Compound Interest

Average Error: 42.9 → 22.4

Time: 19.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -4.140199537422161434777996490013449659769 \cdot 10^{89}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -9.656677881721989431356871738273518499883 \cdot 10^{68}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -526028771461041728:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -5.589864261526092662299462940489906799584 \cdot 10^{-297}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 1.231411710286937002747903660246880103028 \cdot 10^{-203}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \end{array}\]

C

double f(double i, double n) {
        double r154347 = 100.0;
        double r154348 = 1.0;
        double r154349 = i;
        double r154350 = n;
        double r154351 = r154349 / r154350;
        double r154352 = r154348 + r154351;
        double r154353 = pow(r154352, r154350);
        double r154354 = r154353 - r154348;
        double r154355 = r154354 / r154351;
        double r154356 = r154347 * r154355;
        return r154356;
}

↓

double f(double i, double n) {
        double r154357 = n;
        double r154358 = -4.1401995374221614e+89;
        bool r154359 = r154357 <= r154358;
        double r154360 = 100.0;
        double r154361 = 1.0;
        double r154362 = i;
        double r154363 = r154361 * r154362;
        double r154364 = 0.5;
        double r154365 = 2.0;
        double r154366 = pow(r154362, r154365);
        double r154367 = r154364 * r154366;
        double r154368 = log(r154361);
        double r154369 = r154368 * r154357;
        double r154370 = r154367 + r154369;
        double r154371 = r154363 + r154370;
        double r154372 = r154366 * r154368;
        double r154373 = r154364 * r154372;
        double r154374 = r154371 - r154373;
        double r154375 = r154374 / r154362;
        double r154376 = r154360 * r154375;
        double r154377 = r154376 * r154357;
        double r154378 = -9.65667788172199e+68;
        bool r154379 = r154357 <= r154378;
        double r154380 = r154362 / r154357;
        double r154381 = r154361 + r154380;
        double r154382 = pow(r154381, r154357);
        double r154383 = r154382 - r154361;
        double r154384 = r154360 * r154383;
        double r154385 = r154384 / r154380;
        double r154386 = -5.260287714610417e+17;
        bool r154387 = r154357 <= r154386;
        double r154388 = -5.589864261526093e-297;
        bool r154389 = r154357 <= r154388;
        double r154390 = 1.231411710286937e-203;
        bool r154391 = r154357 <= r154390;
        double r154392 = 1.0;
        double r154393 = r154369 + r154392;
        double r154394 = r154363 + r154393;
        double r154395 = r154394 - r154361;
        double r154396 = r154395 / r154380;
        double r154397 = r154360 * r154396;
        double r154398 = r154391 ? r154397 : r154377;
        double r154399 = r154389 ? r154385 : r154398;
        double r154400 = r154387 ? r154377 : r154399;
        double r154401 = r154379 ? r154385 : r154400;
        double r154402 = r154359 ? r154377 : r154401;
        return r154402;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.9
Target	42.6
Herbie	22.4

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

Derivation

1. Split input into 3 regimes

2. if n < -4.1401995374221614e+89 or -9.65667788172199e+68 < n < -5.260287714610417e+17 or 1.231411710286937e-203 < n

   1. Initial program 53.0

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

   2. Taylor expanded around 0 40.5

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

   4. Applied associate-/r/ 24.0

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

   5. Applied associate-*r* 24.0

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

   if -4.1401995374221614e+89 < n < -9.65667788172199e+68 or -5.260287714610417e+17 < n < -5.589864261526093e-297

   1. Initial program 18.8

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

   3. Applied associate-*r/ 18.8

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

   if -5.589864261526093e-297 < n < 1.231411710286937e-203

   1. Initial program 34.9

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

   2. Taylor expanded around 0 20.1

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

4. Final simplification 22.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -4.140199537422161434777996490013449659769 \cdot 10^{89}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -9.656677881721989431356871738273518499883 \cdot 10^{68}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -526028771461041728:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -5.589864261526092662299462940489906799584 \cdot 10^{-297}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 1.231411710286937002747903660246880103028 \cdot 10^{-203}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(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))))