171454 + r171457;
        double r171459 = r171458 / r171416;
        double r171460 = r171459 / r171398;
        double r171461 = r171395 * r171460;
        double r171462 = -3.6453417501860803e-293;
        bool r171463 = r171392 <= r171462;
        double r171464 = 9.279004486906034;
        bool r171465 = r171392 <= r171464;
        double r171466 = r171465 ? r171434 : r171419;
        double r171467 = r171463 ? r171449 : r171466;
        double r171468 = r171451 ? r171461 : r171467;
        double r171469 = r171436 ? r171449 : r171468;
        double r171470 = r171421 ? r171434 : r171469;
        double r171471 = r171394 ? r171419 : r171470;
        return r171471;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.3
Target	42.6
Herbie	32.9

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

Derivation

1. Split input into 4 regimes

2. if i < -0.0004726437827530885 or 9.279004486906034 < i

   1. Initial program 29.1

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

   3. Applied flip-- 29.1

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

   4. Simplified 29.1

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

   6. Applied flip3-+ 29.1

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

   7. Simplified 29.1

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

   if -0.0004726437827530885 < i < -1.9646845915022113e-206 or -3.6453417501860803e-293 < i < 9.279004486906034

   1. Initial program 50.8

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

   2. Taylor expanded around 0 33.1

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

   if -1.9646845915022113e-206 < i < -2.8828508567879816e-226 or -1.0190934966261474e-241 < i < -3.6453417501860803e-293

   1. Initial program 47.7

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

   3. Applied div-inv 47.7

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

   4. Applied add-cube-cbrt 47.7

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

   5. Applied times-frac 47.1

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

   6. Applied associate-*r* 47.1

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

   8. Applied add-log-exp 47.1

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

   if -2.8828508567879816e-226 < i < -1.0190934966261474e-241

   1. Initial program 46.5

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

   3. Applied flip-- 46.5

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

   4. Simplified 46.5

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

   5. Taylor expanded around 0 42.2

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

   6. Simplified 42.2

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

4. Final simplification 32.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -4.726437827530884962234924984159079031087 \cdot 10^{-4}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}^{3} + {\left(-1 \cdot 1\right)}^{3}}{\left(-1 \cdot 1\right) \cdot \left(\left(-1 \cdot 1\right) - {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) + {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)}}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.964684591502211290172320573095869134171 \cdot 10^{-206}:\\ \;\;\;\;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 -2.882850856787981581879145472439421418808 \cdot 10^{-226}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n}}\right) - 1}}{i}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\\ \mathbf{elif}\;i \le -1.019093496626147386492788288826940719188 \cdot 10^{-241}:\\ \;\;\;\;100 \cdot \frac{\frac{2 \cdot \left(i + {i}^{2}\right) + \left(2 \cdot \left(\log 1 \cdot n\right) - 2 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -3.645341750186080323375566759305922898034 \cdot 10^{-293}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n}}\right) - 1}}{i}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 9.27900448690603418810951552586629986763:\\ \;\;\;\;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{else}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}^{3} + {\left(-1 \cdot 1\right)}^{3}}{\left(-1 \cdot 1\right) \cdot \left(\left(-1 \cdot 1\right) - {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) + {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)}}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(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))))