Compound Interest

Average Error: 42.3 → 21.3

Time: 25.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -4.919477946002002 \cdot 10^{+83}:\\ \;\;\;\;100 \cdot \left(\left(n + i \cdot \left(\frac{1}{2} \cdot n\right)\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{6}\right) \cdot n\right)\\ \mathbf{elif}\;n \le -1.1163211071803005 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{i}{n}} \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100\right)\\ \mathbf{elif}\;n \le 6.743108364226336 \cdot 10^{-114}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(n + i \cdot \left(\frac{1}{2} \cdot n\right)\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{6}\right) \cdot n\right)\\ \end{array}\]

C

double f(double i, double n) {
        double r6144268 = 100.0;
        double r6144269 = 1.0;
        double r6144270 = i;
        double r6144271 = n;
        double r6144272 = r6144270 / r6144271;
        double r6144273 = r6144269 + r6144272;
        double r6144274 = pow(r6144273, r6144271);
        double r6144275 = r6144274 - r6144269;
        double r6144276 = r6144275 / r6144272;
        double r6144277 = r6144268 * r6144276;
        return r6144277;
}

↓

double f(double i, double n) {
        double r6144278 = n;
        double r6144279 = -4.919477946002002e+83;
        bool r6144280 = r6144278 <= r6144279;
        double r6144281 = 100.0;
        double r6144282 = i;
        double r6144283 = 0.5;
        double r6144284 = r6144283 * r6144278;
        double r6144285 = r6144282 * r6144284;
        double r6144286 = r6144278 + r6144285;
        double r6144287 = r6144282 * r6144282;
        double r6144288 = 0.16666666666666666;
        double r6144289 = r6144287 * r6144288;
        double r6144290 = r6144289 * r6144278;
        double r6144291 = r6144286 + r6144290;
        double r6144292 = r6144281 * r6144291;
        double r6144293 = -1.1163211071803005e-19;
        bool r6144294 = r6144278 <= r6144293;
        double r6144295 = 1.0;
        double r6144296 = r6144282 / r6144278;
        double r6144297 = r6144295 / r6144296;
        double r6144298 = r6144295 + r6144296;
        double r6144299 = pow(r6144298, r6144278);
        double r6144300 = r6144299 - r6144295;
        double r6144301 = r6144300 * r6144281;
        double r6144302 = r6144297 * r6144301;
        double r6144303 = 6.743108364226336e-114;
        bool r6144304 = r6144278 <= r6144303;
        double r6144305 = 0.0;
        double r6144306 = r6144304 ? r6144305 : r6144292;
        double r6144307 = r6144294 ? r6144302 : r6144306;
        double r6144308 = r6144280 ? r6144292 : r6144307;
        return r6144308;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.3
Target	41.9
Herbie	21.3

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

Derivation

1. Split input into 3 regimes

2. if n < -4.919477946002002e+83 or 6.743108364226336e-114 < n

   1. Initial program 55.1

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

   2. Taylor expanded around 0 38.4

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

   3. Simplified 38.4

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

   4. Taylor expanded around 0 20.5

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

   5. Simplified 20.5

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

   if -4.919477946002002e+83 < n < -1.1163211071803005e-19

   1. Initial program 31.3

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

   3. Applied div-inv 31.3

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

   4. Applied associate-*r* 31.3

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

   if -1.1163211071803005e-19 < n < 6.743108364226336e-114

   1. Initial program 26.3

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

   2. Taylor expanded around 0 19.7

      \[\leadsto \color{blue}{0}\]
3. Recombined 3 regimes into one program.

4. Final simplification 21.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -4.919477946002002 \cdot 10^{+83}:\\ \;\;\;\;100 \cdot \left(\left(n + i \cdot \left(\frac{1}{2} \cdot n\right)\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{6}\right) \cdot n\right)\\ \mathbf{elif}\;n \le -1.1163211071803005 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{i}{n}} \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100\right)\\ \mathbf{elif}\;n \le 6.743108364226336 \cdot 10^{-114}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(n + i \cdot \left(\frac{1}{2} \cdot n\right)\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{6}\right) \cdot n\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (i n)
  :name "Compound Interest"

  :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))))