Compound Interest

Average Error: 42.7 → 21.3

Time: 14.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -0.318831405035280346:\\ \;\;\;\;\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 0.0030890666332557444:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \end{array}\]

C

double f(double i, double n) {
        double r134281 = 100.0;
        double r134282 = 1.0;
        double r134283 = i;
        double r134284 = n;
        double r134285 = r134283 / r134284;
        double r134286 = r134282 + r134285;
        double r134287 = pow(r134286, r134284);
        double r134288 = r134287 - r134282;
        double r134289 = r134288 / r134285;
        double r134290 = r134281 * r134289;
        return r134290;
}

↓

double f(double i, double n) {
        double r134291 = i;
        double r134292 = -0.31883140503528035;
        bool r134293 = r134291 <= r134292;
        double r134294 = 100.0;
        double r134295 = sqrt(r134294);
        double r134296 = 1.0;
        double r134297 = n;
        double r134298 = r134291 / r134297;
        double r134299 = r134296 + r134298;
        double r134300 = pow(r134299, r134297);
        double r134301 = r134300 - r134296;
        double r134302 = r134301 / r134298;
        double r134303 = r134295 * r134302;
        double r134304 = r134295 * r134303;
        double r134305 = 0.0030890666332557444;
        bool r134306 = r134291 <= r134305;
        double r134307 = 0.5;
        double r134308 = 2.0;
        double r134309 = pow(r134291, r134308);
        double r134310 = log(r134296);
        double r134311 = r134310 * r134297;
        double r134312 = fma(r134307, r134309, r134311);
        double r134313 = r134309 * r134310;
        double r134314 = r134307 * r134313;
        double r134315 = r134312 - r134314;
        double r134316 = fma(r134291, r134296, r134315);
        double r134317 = r134316 / r134291;
        double r134318 = r134317 * r134297;
        double r134319 = r134294 * r134318;
        double r134320 = r134301 / r134291;
        double r134321 = r134294 * r134320;
        double r134322 = r134321 * r134297;
        double r134323 = r134306 ? r134319 : r134322;
        double r134324 = r134293 ? r134304 : r134323;
        return r134324;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.7
Target	42.6
Herbie	21.3

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

Derivation

1. Split input into 3 regimes

2. if i < -0.31883140503528035

   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 add-sqr-sqrt 28.1

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

   4. Applied associate-*l* 28.2

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

   if -0.31883140503528035 < i < 0.0030890666332557444

   1. Initial program 50.8

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

   2. Taylor expanded around 0 34.3

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

   3. Simplified 34.3

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

   5. Applied associate-/r/ 17.1

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

   if 0.0030890666332557444 < i

   1. Initial program 29.6

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

   3. Applied associate-/r/ 29.6

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

   4. Applied associate-*r* 29.6

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

4. Final simplification 21.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.318831405035280346:\\ \;\;\;\;\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 0.0030890666332557444:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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))))