Average Error: 47.8 → 16.9
Time: 12.9s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -6.8977356701701615 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 3.9028797563160959 \cdot 10^{-7}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}\right) \cdot 100\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} - \sqrt{1}}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -6.8977356701701615 \cdot 10^{-9}:\\
\;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;i \le 3.9028797563160959 \cdot 10^{-7}:\\
\;\;\;\;\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(\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}\right) \cdot 100\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} - \sqrt{1}}{\frac{i}{n}}\\

\end{array}
double f(double i, double n) {
        double r98636 = 100.0;
        double r98637 = 1.0;
        double r98638 = i;
        double r98639 = n;
        double r98640 = r98638 / r98639;
        double r98641 = r98637 + r98640;
        double r98642 = pow(r98641, r98639);
        double r98643 = r98642 - r98637;
        double r98644 = r98643 / r98640;
        double r98645 = r98636 * r98644;
        return r98645;
}
↓
double f(double i, double n) {
        double r98646 = i;
        double r98647 = -6.8977356701701615e-09;
        bool r98648 = r98646 <= r98647;
        double r98649 = 100.0;
        double r98650 = 1.0;
        double r98651 = n;
        double r98652 = r98646 / r98651;
        double r98653 = r98650 + r98652;
        double r98654 = pow(r98653, r98651);
        double r98655 = r98654 / r98652;
        double r98656 = r98650 / r98652;
        double r98657 = r98655 - r98656;
        double r98658 = r98649 * r98657;
        double r98659 = 3.902879756316096e-07;
        bool r98660 = r98646 <= r98659;
        double r98661 = r98650 * r98646;
        double r98662 = 0.5;
        double r98663 = 2.0;
        double r98664 = pow(r98646, r98663);
        double r98665 = r98662 * r98664;
        double r98666 = log(r98650);
        double r98667 = r98666 * r98651;
        double r98668 = r98665 + r98667;
        double r98669 = r98661 + r98668;
        double r98670 = r98664 * r98666;
        double r98671 = r98662 * r98670;
        double r98672 = r98669 - r98671;
        double r98673 = r98672 / r98646;
        double r98674 = r98649 * r98673;
        double r98675 = r98674 * r98651;
        double r98676 = r98651 / r98663;
        double r98677 = pow(r98653, r98676);
        double r98678 = sqrt(r98650);
        double r98679 = r98677 + r98678;
        double r98680 = r98679 * r98649;
        double r98681 = r98677 - r98678;
        double r98682 = r98681 / r98652;
        double r98683 = r98680 * r98682;
        double r98684 = r98660 ? r98675 : r98683;
        double r98685 = r98648 ? r98658 : r98684;
        return r98685;
}

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
Target47.4
Herbie16.9
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -6.8977356701701615e-09

    1. Initial program 29.3

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

    3. Applied div-sub29.3

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

    if -6.8977356701701615e-09 < i < 3.902879756316096e-07

    1. Initial program 58.5

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

    2. Taylor expanded around 0 26.3

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

    4. Applied associate-/r/8.6

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

    5. Applied associate-*r*8.6

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

    if 3.902879756316096e-07 < i

    1. Initial program 32.1

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

    3. Applied *-un-lft-identity32.1

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

    4. Applied *-un-lft-identity32.1

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

    5. Applied times-frac32.1

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

    6. Applied add-sqr-sqrt32.1

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

    7. Applied sqr-pow32.2

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

    8. Applied difference-of-squares32.2

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

    9. Applied times-frac32.2

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

    10. Applied associate-*r*32.2

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

    11. Simplified32.2

      \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}\right) \cdot 100\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} - \sqrt{1}}{\frac{i}{n}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -6.8977356701701615 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 3.9028797563160959 \cdot 10^{-7}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}\right) \cdot 100\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} - \sqrt{1}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))))