Average Error: 42.4 → 21.4
Time: 17.6s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.49613502984700486:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 2.7217117903300314 \cdot 10^{-9}:\\ \;\;\;\;\left(\left(100 + \left(50 \cdot i + 100 \cdot \frac{\log 1 \cdot n}{i}\right)\right) - 50 \cdot \left(i \cdot \log 1\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{i}\right) \cdot n\\ \end{array}\]
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(\left(100 + \left(50 \cdot i + 100 \cdot \frac{\log 1 \cdot n}{i}\right)\right) - 50 \cdot \left(i \cdot \log 1\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 r120561 = 100.0;
        double r120562 = 1.0;
        double r120563 = i;
        double r120564 = n;
        double r120565 = r120563 / r120564;
        double r120566 = r120562 + r120565;
        double r120567 = pow(r120566, r120564);
        double r120568 = r120567 - r120562;
        double r120569 = r120568 / r120565;
        double r120570 = r120561 * r120569;
        return r120570;
}
↓
double f(double i, double n) {
        double r120571 = i;
        double r120572 = -1.4961350298470049;
        bool r120573 = r120571 <= r120572;
        double r120574 = 100.0;
        double r120575 = 1.0;
        double r120576 = n;
        double r120577 = r120571 / r120576;
        double r120578 = r120575 + r120577;
        double r120579 = pow(r120578, r120576);
        double r120580 = r120579 / r120577;
        double r120581 = r120575 / r120577;
        double r120582 = r120580 - r120581;
        double r120583 = r120574 * r120582;
        double r120584 = 2.7217117903300314e-09;
        bool r120585 = r120571 <= r120584;
        double r120586 = 50.0;
        double r120587 = r120586 * r120571;
        double r120588 = log(r120575);
        double r120589 = r120588 * r120576;
        double r120590 = r120589 / r120571;
        double r120591 = r120574 * r120590;
        double r120592 = r120587 + r120591;
        double r120593 = r120574 + r120592;
        double r120594 = r120571 * r120588;
        double r120595 = r120586 * r120594;
        double r120596 = r120593 - r120595;
        double r120597 = r120596 * r120576;
        double r120598 = 2.0;
        double r120599 = r120598 * r120576;
        double r120600 = pow(r120578, r120599);
        double r120601 = r120575 * r120575;
        double r120602 = r120600 - r120601;
        double r120603 = r120579 + r120575;
        double r120604 = r120602 / r120603;
        double r120605 = r120604 / r120571;
        double r120606 = r120574 * r120605;
        double r120607 = r120606 * r120576;
        double r120608 = r120585 ? r120597 : r120607;
        double r120609 = r120573 ? r120583 : r120608;
        return r120609;
}

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. Taylor expanded around 0 17.1

      \[\leadsto \color{blue}{\left(\left(100 + \left(50 \cdot i + 100 \cdot \frac{\log 1 \cdot n}{i}\right)\right) - 50 \cdot \left(i \cdot \log 1\right)\right)} \cdot n\]

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.49613502984700486:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 2.7217117903300314 \cdot 10^{-9}:\\ \;\;\;\;\left(\left(100 + \left(50 \cdot i + 100 \cdot \frac{\log 1 \cdot n}{i}\right)\right) - 50 \cdot \left(i \cdot \log 1\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{i}\right) \cdot n\\ \end{array}\]

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))))