Average Error: 42.7 → 21.3
Time: 14.9s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.318831405035280346:\\ \;\;\;\;\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 0.0030890666332557444:\\ \;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.318831405035280346:\\
\;\;\;\;\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;i \le 0.0030890666332557444:\\
\;\;\;\;100 \cdot \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)\\

\mathbf{else}:\\
\;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\

\end{array}
double f(double i, double n) {
        double r142554 = 100.0;
        double r142555 = 1.0;
        double r142556 = i;
        double r142557 = n;
        double r142558 = r142556 / r142557;
        double r142559 = r142555 + r142558;
        double r142560 = pow(r142559, r142557);
        double r142561 = r142560 - r142555;
        double r142562 = r142561 / r142558;
        double r142563 = r142554 * r142562;
        return r142563;
}
↓
double f(double i, double n) {
        double r142564 = i;
        double r142565 = -0.31883140503528035;
        bool r142566 = r142564 <= r142565;
        double r142567 = 100.0;
        double r142568 = sqrt(r142567);
        double r142569 = 1.0;
        double r142570 = n;
        double r142571 = r142564 / r142570;
        double r142572 = r142569 + r142571;
        double r142573 = pow(r142572, r142570);
        double r142574 = r142573 - r142569;
        double r142575 = r142574 / r142571;
        double r142576 = r142568 * r142575;
        double r142577 = r142568 * r142576;
        double r142578 = 0.0030890666332557444;
        bool r142579 = r142564 <= r142578;
        double r142580 = r142569 * r142564;
        double r142581 = 0.5;
        double r142582 = 2.0;
        double r142583 = pow(r142564, r142582);
        double r142584 = r142581 * r142583;
        double r142585 = log(r142569);
        double r142586 = r142585 * r142570;
        double r142587 = r142584 + r142586;
        double r142588 = r142580 + r142587;
        double r142589 = r142583 * r142585;
        double r142590 = r142581 * r142589;
        double r142591 = r142588 - r142590;
        double r142592 = r142591 / r142564;
        double r142593 = r142592 * r142570;
        double r142594 = r142567 * r142593;
        double r142595 = r142574 / r142564;
        double r142596 = r142567 * r142595;
        double r142597 = r142596 * r142570;
        double r142598 = r142579 ? r142594 : r142597;
        double r142599 = r142566 ? r142577 : r142598;
        return r142599;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.7
Target42.6
Herbie21.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -0.31883140503528035

    1. Initial program 28.1

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

    3. Applied add-sqr-sqrt28.1

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

    4. Applied associate-*l*28.2

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

    if -0.31883140503528035 < i < 0.0030890666332557444

    1. Initial program 50.8

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

    2. Taylor expanded around 0 34.3

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

    4. Applied associate-/r/17.1

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

    if 0.0030890666332557444 < i

    1. Initial program 29.6

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

    3. Applied associate-/r/29.6

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

    4. Applied associate-*r*29.6

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

  4. Final simplification21.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.318831405035280346:\\ \;\;\;\;\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 0.0030890666332557444:\\ \;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(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))))