Compound Interest

Average Error: 42.9 → 23.4

Time: 37.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -5.45421367555386284 \cdot 10^{117}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{elif}\;n \le -8.94632381714276356 \cdot 10^{69}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{elif}\;n \le -50247890715477128:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{elif}\;n \le -5.80539647643117144 \cdot 10^{-192}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, {\left(\sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, -1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 2.12821304689236942 \cdot 10^{-175}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \end{array}\]

C

double f(double i, double n) {
        double r111310 = 100.0;
        double r111311 = 1.0;
        double r111312 = i;
        double r111313 = n;
        double r111314 = r111312 / r111313;
        double r111315 = r111311 + r111314;
        double r111316 = pow(r111315, r111313);
        double r111317 = r111316 - r111311;
        double r111318 = r111317 / r111314;
        double r111319 = r111310 * r111318;
        return r111319;
}

↓

double f(double i, double n) {
        double r111320 = n;
        double r111321 = -5.454213675553863e+117;
        bool r111322 = r111320 <= r111321;
        double r111323 = 100.0;
        double r111324 = 1.0;
        double r111325 = i;
        double r111326 = 0.5;
        double r111327 = 2.0;
        double r111328 = pow(r111325, r111327);
        double r111329 = log(r111324);
        double r111330 = r111329 * r111320;
        double r111331 = fma(r111326, r111328, r111330);
        double r111332 = fma(r111324, r111325, r111331);
        double r111333 = r111328 * r111329;
        double r111334 = r111326 * r111333;
        double r111335 = r111332 - r111334;
        double r111336 = r111335 / r111325;
        double r111337 = r111336 * r111320;
        double r111338 = r111323 * r111337;
        double r111339 = -8.946323817142764e+69;
        bool r111340 = r111320 <= r111339;
        double r111341 = r111323 / r111325;
        double r111342 = r111325 / r111320;
        double r111343 = r111324 + r111342;
        double r111344 = pow(r111343, r111320);
        double r111345 = r111344 - r111324;
        double r111346 = 1.0;
        double r111347 = r111346 / r111320;
        double r111348 = r111345 / r111347;
        double r111349 = r111341 * r111348;
        double r111350 = -5.024789071547713e+16;
        bool r111351 = r111320 <= r111350;
        double r111352 = -5.8053964764311714e-192;
        bool r111353 = r111320 <= r111352;
        double r111354 = cbrt(r111343);
        double r111355 = r111354 * r111354;
        double r111356 = pow(r111355, r111320);
        double r111357 = pow(r111354, r111320);
        double r111358 = -r111324;
        double r111359 = fma(r111356, r111357, r111358);
        double r111360 = r111359 / r111342;
        double r111361 = r111323 * r111360;
        double r111362 = 2.1282130468923694e-175;
        bool r111363 = r111320 <= r111362;
        double r111364 = fma(r111329, r111320, r111346);
        double r111365 = fma(r111324, r111325, r111364);
        double r111366 = r111365 - r111324;
        double r111367 = r111366 / r111342;
        double r111368 = r111323 * r111367;
        double r111369 = r111363 ? r111368 : r111338;
        double r111370 = r111353 ? r111361 : r111369;
        double r111371 = r111351 ? r111338 : r111370;
        double r111372 = r111340 ? r111349 : r111371;
        double r111373 = r111322 ? r111338 : r111372;
        return r111373;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.9
Target	43.3
Herbie	23.4

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

Derivation

1. Split input into 4 regimes

2. if n < -5.454213675553863e+117 or -8.946323817142764e+69 < n < -5.024789071547713e+16 or 2.1282130468923694e-175 < n

   1. Initial program 53.5

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

   2. Taylor expanded around 0 40.2

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

   3. Simplified 40.2

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

   5. Applied associate-/r/ 23.2

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

   if -5.454213675553863e+117 < n < -8.946323817142764e+69

   1. Initial program 38.0

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

   3. Applied div-inv 38.0

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

   4. Applied *-un-lft-identity 38.0

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

   5. Applied times-frac 37.8

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

   6. Applied associate-*r* 37.8

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

   7. Simplified 37.8

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

   if -5.024789071547713e+16 < n < -5.8053964764311714e-192

   1. Initial program 20.8

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

   3. Applied add-cube-cbrt 20.8

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

   4. Applied unpow-prod-down 20.8

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

   5. Applied fma-neg 20.8

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

   if -5.8053964764311714e-192 < n < 2.1282130468923694e-175

   1. Initial program 26.9

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

   2. Taylor expanded around 0 22.3

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

   3. Simplified 22.3

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

4. Final simplification 23.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -5.45421367555386284 \cdot 10^{117}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{elif}\;n \le -8.94632381714276356 \cdot 10^{69}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{elif}\;n \le -50247890715477128:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{elif}\;n \le -5.80539647643117144 \cdot 10^{-192}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, {\left(\sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, -1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 2.12821304689236942 \cdot 10^{-175}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +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))))