Compound Interest

Average Error: 43.1 → 21.7

Time: 29.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -0.008529206764181104793998144941724603995681:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 4.039962480132392563803023222135379910469:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(1, i, \left(i \cdot i\right) \cdot 0.5\right)\right) - \left(\left(i \cdot i\right) \cdot \log 1\right) \cdot 0.5}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r9543179 = 100.0;
        double r9543180 = 1.0;
        double r9543181 = i;
        double r9543182 = n;
        double r9543183 = r9543181 / r9543182;
        double r9543184 = r9543180 + r9543183;
        double r9543185 = pow(r9543184, r9543182);
        double r9543186 = r9543185 - r9543180;
        double r9543187 = r9543186 / r9543183;
        double r9543188 = r9543179 * r9543187;
        return r9543188;
}

↓

double f(double i, double n) {
        double r9543189 = i;
        double r9543190 = -0.008529206764181105;
        bool r9543191 = r9543189 <= r9543190;
        double r9543192 = 100.0;
        double r9543193 = r9543192 / r9543189;
        double r9543194 = 1.0;
        double r9543195 = n;
        double r9543196 = r9543189 / r9543195;
        double r9543197 = r9543194 + r9543196;
        double r9543198 = pow(r9543197, r9543195);
        double r9543199 = r9543198 - r9543194;
        double r9543200 = 1.0;
        double r9543201 = r9543200 / r9543195;
        double r9543202 = r9543199 / r9543201;
        double r9543203 = r9543193 * r9543202;
        double r9543204 = 4.039962480132393;
        bool r9543205 = r9543189 <= r9543204;
        double r9543206 = log(r9543194);
        double r9543207 = r9543189 * r9543189;
        double r9543208 = 0.5;
        double r9543209 = r9543207 * r9543208;
        double r9543210 = fma(r9543194, r9543189, r9543209);
        double r9543211 = fma(r9543206, r9543195, r9543210);
        double r9543212 = r9543207 * r9543206;
        double r9543213 = r9543212 * r9543208;
        double r9543214 = r9543211 - r9543213;
        double r9543215 = r9543214 / r9543189;
        double r9543216 = r9543215 * r9543195;
        double r9543217 = r9543192 * r9543216;
        double r9543218 = r9543205 ? r9543217 : r9543203;
        double r9543219 = r9543191 ? r9543203 : r9543218;
        return r9543219;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.1
Target	42.9
Herbie	21.7

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

Derivation

1. Split input into 2 regimes

2. if i < -0.008529206764181105 or 4.039962480132393 < i

   1. Initial program 30.1

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

   3. Applied div-inv 30.1

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

   4. Applied *-un-lft-identity 30.1

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

   5. Applied times-frac 30.6

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

   6. Applied associate-*r* 30.6

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

   7. Simplified 30.6

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

   if -0.008529206764181105 < i < 4.039962480132393

   1. Initial program 50.8

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

   2. Taylor expanded around 0 33.6

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

   3. Simplified 33.6

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(1, i, \left(i \cdot i\right) \cdot 0.5\right)\right) - \left(\left(i \cdot i\right) \cdot \log 1\right) \cdot 0.5}}{\frac{i}{n}}\]
   4. Using strategy rm

   5. Applied associate-/r/ 16.4

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(1, i, \left(i \cdot i\right) \cdot 0.5\right)\right) - \left(\left(i \cdot i\right) \cdot \log 1\right) \cdot 0.5}{i} \cdot n\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 21.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.008529206764181104793998144941724603995681:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 4.039962480132392563803023222135379910469:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(1, i, \left(i \cdot i\right) \cdot 0.5\right)\right) - \left(\left(i \cdot i\right) \cdot \log 1\right) \cdot 0.5}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (i n)
  :name "Compound Interest"

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))