Average Error: 42.4 → 21.4
Time: 19.5s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.49613502984700486:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 2.7217117903300314 \cdot 10^{-9}:\\ \;\;\;\;\left(\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{i}\right) \cdot n\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -1.49613502984700486:\\
\;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;i \le 2.7217117903300314 \cdot 10^{-9}:\\
\;\;\;\;\left(\sqrt{100} \cdot \left(\sqrt{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)\right) \cdot n\\

\mathbf{else}:\\
\;\;\;\;\left(100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{i}\right) \cdot n\\

\end{array}
double f(double i, double n) {
        double r179493 = 100.0;
        double r179494 = 1.0;
        double r179495 = i;
        double r179496 = n;
        double r179497 = r179495 / r179496;
        double r179498 = r179494 + r179497;
        double r179499 = pow(r179498, r179496);
        double r179500 = r179499 - r179494;
        double r179501 = r179500 / r179497;
        double r179502 = r179493 * r179501;
        return r179502;
}
↓
double f(double i, double n) {
        double r179503 = i;
        double r179504 = -1.4961350298470049;
        bool r179505 = r179503 <= r179504;
        double r179506 = 100.0;
        double r179507 = 1.0;
        double r179508 = n;
        double r179509 = r179503 / r179508;
        double r179510 = r179507 + r179509;
        double r179511 = pow(r179510, r179508);
        double r179512 = r179511 / r179509;
        double r179513 = r179507 / r179509;
        double r179514 = r179512 - r179513;
        double r179515 = r179506 * r179514;
        double r179516 = 2.7217117903300314e-09;
        bool r179517 = r179503 <= r179516;
        double r179518 = sqrt(r179506);
        double r179519 = r179507 * r179503;
        double r179520 = 0.5;
        double r179521 = 2.0;
        double r179522 = pow(r179503, r179521);
        double r179523 = r179520 * r179522;
        double r179524 = log(r179507);
        double r179525 = r179524 * r179508;
        double r179526 = r179523 + r179525;
        double r179527 = r179519 + r179526;
        double r179528 = r179522 * r179524;
        double r179529 = r179520 * r179528;
        double r179530 = r179527 - r179529;
        double r179531 = r179530 / r179503;
        double r179532 = r179518 * r179531;
        double r179533 = r179518 * r179532;
        double r179534 = r179533 * r179508;
        double r179535 = r179521 * r179508;
        double r179536 = pow(r179510, r179535);
        double r179537 = r179507 * r179507;
        double r179538 = r179536 - r179537;
        double r179539 = r179511 + r179507;
        double r179540 = r179538 / r179539;
        double r179541 = r179540 / r179503;
        double r179542 = r179506 * r179541;
        double r179543 = r179542 * r179508;
        double r179544 = r179517 ? r179534 : r179543;
        double r179545 = r179505 ? r179515 : r179544;
        return r179545;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.4
Target42.5
Herbie21.4
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -1.4961350298470049

    1. Initial program 26.7

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

    3. Applied div-sub26.8

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

    if -1.4961350298470049 < i < 2.7217117903300314e-09

    1. Initial program 50.4

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

    3. Applied associate-/r/50.1

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

    4. Applied associate-*r*50.1

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

    5. Taylor expanded around 0 17.1

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

    7. Applied add-sqr-sqrt17.1

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

    8. Applied associate-*l*17.1

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

    if 2.7217117903300314e-09 < i

    1. Initial program 32.0

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

    3. Applied associate-/r/32.0

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

    4. Applied associate-*r*32.0

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

    6. Applied flip--32.0

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

    7. Simplified32.0

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

  4. Final simplification21.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.49613502984700486:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 2.7217117903300314 \cdot 10^{-9}:\\ \;\;\;\;\left(\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{i}\right) \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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))))