Compound Interest

Average Error: 43.0 → 21.1

Time: 32.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -0.01274350772906004958251457992446376010776:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 8.942303972784239462012963121098640220187 \cdot 10^{-300}:\\ \;\;\;\;\frac{\left(\left(\left(-\log 1\right) \cdot \left(\left(i \cdot i\right) \cdot 0.5\right) + \mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(1, i, \left(i \cdot i\right) \cdot 0.5\right)\right)\right) \cdot n\right) \cdot 100}{i}\\ \mathbf{elif}\;i \le 1.150450650179924164542989794445145639508 \cdot 10^{-111}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, 1 \cdot i\right)\right) - \left(\left(i \cdot i\right) \cdot \log 1\right) \cdot 0.5}{i}\right) \cdot n\\ \mathbf{elif}\;i \le 4586135949800822790097416960868352:\\ \;\;\;\;\frac{\left(\left(\left(-\log 1\right) \cdot \left(\left(i \cdot i\right) \cdot 0.5\right) + \mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(1, i, \left(i \cdot i\right) \cdot 0.5\right)\right)\right) \cdot n\right) \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{1}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r5385312 = 100.0;
        double r5385313 = 1.0;
        double r5385314 = i;
        double r5385315 = n;
        double r5385316 = r5385314 / r5385315;
        double r5385317 = r5385313 + r5385316;
        double r5385318 = pow(r5385317, r5385315);
        double r5385319 = r5385318 - r5385313;
        double r5385320 = r5385319 / r5385316;
        double r5385321 = r5385312 * r5385320;
        return r5385321;
}

↓

double f(double i, double n) {
        double r5385322 = i;
        double r5385323 = -0.01274350772906005;
        bool r5385324 = r5385322 <= r5385323;
        double r5385325 = 100.0;
        double r5385326 = n;
        double r5385327 = r5385322 / r5385326;
        double r5385328 = 1.0;
        double r5385329 = r5385327 + r5385328;
        double r5385330 = pow(r5385329, r5385326);
        double r5385331 = r5385330 / r5385327;
        double r5385332 = r5385328 / r5385327;
        double r5385333 = r5385331 - r5385332;
        double r5385334 = r5385325 * r5385333;
        double r5385335 = 8.94230397278424e-300;
        bool r5385336 = r5385322 <= r5385335;
        double r5385337 = log(r5385328);
        double r5385338 = -r5385337;
        double r5385339 = r5385322 * r5385322;
        double r5385340 = 0.5;
        double r5385341 = r5385339 * r5385340;
        double r5385342 = r5385338 * r5385341;
        double r5385343 = fma(r5385328, r5385322, r5385341);
        double r5385344 = fma(r5385337, r5385326, r5385343);
        double r5385345 = r5385342 + r5385344;
        double r5385346 = r5385345 * r5385326;
        double r5385347 = r5385346 * r5385325;
        double r5385348 = r5385347 / r5385322;
        double r5385349 = 1.1504506501799242e-111;
        bool r5385350 = r5385322 <= r5385349;
        double r5385351 = r5385328 * r5385322;
        double r5385352 = fma(r5385339, r5385340, r5385351);
        double r5385353 = fma(r5385326, r5385337, r5385352);
        double r5385354 = r5385339 * r5385337;
        double r5385355 = r5385354 * r5385340;
        double r5385356 = r5385353 - r5385355;
        double r5385357 = r5385356 / r5385322;
        double r5385358 = r5385325 * r5385357;
        double r5385359 = r5385358 * r5385326;
        double r5385360 = 4.586135949800823e+33;
        bool r5385361 = r5385322 <= r5385360;
        double r5385362 = r5385325 / r5385322;
        double r5385363 = r5385330 - r5385328;
        double r5385364 = 1.0;
        double r5385365 = r5385364 / r5385326;
        double r5385366 = r5385363 / r5385365;
        double r5385367 = r5385362 * r5385366;
        double r5385368 = r5385361 ? r5385348 : r5385367;
        double r5385369 = r5385350 ? r5385359 : r5385368;
        double r5385370 = r5385336 ? r5385348 : r5385369;
        double r5385371 = r5385324 ? r5385334 : r5385370;
        return r5385371;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.0
Target	42.8
Herbie	21.1

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

Derivation

1. Split input into 4 regimes

2. if i < -0.01274350772906005

   1. Initial program 28.2

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

   3. Applied div-sub 28.3

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

   if -0.01274350772906005 < i < 8.94230397278424e-300 or 1.1504506501799242e-111 < i < 4.586135949800823e+33

   1. Initial program 50.8

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

   2. Taylor expanded around 0 32.6

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

   3. Simplified 32.6

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

   5. Applied associate-/r/ 18.5

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

   6. Applied associate-*r* 18.5

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

   8. Applied div-inv 18.7

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

   10. Applied associate-*r/ 18.5

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

   11. Applied associate-*r/ 18.6

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

   12. Applied associate-*l/ 17.0

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

   13. Simplified 16.9

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

   if 8.94230397278424e-300 < i < 1.1504506501799242e-111

   1. Initial program 49.8

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

   2. Taylor expanded around 0 36.9

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

   3. Simplified 36.9

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

   5. Applied associate-/r/ 15.9

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

   6. Applied associate-*r* 15.9

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

   if 4.586135949800823e+33 < i

   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

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

   4. Applied *-un-lft-identity 31.3

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

   5. Applied times-frac 31.3

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

   6. Applied associate-*r* 31.3

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

   7. Simplified 31.3

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

4. Final simplification 21.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.01274350772906004958251457992446376010776:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 8.942303972784239462012963121098640220187 \cdot 10^{-300}:\\ \;\;\;\;\frac{\left(\left(\left(-\log 1\right) \cdot \left(\left(i \cdot i\right) \cdot 0.5\right) + \mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(1, i, \left(i \cdot i\right) \cdot 0.5\right)\right)\right) \cdot n\right) \cdot 100}{i}\\ \mathbf{elif}\;i \le 1.150450650179924164542989794445145639508 \cdot 10^{-111}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, 1 \cdot i\right)\right) - \left(\left(i \cdot i\right) \cdot \log 1\right) \cdot 0.5}{i}\right) \cdot n\\ \mathbf{elif}\;i \le 4586135949800822790097416960868352:\\ \;\;\;\;\frac{\left(\left(\left(-\log 1\right) \cdot \left(\left(i \cdot i\right) \cdot 0.5\right) + \mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(1, i, \left(i \cdot i\right) \cdot 0.5\right)\right)\right) \cdot n\right) \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{1}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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))))