Compound Interest

Average Error: 42.7 → 32.4

Time: 36.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -7.3291384716741002 \cdot 10^{181}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -9.8037587964115943 \cdot 10^{-219}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\frac{1}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}} \cdot \frac{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\sqrt[3]{\frac{i}{n}}}\right)\\ \mathbf{elif}\;n \le 1.54725910109650585 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 6.44601937092272336 \cdot 10^{168}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)\\ \end{array}\]

C

double f(double i, double n) {
        double r277260 = 100.0;
        double r277261 = 1.0;
        double r277262 = i;
        double r277263 = n;
        double r277264 = r277262 / r277263;
        double r277265 = r277261 + r277264;
        double r277266 = pow(r277265, r277263);
        double r277267 = r277266 - r277261;
        double r277268 = r277267 / r277264;
        double r277269 = r277260 * r277268;
        return r277269;
}

↓

double f(double i, double n) {
        double r277270 = n;
        double r277271 = -7.3291384716741e+181;
        bool r277272 = r277270 <= r277271;
        double r277273 = 100.0;
        double r277274 = 1.0;
        double r277275 = i;
        double r277276 = r277274 * r277275;
        double r277277 = 0.5;
        double r277278 = 2.0;
        double r277279 = pow(r277275, r277278);
        double r277280 = r277277 * r277279;
        double r277281 = log(r277274);
        double r277282 = r277281 * r277270;
        double r277283 = r277280 + r277282;
        double r277284 = r277276 + r277283;
        double r277285 = r277279 * r277281;
        double r277286 = r277277 * r277285;
        double r277287 = r277284 - r277286;
        double r277288 = r277275 / r277270;
        double r277289 = r277287 / r277288;
        double r277290 = r277273 * r277289;
        double r277291 = -9.803758796411594e-219;
        bool r277292 = r277270 <= r277291;
        double r277293 = 1.0;
        double r277294 = r277274 + r277288;
        double r277295 = r277278 * r277270;
        double r277296 = pow(r277294, r277295);
        double r277297 = r277274 * r277274;
        double r277298 = r277296 + r277297;
        double r277299 = cbrt(r277298);
        double r277300 = r277299 * r277299;
        double r277301 = r277293 / r277300;
        double r277302 = pow(r277294, r277270);
        double r277303 = r277302 + r277274;
        double r277304 = sqrt(r277303);
        double r277305 = r277301 / r277304;
        double r277306 = cbrt(r277288);
        double r277307 = r277306 * r277306;
        double r277308 = r277305 / r277307;
        double r277309 = r277278 * r277295;
        double r277310 = pow(r277294, r277309);
        double r277311 = r277297 * r277297;
        double r277312 = -r277311;
        double r277313 = r277310 + r277312;
        double r277314 = r277313 / r277299;
        double r277315 = r277314 / r277304;
        double r277316 = r277315 / r277306;
        double r277317 = r277308 * r277316;
        double r277318 = r277273 * r277317;
        double r277319 = 1.5472591010965058e-128;
        bool r277320 = r277270 <= r277319;
        double r277321 = sqrt(r277273);
        double r277322 = r277282 + r277293;
        double r277323 = r277276 + r277322;
        double r277324 = r277323 - r277274;
        double r277325 = r277324 / r277288;
        double r277326 = r277321 * r277325;
        double r277327 = r277321 * r277326;
        double r277328 = 6.446019370922723e+168;
        bool r277329 = r277270 <= r277328;
        double r277330 = pow(r277288, r277270);
        double r277331 = r277330 - r277274;
        double r277332 = r277331 / r277288;
        double r277333 = r277321 * r277332;
        double r277334 = r277321 * r277333;
        double r277335 = r277329 ? r277290 : r277334;
        double r277336 = r277320 ? r277327 : r277335;
        double r277337 = r277292 ? r277318 : r277336;
        double r277338 = r277272 ? r277290 : r277337;
        return r277338;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.7
Target	42.3
Herbie	32.4

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

Derivation

1. Split input into 4 regimes

2. if n < -7.3291384716741e+181 or 1.5472591010965058e-128 < n < 6.446019370922723e+168

   1. Initial program 57.1

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

   2. Taylor expanded around 0 35.6

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

   if -7.3291384716741e+181 < n < -9.803758796411594e-219

   1. Initial program 28.1

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

   3. Applied flip-- 28.1

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

   4. Simplified 28.0

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

   6. Applied flip-+ 28.0

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

   7. Simplified 28.0

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

   8. Simplified 28.0

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

   10. Applied add-cube-cbrt 28.2

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

   11. Applied add-sqr-sqrt 28.2

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

   12. Applied add-cube-cbrt 28.2

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

   13. Applied *-un-lft-identity 28.2

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

   14. Applied times-frac 28.2

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

   15. Applied times-frac 28.2

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

   16. Applied times-frac 28.2

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

   if -9.803758796411594e-219 < n < 1.5472591010965058e-128

   1. Initial program 32.9

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

   3. Applied add-sqr-sqrt 32.9

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

   4. Applied associate-*l* 32.9

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

   5. Taylor expanded around 0 27.2

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

   if 6.446019370922723e+168 < n

   1. Initial program 61.9

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

   3. Applied add-sqr-sqrt 61.9

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

   4. Applied associate-*l* 61.9

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

   5. Taylor expanded around inf 64.0

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

   6. Simplified 44.6

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

4. Final simplification 32.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -7.3291384716741002 \cdot 10^{181}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -9.8037587964115943 \cdot 10^{-219}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\frac{1}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}} \cdot \frac{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\sqrt[3]{\frac{i}{n}}}\right)\\ \mathbf{elif}\;n \le 1.54725910109650585 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 6.44601937092272336 \cdot 10^{168}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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))))