Average Error: 47.8 → 17.3
Time: 13.1s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.0281548125542239701:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 9.41291569922092328 \cdot 10^{-13}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + \sqrt{1}}{i}\right) \cdot \frac{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} - \sqrt{1}}{\frac{1}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.0281548125542239701:\\
\;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 9.41291569922092328 \cdot 10^{-13}:\\
\;\;\;\;\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{else}:\\
\;\;\;\;\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}}\\

\end{array}
double f(double i, double n) {
        double r112602 = 100.0;
        double r112603 = 1.0;
        double r112604 = i;
        double r112605 = n;
        double r112606 = r112604 / r112605;
        double r112607 = r112603 + r112606;
        double r112608 = pow(r112607, r112605);
        double r112609 = r112608 - r112603;
        double r112610 = r112609 / r112606;
        double r112611 = r112602 * r112610;
        return r112611;
}
↓
double f(double i, double n) {
        double r112612 = i;
        double r112613 = -0.02815481255422397;
        bool r112614 = r112612 <= r112613;
        double r112615 = 100.0;
        double r112616 = 1.0;
        double r112617 = n;
        double r112618 = r112612 / r112617;
        double r112619 = r112616 + r112618;
        double r112620 = pow(r112619, r112617);
        double r112621 = r112620 - r112616;
        double r112622 = r112615 * r112621;
        double r112623 = r112622 / r112618;
        double r112624 = 9.412915699220923e-13;
        bool r112625 = r112612 <= r112624;
        double r112626 = r112616 * r112612;
        double r112627 = 0.5;
        double r112628 = 2.0;
        double r112629 = pow(r112612, r112628);
        double r112630 = r112627 * r112629;
        double r112631 = log(r112616);
        double r112632 = r112631 * r112617;
        double r112633 = r112630 + r112632;
        double r112634 = r112626 + r112633;
        double r112635 = r112629 * r112631;
        double r112636 = r112627 * r112635;
        double r112637 = r112634 - r112636;
        double r112638 = r112637 / r112612;
        double r112639 = r112615 * r112638;
        double r112640 = r112639 * r112617;
        double r112641 = sqrt(r112620);
        double r112642 = sqrt(r112616);
        double r112643 = r112641 + r112642;
        double r112644 = r112643 / r112612;
        double r112645 = r112615 * r112644;
        double r112646 = r112641 - r112642;
        double r112647 = 1.0;
        double r112648 = r112647 / r112617;
        double r112649 = r112646 / r112648;
        double r112650 = r112645 * r112649;
        double r112651 = r112625 ? r112640 : r112650;
        double r112652 = r112614 ? r112623 : r112651;
        return r112652;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.8
Target47.2
Herbie17.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -0.02815481255422397

    1. Initial program 28.7

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

    3. Applied associate-*r/28.7

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

    if -0.02815481255422397 < i < 9.412915699220923e-13

    1. Initial program 58.2

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

    2. Taylor expanded around 0 26.5

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

    4. Applied associate-/r/9.2

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

    5. Applied associate-*r*9.2

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

    if 9.412915699220923e-13 < i

    1. Initial program 34.0

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

    3. Applied div-inv34.0

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

    4. Applied add-sqr-sqrt34.0

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

    5. Applied add-sqr-sqrt34.1

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

    6. Applied difference-of-squares34.0

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

    7. Applied times-frac34.1

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

    8. Applied associate-*r*34.1

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

  4. Final simplification17.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.0281548125542239701:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 9.41291569922092328 \cdot 10^{-13}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + \sqrt{1}}{i}\right) \cdot \frac{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} - \sqrt{1}}{\frac{1}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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))))