Average Error: 42.9 → 22.5
Time: 33.6s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -743560.0364592294208705425262451171875:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(\left(i \cdot i\right) \cdot \log 1\right) \cdot 0.5}{i} \cdot n\right)\\ \mathbf{elif}\;n \le -2.730403974900932650671053560507976027298 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n}} - \sqrt{1}}{\frac{1}{n}} \cdot \left(100 \cdot \frac{\sqrt{1} + \sqrt{{\left(\frac{i}{n} + 1\right)}^{n}}}{i}\right)\\ \mathbf{elif}\;n \le 5.492513693153940784790669261791516735088 \cdot 10^{-150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(1, i, 1\right)\right) - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(\left(i \cdot i\right) \cdot \log 1\right) \cdot 0.5}{i} \cdot n\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -743560.0364592294208705425262451171875:\\
\;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(\left(i \cdot i\right) \cdot \log 1\right) \cdot 0.5}{i} \cdot n\right)\\

\mathbf{elif}\;n \le -2.730403974900932650671053560507976027298 \cdot 10^{-303}:\\
\;\;\;\;\frac{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n}} - \sqrt{1}}{\frac{1}{n}} \cdot \left(100 \cdot \frac{\sqrt{1} + \sqrt{{\left(\frac{i}{n} + 1\right)}^{n}}}{i}\right)\\

\mathbf{elif}\;n \le 5.492513693153940784790669261791516735088 \cdot 10^{-150}:\\
\;\;\;\;\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(1, i, 1\right)\right) - 1}{\frac{i}{n}} \cdot 100\\

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

\end{array}
double f(double i, double n) {
        double r6381625 = 100.0;
        double r6381626 = 1.0;
        double r6381627 = i;
        double r6381628 = n;
        double r6381629 = r6381627 / r6381628;
        double r6381630 = r6381626 + r6381629;
        double r6381631 = pow(r6381630, r6381628);
        double r6381632 = r6381631 - r6381626;
        double r6381633 = r6381632 / r6381629;
        double r6381634 = r6381625 * r6381633;
        return r6381634;
}
↓
double f(double i, double n) {
        double r6381635 = n;
        double r6381636 = -743560.0364592294;
        bool r6381637 = r6381635 <= r6381636;
        double r6381638 = 100.0;
        double r6381639 = 1.0;
        double r6381640 = log(r6381639);
        double r6381641 = i;
        double r6381642 = r6381641 * r6381641;
        double r6381643 = 0.5;
        double r6381644 = r6381641 * r6381639;
        double r6381645 = fma(r6381642, r6381643, r6381644);
        double r6381646 = fma(r6381635, r6381640, r6381645);
        double r6381647 = r6381642 * r6381640;
        double r6381648 = r6381647 * r6381643;
        double r6381649 = r6381646 - r6381648;
        double r6381650 = r6381649 / r6381641;
        double r6381651 = r6381650 * r6381635;
        double r6381652 = r6381638 * r6381651;
        double r6381653 = -2.7304039749009327e-303;
        bool r6381654 = r6381635 <= r6381653;
        double r6381655 = r6381641 / r6381635;
        double r6381656 = r6381655 + r6381639;
        double r6381657 = pow(r6381656, r6381635);
        double r6381658 = sqrt(r6381657);
        double r6381659 = sqrt(r6381639);
        double r6381660 = r6381658 - r6381659;
        double r6381661 = 1.0;
        double r6381662 = r6381661 / r6381635;
        double r6381663 = r6381660 / r6381662;
        double r6381664 = r6381659 + r6381658;
        double r6381665 = r6381664 / r6381641;
        double r6381666 = r6381638 * r6381665;
        double r6381667 = r6381663 * r6381666;
        double r6381668 = 5.492513693153941e-150;
        bool r6381669 = r6381635 <= r6381668;
        double r6381670 = fma(r6381639, r6381641, r6381661);
        double r6381671 = fma(r6381635, r6381640, r6381670);
        double r6381672 = r6381671 - r6381639;
        double r6381673 = r6381672 / r6381655;
        double r6381674 = r6381673 * r6381638;
        double r6381675 = r6381669 ? r6381674 : r6381652;
        double r6381676 = r6381654 ? r6381667 : r6381675;
        double r6381677 = r6381637 ? r6381652 : r6381676;
        return r6381677;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.9
Target42.3
Herbie22.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if n < -743560.0364592294 or 5.492513693153941e-150 < n

    1. Initial program 52.4

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

    2. Taylor expanded around 0 39.6

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

    3. Simplified39.6

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

    5. Applied associate-/r/23.5

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

    if -743560.0364592294 < n < -2.7304039749009327e-303

    1. Initial program 15.9

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

    3. Applied div-inv15.9

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

    4. Applied add-sqr-sqrt15.9

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

    5. Applied add-sqr-sqrt15.9

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

    6. Applied difference-of-squares15.9

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

    7. Applied times-frac16.5

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

    8. Applied associate-*r*17.0

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

    if -2.7304039749009327e-303 < n < 5.492513693153941e-150

    1. Initial program 42.1

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

    2. Taylor expanded around 0 28.1

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

    3. Simplified28.1

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

  4. Final simplification22.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -743560.0364592294208705425262451171875:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(\left(i \cdot i\right) \cdot \log 1\right) \cdot 0.5}{i} \cdot n\right)\\ \mathbf{elif}\;n \le -2.730403974900932650671053560507976027298 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n}} - \sqrt{1}}{\frac{1}{n}} \cdot \left(100 \cdot \frac{\sqrt{1} + \sqrt{{\left(\frac{i}{n} + 1\right)}^{n}}}{i}\right)\\ \mathbf{elif}\;n \le 5.492513693153940784790669261791516735088 \cdot 10^{-150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(1, i, 1\right)\right) - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(\left(i \cdot i\right) \cdot \log 1\right) \cdot 0.5}{i} \cdot n\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +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))))