Average Error: 43.0 → 22.9
Time: 13.8s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -7.76693767745361788861631953913126532955 \cdot 10^{94}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(i, 1, \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -9.519350012649904306163831713839410413139 \cdot 10^{-251}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 1.613414883038850832801467887537631845006 \cdot 10^{-130}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(i, 1, \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\right) \cdot n\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -7.76693767745361788861631953913126532955 \cdot 10^{94}:\\
\;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(i, 1, \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\right) \cdot n\\

\mathbf{elif}\;n \le -9.519350012649904306163831713839410413139 \cdot 10^{-251}:\\
\;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;n \le 1.613414883038850832801467887537631845006 \cdot 10^{-130}:\\
\;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\

\mathbf{else}:\\
\;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(i, 1, \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\right) \cdot n\\

\end{array}
double f(double i, double n) {
        double r137441 = 100.0;
        double r137442 = 1.0;
        double r137443 = i;
        double r137444 = n;
        double r137445 = r137443 / r137444;
        double r137446 = r137442 + r137445;
        double r137447 = pow(r137446, r137444);
        double r137448 = r137447 - r137442;
        double r137449 = r137448 / r137445;
        double r137450 = r137441 * r137449;
        return r137450;
}
↓
double f(double i, double n) {
        double r137451 = n;
        double r137452 = -7.766937677453618e+94;
        bool r137453 = r137451 <= r137452;
        double r137454 = 100.0;
        double r137455 = i;
        double r137456 = 1.0;
        double r137457 = 0.5;
        double r137458 = 2.0;
        double r137459 = pow(r137455, r137458);
        double r137460 = r137457 * r137459;
        double r137461 = log(r137456);
        double r137462 = r137461 * r137451;
        double r137463 = r137460 + r137462;
        double r137464 = r137459 * r137461;
        double r137465 = r137457 * r137464;
        double r137466 = r137463 - r137465;
        double r137467 = fma(r137455, r137456, r137466);
        double r137468 = r137467 / r137455;
        double r137469 = r137454 * r137468;
        double r137470 = r137469 * r137451;
        double r137471 = -9.519350012649904e-251;
        bool r137472 = r137451 <= r137471;
        double r137473 = r137455 / r137451;
        double r137474 = r137456 + r137473;
        double r137475 = pow(r137474, r137451);
        double r137476 = r137475 / r137473;
        double r137477 = r137456 / r137473;
        double r137478 = r137476 - r137477;
        double r137479 = r137454 * r137478;
        double r137480 = 1.6134148830388508e-130;
        bool r137481 = r137451 <= r137480;
        double r137482 = 1.0;
        double r137483 = fma(r137461, r137451, r137482);
        double r137484 = fma(r137456, r137455, r137483);
        double r137485 = r137484 - r137456;
        double r137486 = r137485 / r137473;
        double r137487 = r137454 * r137486;
        double r137488 = r137481 ? r137487 : r137470;
        double r137489 = r137472 ? r137479 : r137488;
        double r137490 = r137453 ? r137470 : r137489;
        return r137490;
}

Error

Bits error versus i

Bits error versus n

Target

Original43.0
Target42.8
Herbie22.9
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if n < -7.766937677453618e+94 or 1.6134148830388508e-130 < n

    1. Initial program 55.3

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

    3. Applied associate-/r/55.0

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

    4. Applied associate-*r*55.0

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

    5. Taylor expanded around 0 21.5

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

    6. Simplified21.5

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

    8. Applied fma-udef21.5

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

    if -7.766937677453618e+94 < n < -9.519350012649904e-251

    1. Initial program 23.8

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

    3. Applied div-sub23.9

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

    if -9.519350012649904e-251 < n < 1.6134148830388508e-130

    1. Initial program 35.4

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

    2. Taylor expanded around 0 26.0

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

    3. Simplified26.0

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

  4. Final simplification22.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -7.76693767745361788861631953913126532955 \cdot 10^{94}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(i, 1, \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -9.519350012649904306163831713839410413139 \cdot 10^{-251}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 1.613414883038850832801467887537631845006 \cdot 10^{-130}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(i, 1, \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\right) \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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))))