Average Error: 48.0 → 17.6
Time: 12.9s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -2.2717441078324636 \cdot 10^{-10}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 2.5704520336039343 \cdot 10^{-23}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\right) \cdot n\\ \mathbf{elif}\;i \le 1.5471622621860444 \cdot 10^{204}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -2.2717441078324636 \cdot 10^{-10}:\\
\;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;i \le 2.5704520336039343 \cdot 10^{-23}:\\
\;\;\;\;\left(100 \cdot \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}\right) \cdot n\\

\mathbf{elif}\;i \le 1.5471622621860444 \cdot 10^{204}:\\
\;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\

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

\end{array}
double f(double i, double n) {
        double r131465 = 100.0;
        double r131466 = 1.0;
        double r131467 = i;
        double r131468 = n;
        double r131469 = r131467 / r131468;
        double r131470 = r131466 + r131469;
        double r131471 = pow(r131470, r131468);
        double r131472 = r131471 - r131466;
        double r131473 = r131472 / r131469;
        double r131474 = r131465 * r131473;
        return r131474;
}
↓
double f(double i, double n) {
        double r131475 = i;
        double r131476 = -2.2717441078324636e-10;
        bool r131477 = r131475 <= r131476;
        double r131478 = 100.0;
        double r131479 = 1.0;
        double r131480 = n;
        double r131481 = r131475 / r131480;
        double r131482 = r131479 + r131481;
        double r131483 = pow(r131482, r131480);
        double r131484 = r131483 / r131481;
        double r131485 = r131479 / r131481;
        double r131486 = r131484 - r131485;
        double r131487 = r131478 * r131486;
        double r131488 = 2.5704520336039343e-23;
        bool r131489 = r131475 <= r131488;
        double r131490 = 0.5;
        double r131491 = 2.0;
        double r131492 = pow(r131475, r131491);
        double r131493 = log(r131479);
        double r131494 = r131493 * r131480;
        double r131495 = fma(r131490, r131492, r131494);
        double r131496 = r131492 * r131493;
        double r131497 = r131490 * r131496;
        double r131498 = r131495 - r131497;
        double r131499 = fma(r131475, r131479, r131498);
        double r131500 = r131499 / r131475;
        double r131501 = r131478 * r131500;
        double r131502 = r131501 * r131480;
        double r131503 = 1.5471622621860444e+204;
        bool r131504 = r131475 <= r131503;
        double r131505 = 1.0;
        double r131506 = fma(r131493, r131480, r131505);
        double r131507 = fma(r131479, r131475, r131506);
        double r131508 = r131507 - r131479;
        double r131509 = r131508 / r131481;
        double r131510 = r131478 * r131509;
        double r131511 = r131504 ? r131487 : r131510;
        double r131512 = r131489 ? r131502 : r131511;
        double r131513 = r131477 ? r131487 : r131512;
        return r131513;
}

Error

Bits error versus i

Bits error versus n

Target

Original48.0
Target48.1
Herbie17.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -2.2717441078324636e-10 or 2.5704520336039343e-23 < i < 1.5471622621860444e+204

    1. Initial program 31.5

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

    3. Applied div-sub31.5

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

    if -2.2717441078324636e-10 < i < 2.5704520336039343e-23

    1. Initial program 58.5

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

    2. Taylor expanded around 0 26.1

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

    3. Simplified26.1

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

    5. Applied associate-/r/8.5

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

    6. Applied associate-*r*8.5

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

    if 1.5471622621860444e+204 < i

    1. Initial program 33.2

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

    2. Taylor expanded around 0 32.5

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

    3. Simplified32.5

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

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.2717441078324636 \cdot 10^{-10}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 2.5704520336039343 \cdot 10^{-23}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\right) \cdot n\\ \mathbf{elif}\;i \le 1.5471622621860444 \cdot 10^{204}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +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))))