Compound Interest

Average Error: 43.1 → 30.0

Time: 30.8s

Precision: 64

Math

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

↓

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

C

double f(double i, double n) {
        double r130334 = 100.0;
        double r130335 = 1.0;
        double r130336 = i;
        double r130337 = n;
        double r130338 = r130336 / r130337;
        double r130339 = r130335 + r130338;
        double r130340 = pow(r130339, r130337);
        double r130341 = r130340 - r130335;
        double r130342 = r130341 / r130338;
        double r130343 = r130334 * r130342;
        return r130343;
}

↓

double f(double i, double n) {
        double r130344 = i;
        double r130345 = -42606.74875587942;
        bool r130346 = r130344 <= r130345;
        double r130347 = 100.0;
        double r130348 = n;
        double r130349 = r130344 / r130348;
        double r130350 = pow(r130349, r130348);
        double r130351 = 1.0;
        double r130352 = r130350 - r130351;
        double r130353 = r130344 / r130352;
        double r130354 = r130348 / r130353;
        double r130355 = r130347 * r130354;
        double r130356 = 3.332988420430884e-27;
        bool r130357 = r130344 <= r130356;
        double r130358 = 0.5;
        double r130359 = 2.0;
        double r130360 = pow(r130344, r130359);
        double r130361 = log(r130351);
        double r130362 = r130361 * r130348;
        double r130363 = fma(r130358, r130360, r130362);
        double r130364 = fma(r130351, r130344, r130363);
        double r130365 = r130360 * r130361;
        double r130366 = r130358 * r130365;
        double r130367 = r130364 - r130366;
        double r130368 = r130367 / r130349;
        double r130369 = r130347 * r130368;
        double r130370 = 9.944860624286458e+141;
        bool r130371 = r130344 <= r130370;
        double r130372 = r130347 * r130348;
        double r130373 = 1.0;
        double r130374 = 0.5;
        double r130375 = log(r130344);
        double r130376 = pow(r130375, r130359);
        double r130377 = pow(r130348, r130359);
        double r130378 = r130376 * r130377;
        double r130379 = log(r130348);
        double r130380 = pow(r130379, r130359);
        double r130381 = r130377 * r130380;
        double r130382 = 0.16666666666666666;
        double r130383 = 3.0;
        double r130384 = pow(r130375, r130383);
        double r130385 = pow(r130348, r130383);
        double r130386 = r130384 * r130385;
        double r130387 = r130385 * r130380;
        double r130388 = r130375 * r130387;
        double r130389 = r130374 * r130388;
        double r130390 = fma(r130375, r130348, r130389);
        double r130391 = fma(r130382, r130386, r130390);
        double r130392 = fma(r130374, r130381, r130391);
        double r130393 = fma(r130374, r130378, r130392);
        double r130394 = r130348 * r130348;
        double r130395 = fma(r130394, r130375, r130348);
        double r130396 = r130379 * r130395;
        double r130397 = r130393 - r130396;
        double r130398 = r130385 * r130379;
        double r130399 = r130376 * r130398;
        double r130400 = pow(r130379, r130383);
        double r130401 = r130385 * r130400;
        double r130402 = r130382 * r130401;
        double r130403 = fma(r130374, r130399, r130402);
        double r130404 = r130397 - r130403;
        double r130405 = r130344 / r130404;
        double r130406 = r130373 / r130405;
        double r130407 = r130372 * r130406;
        double r130408 = 2.568245662391043e+231;
        bool r130409 = r130344 <= r130408;
        double r130410 = r130351 + r130349;
        double r130411 = pow(r130410, r130348);
        double r130412 = r130411 - r130351;
        double r130413 = cbrt(r130412);
        double r130414 = r130413 * r130413;
        double r130415 = r130414 / r130344;
        double r130416 = cbrt(r130410);
        double r130417 = r130416 * r130416;
        double r130418 = pow(r130417, r130348);
        double r130419 = pow(r130416, r130348);
        double r130420 = r130418 * r130419;
        double r130421 = r130420 - r130351;
        double r130422 = cbrt(r130421);
        double r130423 = r130422 * r130348;
        double r130424 = r130415 * r130423;
        double r130425 = r130347 * r130424;
        double r130426 = fma(r130361, r130348, r130373);
        double r130427 = fma(r130351, r130344, r130426);
        double r130428 = r130427 - r130351;
        double r130429 = r130428 / r130349;
        double r130430 = r130347 * r130429;
        double r130431 = r130409 ? r130425 : r130430;
        double r130432 = r130371 ? r130407 : r130431;
        double r130433 = r130357 ? r130369 : r130432;
        double r130434 = r130346 ? r130355 : r130433;
        return r130434;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.1
Target	42.8
Herbie	30.0

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

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

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

   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

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

   3. Simplified 34.2

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

   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

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

   5. Applied div-inv 39.2

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

   6. Applied associate-*r* 39.2

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

   7. Taylor expanded around 0 22.8

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

   8. Simplified 22.8

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

   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-cube-cbrt 32.0

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

   5. Applied times-frac 32.0

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

   6. Simplified 32.0

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

   8. Applied add-cube-cbrt 32.0

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

   9. Applied unpow-prod-down 31.9

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

   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. Simplified 36.1

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

4. Final simplification 30.0

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(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))))