Compound Interest

Average Error: 42.2 → 24.2

Time: 36.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -2.260823056891884243812192182030091462802 \cdot 10^{130}:\\ \;\;\;\;\left(100 \cdot \frac{i \cdot \left(1 + 0.5 \cdot i\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -4.971792173734567753657378996256742049344 \cdot 10^{92}:\\ \;\;\;\;\frac{\left(100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \cdot n}{i}\\ \mathbf{elif}\;n \le -3.460770537187881137122290954175863408168 \cdot 10^{79} \lor \neg \left(n \le 6.434758453518077622758166934880438333831 \cdot 10^{-141}\right):\\ \;\;\;\;\left(100 \cdot \frac{i \cdot \left(1 + 0.5 \cdot i\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right)}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r240364 = 100.0;
        double r240365 = 1.0;
        double r240366 = i;
        double r240367 = n;
        double r240368 = r240366 / r240367;
        double r240369 = r240365 + r240368;
        double r240370 = pow(r240369, r240367);
        double r240371 = r240370 - r240365;
        double r240372 = r240371 / r240368;
        double r240373 = r240364 * r240372;
        return r240373;
}

↓

double f(double i, double n) {
        double r240374 = n;
        double r240375 = -2.260823056891884e+130;
        bool r240376 = r240374 <= r240375;
        double r240377 = 100.0;
        double r240378 = i;
        double r240379 = 1.0;
        double r240380 = 0.5;
        double r240381 = r240380 * r240378;
        double r240382 = r240379 + r240381;
        double r240383 = r240378 * r240382;
        double r240384 = log(r240379);
        double r240385 = 2.0;
        double r240386 = pow(r240378, r240385);
        double r240387 = r240380 * r240386;
        double r240388 = r240374 - r240387;
        double r240389 = r240384 * r240388;
        double r240390 = r240383 + r240389;
        double r240391 = r240390 / r240378;
        double r240392 = r240377 * r240391;
        double r240393 = r240392 * r240374;
        double r240394 = -4.971792173734568e+92;
        bool r240395 = r240374 <= r240394;
        double r240396 = r240378 / r240374;
        double r240397 = r240379 + r240396;
        double r240398 = pow(r240397, r240374);
        double r240399 = r240398 - r240379;
        double r240400 = r240377 * r240399;
        double r240401 = r240400 * r240374;
        double r240402 = r240401 / r240378;
        double r240403 = -3.460770537187881e+79;
        bool r240404 = r240374 <= r240403;
        double r240405 = 6.434758453518078e-141;
        bool r240406 = r240374 <= r240405;
        double r240407 = !r240406;
        bool r240408 = r240404 || r240407;
        double r240409 = exp(r240399);
        double r240410 = log(r240409);
        double r240411 = r240410 / r240396;
        double r240412 = r240377 * r240411;
        double r240413 = r240408 ? r240393 : r240412;
        double r240414 = r240395 ? r240402 : r240413;
        double r240415 = r240376 ? r240393 : r240414;
        return r240415;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.2
Target	42.3
Herbie	24.2

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

Derivation

1. Split input into 3 regimes

2. if n < -2.260823056891884e+130 or -4.971792173734568e+92 < n < -3.460770537187881e+79 or 6.434758453518078e-141 < n

   1. Initial program 55.8

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

   3. Applied associate-/r/ 55.4

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

   4. Applied associate-*r* 55.4

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

   5. Taylor expanded around 0 21.1

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

   6. Simplified 21.1

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

   if -2.260823056891884e+130 < n < -4.971792173734568e+92

   1. Initial program 38.6

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

   3. Applied associate-/r/ 38.4

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

   4. Applied associate-*r* 38.3

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

   6. Applied associate-*r/ 38.3

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

   7. Applied associate-*l/ 38.3

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

   if -3.460770537187881e+79 < n < 6.434758453518078e-141

   1. Initial program 26.5

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

   3. Applied add-log-exp 26.5

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

   4. Applied add-log-exp 26.5

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

   5. Applied diff-log 26.5

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

   6. Simplified 26.5

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

4. Final simplification 24.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.260823056891884243812192182030091462802 \cdot 10^{130}:\\ \;\;\;\;\left(100 \cdot \frac{i \cdot \left(1 + 0.5 \cdot i\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -4.971792173734567753657378996256742049344 \cdot 10^{92}:\\ \;\;\;\;\frac{\left(100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \cdot n}{i}\\ \mathbf{elif}\;n \le -3.460770537187881137122290954175863408168 \cdot 10^{79} \lor \neg \left(n \le 6.434758453518077622758166934880438333831 \cdot 10^{-141}\right):\\ \;\;\;\;\left(100 \cdot \frac{i \cdot \left(1 + 0.5 \cdot i\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(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))))