Compound Interest

Average Error: 43.2 → 21.0

Time: 22.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -1.39612767473641797:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.4029318248824242 \cdot 10^{-247}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{1}{\frac{\frac{1}{n}}{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}\\ \mathbf{elif}\;i \le 6.7609796441213635 \cdot 10^{-279}:\\ \;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 2.3133805028339562 \cdot 10^{-220}:\\ \;\;\;\;\frac{100}{i} \cdot \left(\frac{1}{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}} \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\sqrt[3]{\frac{1}{n}}}\right)\\ \mathbf{elif}\;i \le 4.6153207705272044 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r110254 = 100.0;
        double r110255 = 1.0;
        double r110256 = i;
        double r110257 = n;
        double r110258 = r110256 / r110257;
        double r110259 = r110255 + r110258;
        double r110260 = pow(r110259, r110257);
        double r110261 = r110260 - r110255;
        double r110262 = r110261 / r110258;
        double r110263 = r110254 * r110262;
        return r110263;
}

↓

double f(double i, double n) {
        double r110264 = i;
        double r110265 = -1.396127674736418;
        bool r110266 = r110264 <= r110265;
        double r110267 = 100.0;
        double r110268 = 1.0;
        double r110269 = n;
        double r110270 = r110264 / r110269;
        double r110271 = r110268 + r110270;
        double r110272 = 2.0;
        double r110273 = r110272 * r110269;
        double r110274 = pow(r110271, r110273);
        double r110275 = r110268 * r110268;
        double r110276 = r110274 - r110275;
        double r110277 = pow(r110271, r110269);
        double r110278 = r110277 + r110268;
        double r110279 = r110276 / r110278;
        double r110280 = r110279 / r110270;
        double r110281 = r110267 * r110280;
        double r110282 = -1.4029318248824242e-247;
        bool r110283 = r110264 <= r110282;
        double r110284 = r110267 / r110264;
        double r110285 = 1.0;
        double r110286 = r110285 / r110269;
        double r110287 = r110268 * r110264;
        double r110288 = 0.5;
        double r110289 = pow(r110264, r110272);
        double r110290 = r110288 * r110289;
        double r110291 = log(r110268);
        double r110292 = r110291 * r110269;
        double r110293 = r110290 + r110292;
        double r110294 = r110287 + r110293;
        double r110295 = r110289 * r110291;
        double r110296 = r110288 * r110295;
        double r110297 = r110294 - r110296;
        double r110298 = r110286 / r110297;
        double r110299 = r110285 / r110298;
        double r110300 = r110284 * r110299;
        double r110301 = 6.7609796441213635e-279;
        bool r110302 = r110264 <= r110301;
        double r110303 = r110297 / r110264;
        double r110304 = r110303 * r110269;
        double r110305 = r110267 * r110304;
        double r110306 = 2.3133805028339562e-220;
        bool r110307 = r110264 <= r110306;
        double r110308 = cbrt(r110286);
        double r110309 = r110308 * r110308;
        double r110310 = r110285 / r110309;
        double r110311 = r110297 / r110308;
        double r110312 = r110310 * r110311;
        double r110313 = r110284 * r110312;
        double r110314 = 4.6153207705272044e-07;
        bool r110315 = r110264 <= r110314;
        double r110316 = r110315 ? r110305 : r110281;
        double r110317 = r110307 ? r110313 : r110316;
        double r110318 = r110302 ? r110305 : r110317;
        double r110319 = r110283 ? r110300 : r110318;
        double r110320 = r110266 ? r110281 : r110319;
        return r110320;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.2
Target	42.8
Herbie	21.0

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

Derivation

1. Split input into 4 regimes

2. if i < -1.396127674736418 or 4.6153207705272044e-07 < i

   1. Initial program 30.2

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

   3. Applied flip-- 30.2

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

   4. Simplified 30.1

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

   if -1.396127674736418 < i < -1.4029318248824242e-247

   1. Initial program 51.3

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

   3. Applied div-inv 51.3

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

   4. Applied *-un-lft-identity 51.3

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

   5. Applied times-frac 51.0

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

   6. Applied associate-*r* 51.0

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

   7. Simplified 51.0

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

   8. Taylor expanded around 0 15.3

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

   10. Applied clear-num 15.0

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

   if -1.4029318248824242e-247 < i < 6.7609796441213635e-279 or 2.3133805028339562e-220 < i < 4.6153207705272044e-07

   1. Initial program 51.3

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

   3. Applied div-inv 51.3

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

   4. Applied *-un-lft-identity 51.3

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

   5. Applied times-frac 51.0

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

   6. Applied associate-*r* 51.2

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

   7. Simplified 51.2

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

   8. Taylor expanded around 0 15.8

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

   10. Applied div-inv 15.8

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

   11. Applied associate-*l* 15.2

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

   12. Simplified 15.7

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

   if 6.7609796441213635e-279 < i < 2.3133805028339562e-220

   1. Initial program 48.3

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

   3. Applied div-inv 48.3

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

   4. Applied *-un-lft-identity 48.3

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

   5. Applied times-frac 47.7

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

   6. Applied associate-*r* 47.7

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

   7. Simplified 47.7

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

   8. Taylor expanded around 0 16.7

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

   10. Applied add-cube-cbrt 17.3

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

   11. Applied *-un-lft-identity 17.3

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

   12. Applied times-frac 17.3

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

4. Final simplification 21.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.39612767473641797:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.4029318248824242 \cdot 10^{-247}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{1}{\frac{\frac{1}{n}}{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}\\ \mathbf{elif}\;i \le 6.7609796441213635 \cdot 10^{-279}:\\ \;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 2.3133805028339562 \cdot 10^{-220}:\\ \;\;\;\;\frac{100}{i} \cdot \left(\frac{1}{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}} \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\sqrt[3]{\frac{1}{n}}}\right)\\ \mathbf{elif}\;i \le 4.6153207705272044 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))))