Compound Interest

Average Error: 41.1 → 24.6

Time: 17.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1.8403630755190044 \cdot 10^{260}:\\ \;\;\;\;\frac{\frac{100 \cdot \mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}}{\frac{1}{n}}\\ \mathbf{elif}\;n \le 1.2344778126423281 \cdot 10^{-267}:\\ \;\;\;\;\frac{100}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot n - 1 \cdot n\right)\\ \mathbf{elif}\;n \le 4.27971623982010377 \cdot 10^{-165}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\\ \end{array}\]

C

double f(double i, double n) {
        double r147178 = 100.0;
        double r147179 = 1.0;
        double r147180 = i;
        double r147181 = n;
        double r147182 = r147180 / r147181;
        double r147183 = r147179 + r147182;
        double r147184 = pow(r147183, r147181);
        double r147185 = r147184 - r147179;
        double r147186 = r147185 / r147182;
        double r147187 = r147178 * r147186;
        return r147187;
}

↓

double f(double i, double n) {
        double r147188 = n;
        double r147189 = -1.8403630755190044e+260;
        bool r147190 = r147188 <= r147189;
        double r147191 = 100.0;
        double r147192 = i;
        double r147193 = 1.0;
        double r147194 = 0.5;
        double r147195 = 2.0;
        double r147196 = pow(r147192, r147195);
        double r147197 = log(r147193);
        double r147198 = r147197 * r147188;
        double r147199 = fma(r147194, r147196, r147198);
        double r147200 = r147196 * r147197;
        double r147201 = r147194 * r147200;
        double r147202 = r147199 - r147201;
        double r147203 = fma(r147192, r147193, r147202);
        double r147204 = r147191 * r147203;
        double r147205 = r147204 / r147192;
        double r147206 = 1.0;
        double r147207 = r147206 / r147188;
        double r147208 = r147205 / r147207;
        double r147209 = 1.234477812642328e-267;
        bool r147210 = r147188 <= r147209;
        double r147211 = r147191 / r147192;
        double r147212 = r147192 / r147188;
        double r147213 = r147193 + r147212;
        double r147214 = pow(r147213, r147188);
        double r147215 = r147214 * r147188;
        double r147216 = r147193 * r147188;
        double r147217 = r147215 - r147216;
        double r147218 = r147211 * r147217;
        double r147219 = 4.279716239820104e-165;
        bool r147220 = r147188 <= r147219;
        double r147221 = fma(r147197, r147188, r147206);
        double r147222 = fma(r147193, r147192, r147221);
        double r147223 = r147222 - r147193;
        double r147224 = r147223 / r147212;
        double r147225 = r147191 * r147224;
        double r147226 = r147205 * r147188;
        double r147227 = r147220 ? r147225 : r147226;
        double r147228 = r147210 ? r147218 : r147227;
        double r147229 = r147190 ? r147208 : r147228;
        return r147229;
}

Error

Bits error versus i

Bits error versus n

Target

Original	41.1
Target	40.9
Herbie	24.6

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

Derivation

1. Split input into 4 regimes

2. if n < -1.8403630755190044e+260

   1. Initial program 56.0

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

   3. Applied div-inv 56.1

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

   4. Applied *-un-lft-identity 56.1

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

   5. Applied times-frac 55.6

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

   6. Applied associate-*r* 55.6

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

   7. Simplified 55.6

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

   8. Taylor expanded around 0 37.0

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

   9. Simplified 37.0

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

   11. Applied associate-*r/ 37.0

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

   12. Simplified 37.0

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

   if -1.8403630755190044e+260 < n < 1.234477812642328e-267

   1. Initial program 26.1

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

   3. Applied div-inv 26.1

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

   4. Applied *-un-lft-identity 26.1

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

   5. Applied times-frac 26.3

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

   6. Applied associate-*r* 26.4

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

   7. Simplified 26.4

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

   9. Applied div-sub 26.4

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

   10. Simplified 29.0

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

   11. Simplified 26.4

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

   if 1.234477812642328e-267 < n < 4.279716239820104e-165

   1. Initial program 42.2

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

   2. Taylor expanded around 0 33.4

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

   3. Simplified 33.4

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

   if 4.279716239820104e-165 < n

   1. Initial program 58.8

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

   3. Applied div-inv 58.8

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

   4. Applied *-un-lft-identity 58.8

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

   5. Applied times-frac 58.5

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

   6. Applied associate-*r* 58.5

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

   7. Simplified 58.5

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

   8. Taylor expanded around 0 23.5

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

   9. Simplified 23.5

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

   11. Applied associate-/r/ 23.4

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

   12. Applied associate-*r* 19.6

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

   13. Simplified 19.4

       \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}} \cdot n\]
3. Recombined 4 regimes into one program.

4. Final simplification 24.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.8403630755190044 \cdot 10^{260}:\\ \;\;\;\;\frac{\frac{100 \cdot \mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}}{\frac{1}{n}}\\ \mathbf{elif}\;n \le 1.2344778126423281 \cdot 10^{-267}:\\ \;\;\;\;\frac{100}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot n - 1 \cdot n\right)\\ \mathbf{elif}\;n \le 4.27971623982010377 \cdot 10^{-165}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +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))))