Average Error: 43.1 → 23.6
Time: 19.3s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -1.00297232240918336 \cdot 10^{174}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -4.03699116408299863 \cdot 10^{85}:\\ \;\;\;\;\frac{100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot n\right)}{i}\\ \mathbf{elif}\;n \le -3.31747608845659503 \cdot 10^{63}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -1.0912719332224593 \cdot 10^{26}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -934893.903368213796:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le 1.4472641457797862 \cdot 10^{-291}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 6.25395219693272548 \cdot 10^{-155}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -1.00297232240918336 \cdot 10^{174}:\\
\;\;\;\;\left(100 \cdot \frac{\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\\

\mathbf{elif}\;n \le -4.03699116408299863 \cdot 10^{85}:\\
\;\;\;\;\frac{100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot n\right)}{i}\\

\mathbf{elif}\;n \le -3.31747608845659503 \cdot 10^{63}:\\
\;\;\;\;\left(100 \cdot \frac{\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\\

\mathbf{elif}\;n \le -1.0912719332224593 \cdot 10^{26}:\\
\;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\

\mathbf{elif}\;n \le -934893.903368213796:\\
\;\;\;\;\left(100 \cdot \frac{\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\\

\mathbf{elif}\;n \le 1.4472641457797862 \cdot 10^{-291}:\\
\;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;n \le 6.25395219693272548 \cdot 10^{-155}:\\
\;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\

\mathbf{else}:\\
\;\;\;\;\left(100 \cdot \frac{\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\\

\end{array}
double f(double i, double n) {
        double r119580 = 100.0;
        double r119581 = 1.0;
        double r119582 = i;
        double r119583 = n;
        double r119584 = r119582 / r119583;
        double r119585 = r119581 + r119584;
        double r119586 = pow(r119585, r119583);
        double r119587 = r119586 - r119581;
        double r119588 = r119587 / r119584;
        double r119589 = r119580 * r119588;
        return r119589;
}
↓
double f(double i, double n) {
        double r119590 = n;
        double r119591 = -1.0029723224091834e+174;
        bool r119592 = r119590 <= r119591;
        double r119593 = 100.0;
        double r119594 = 1.0;
        double r119595 = i;
        double r119596 = r119594 * r119595;
        double r119597 = 0.5;
        double r119598 = 2.0;
        double r119599 = pow(r119595, r119598);
        double r119600 = r119597 * r119599;
        double r119601 = log(r119594);
        double r119602 = r119601 * r119590;
        double r119603 = r119600 + r119602;
        double r119604 = r119596 + r119603;
        double r119605 = r119599 * r119601;
        double r119606 = r119597 * r119605;
        double r119607 = r119604 - r119606;
        double r119608 = r119607 / r119595;
        double r119609 = r119593 * r119608;
        double r119610 = r119609 * r119590;
        double r119611 = -4.0369911640829986e+85;
        bool r119612 = r119590 <= r119611;
        double r119613 = r119595 / r119590;
        double r119614 = r119594 + r119613;
        double r119615 = pow(r119614, r119590);
        double r119616 = r119615 - r119594;
        double r119617 = r119616 * r119590;
        double r119618 = r119593 * r119617;
        double r119619 = r119618 / r119595;
        double r119620 = -3.317476088456595e+63;
        bool r119621 = r119590 <= r119620;
        double r119622 = -1.0912719332224593e+26;
        bool r119623 = r119590 <= r119622;
        double r119624 = r119593 * r119616;
        double r119625 = r119624 / r119613;
        double r119626 = -934893.9033682138;
        bool r119627 = r119590 <= r119626;
        double r119628 = 1.4472641457797862e-291;
        bool r119629 = r119590 <= r119628;
        double r119630 = r119615 / r119613;
        double r119631 = r119594 / r119613;
        double r119632 = r119630 - r119631;
        double r119633 = r119593 * r119632;
        double r119634 = 6.2539521969327255e-155;
        bool r119635 = r119590 <= r119634;
        double r119636 = 1.0;
        double r119637 = r119602 + r119636;
        double r119638 = r119596 + r119637;
        double r119639 = r119638 - r119594;
        double r119640 = r119639 / r119613;
        double r119641 = r119593 * r119640;
        double r119642 = r119635 ? r119641 : r119610;
        double r119643 = r119629 ? r119633 : r119642;
        double r119644 = r119627 ? r119610 : r119643;
        double r119645 = r119623 ? r119625 : r119644;
        double r119646 = r119621 ? r119610 : r119645;
        double r119647 = r119612 ? r119619 : r119646;
        double r119648 = r119592 ? r119610 : r119647;
        return r119648;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.1
Target43.0
Herbie23.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 5 regimes

  2. if n < -1.0029723224091834e+174 or -4.0369911640829986e+85 < n < -3.317476088456595e+63 or -1.0912719332224593e+26 < n < -934893.9033682138 or 6.2539521969327255e-155 < n

    1. Initial program 56.2

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

    3. Applied associate-/r/55.9

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

    4. Taylor expanded around 0 21.3

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

    6. Applied associate-*r*21.3

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

    if -1.0029723224091834e+174 < n < -4.0369911640829986e+85

    1. Initial program 40.6

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

    3. Applied associate-/r/40.3

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

    5. Applied associate-*l/40.2

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

    6. Applied associate-*r/40.2

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

    if -3.317476088456595e+63 < n < -1.0912719332224593e+26

    1. Initial program 30.6

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

    3. Applied associate-*r/30.5

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

    if -934893.9033682138 < n < 1.4472641457797862e-291

    1. Initial program 18.0

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

    3. Applied div-sub18.0

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

    if 1.4472641457797862e-291 < n < 6.2539521969327255e-155

    1. Initial program 43.0

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

    2. Taylor expanded around 0 32.4

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right)} - 1}{\frac{i}{n}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification23.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.00297232240918336 \cdot 10^{174}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -4.03699116408299863 \cdot 10^{85}:\\ \;\;\;\;\frac{100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot n\right)}{i}\\ \mathbf{elif}\;n \le -3.31747608845659503 \cdot 10^{63}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -1.0912719332224593 \cdot 10^{26}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -934893.903368213796:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le 1.4472641457797862 \cdot 10^{-291}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 6.25395219693272548 \cdot 10^{-155}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))