Average Error: 43.4 → 20.5
Time: 25.2s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.9893212200252493593310987307631876319647:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.508652436303369411416630768794228112384 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \left(\left(\left(n - \left(i \cdot 0.5\right) \cdot i\right) \cdot \log 1 + i \cdot \left(i \cdot 0.5 + 1\right)\right) \cdot n\right)\right)\\ \mathbf{elif}\;i \le 8.608246042115979009006453659316669705257 \cdot 10^{235}:\\ \;\;\;\;\frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i}{n} \cdot \left(\left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot 1 + {\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)}\right)} \cdot 100\\ \mathbf{elif}\;i \le 1.844738975002478002763642496553644874398 \cdot 10^{296}:\\ \;\;\;\;\frac{\left(\left(1 + i \cdot 1\right) + n \cdot \log 1\right) - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n - \frac{1}{\frac{i}{n}}\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.9893212200252493593310987307631876319647:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 1.508652436303369411416630768794228112384 \cdot 10^{-9}:\\
\;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \left(\left(\left(n - \left(i \cdot 0.5\right) \cdot i\right) \cdot \log 1 + i \cdot \left(i \cdot 0.5 + 1\right)\right) \cdot n\right)\right)\\

\mathbf{elif}\;i \le 8.608246042115979009006453659316669705257 \cdot 10^{235}:\\
\;\;\;\;\frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i}{n} \cdot \left(\left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot 1 + {\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)}\right)} \cdot 100\\

\mathbf{elif}\;i \le 1.844738975002478002763642496553644874398 \cdot 10^{296}:\\
\;\;\;\;\frac{\left(\left(1 + i \cdot 1\right) + n \cdot \log 1\right) - 1}{\frac{i}{n}} \cdot 100\\

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

\end{array}
double f(double i, double n) {
        double r105544 = 100.0;
        double r105545 = 1.0;
        double r105546 = i;
        double r105547 = n;
        double r105548 = r105546 / r105547;
        double r105549 = r105545 + r105548;
        double r105550 = pow(r105549, r105547);
        double r105551 = r105550 - r105545;
        double r105552 = r105551 / r105548;
        double r105553 = r105544 * r105552;
        return r105553;
}
↓
double f(double i, double n) {
        double r105554 = i;
        double r105555 = -0.9893212200252494;
        bool r105556 = r105554 <= r105555;
        double r105557 = 100.0;
        double r105558 = n;
        double r105559 = r105554 / r105558;
        double r105560 = 1.0;
        double r105561 = r105559 + r105560;
        double r105562 = 2.0;
        double r105563 = r105558 * r105562;
        double r105564 = pow(r105561, r105563);
        double r105565 = r105560 * r105560;
        double r105566 = r105564 - r105565;
        double r105567 = pow(r105561, r105558);
        double r105568 = r105560 + r105567;
        double r105569 = r105566 / r105568;
        double r105570 = r105569 / r105559;
        double r105571 = r105557 * r105570;
        double r105572 = 1.5086524363033694e-09;
        bool r105573 = r105554 <= r105572;
        double r105574 = 1.0;
        double r105575 = r105574 / r105554;
        double r105576 = 0.5;
        double r105577 = r105554 * r105576;
        double r105578 = r105577 * r105554;
        double r105579 = r105558 - r105578;
        double r105580 = log(r105560);
        double r105581 = r105579 * r105580;
        double r105582 = r105577 + r105560;
        double r105583 = r105554 * r105582;
        double r105584 = r105581 + r105583;
        double r105585 = r105584 * r105558;
        double r105586 = r105575 * r105585;
        double r105587 = r105557 * r105586;
        double r105588 = 8.608246042115979e+235;
        bool r105589 = r105554 <= r105588;
        double r105590 = 3.0;
        double r105591 = pow(r105567, r105590);
        double r105592 = pow(r105560, r105590);
        double r105593 = r105591 - r105592;
        double r105594 = r105568 * r105560;
        double r105595 = r105594 + r105564;
        double r105596 = r105559 * r105595;
        double r105597 = r105593 / r105596;
        double r105598 = r105597 * r105557;
        double r105599 = 1.844738975002478e+296;
        bool r105600 = r105554 <= r105599;
        double r105601 = r105554 * r105560;
        double r105602 = r105574 + r105601;
        double r105603 = r105558 * r105580;
        double r105604 = r105602 + r105603;
        double r105605 = r105604 - r105560;
        double r105606 = r105605 / r105559;
        double r105607 = r105606 * r105557;
        double r105608 = r105567 / r105554;
        double r105609 = r105608 * r105558;
        double r105610 = r105560 / r105559;
        double r105611 = r105609 - r105610;
        double r105612 = r105557 * r105611;
        double r105613 = r105600 ? r105607 : r105612;
        double r105614 = r105589 ? r105598 : r105613;
        double r105615 = r105573 ? r105587 : r105614;
        double r105616 = r105556 ? r105571 : r105615;
        return r105616;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.4
Target43.3
Herbie20.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 5 regimes

  2. if i < -0.9893212200252494

    1. Initial program 27.9

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

    3. Applied flip--27.9

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

    4. Simplified27.9

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

    5. Simplified27.9

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

    if -0.9893212200252494 < i < 1.5086524363033694e-09

    1. Initial program 51.2

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

    2. Taylor expanded around 0 32.6

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

    3. Simplified32.6

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

    5. Applied div-inv32.7

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

    6. Applied *-un-lft-identity32.7

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

    7. Applied times-frac15.1

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

    8. Simplified15.1

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

    if 1.5086524363033694e-09 < i < 8.608246042115979e+235

    1. Initial program 33.3

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

    3. Applied flip3--33.3

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}{\frac{i}{n}}\]

    4. Applied associate-/l/33.3

      \[\leadsto 100 \cdot \color{blue}{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i}{n} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)\right)}}\]

    5. Simplified33.3

      \[\leadsto 100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\color{blue}{\frac{i}{n} \cdot \left({\left(\frac{i}{n} + 1\right)}^{\left(2 \cdot n\right)} + 1 \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} + 1\right)\right)}}\]

    if 8.608246042115979e+235 < i < 1.844738975002478e+296

    1. Initial program 32.1

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

    2. Taylor expanded around 0 34.0

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

    3. Simplified34.0

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

    if 1.844738975002478e+296 < i

    1. Initial program 28.4

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

    3. Applied div-sub28.4

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

    4. Simplified28.4

      \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n} - \frac{1}{\frac{i}{n}}\right)\]
  3. Recombined 5 regimes into one program.

  4. Final simplification20.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.9893212200252493593310987307631876319647:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.508652436303369411416630768794228112384 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \left(\left(\left(n - \left(i \cdot 0.5\right) \cdot i\right) \cdot \log 1 + i \cdot \left(i \cdot 0.5 + 1\right)\right) \cdot n\right)\right)\\ \mathbf{elif}\;i \le 8.608246042115979009006453659316669705257 \cdot 10^{235}:\\ \;\;\;\;\frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i}{n} \cdot \left(\left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot 1 + {\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)}\right)} \cdot 100\\ \mathbf{elif}\;i \le 1.844738975002478002763642496553644874398 \cdot 10^{296}:\\ \;\;\;\;\frac{\left(\left(1 + i \cdot 1\right) + n \cdot \log 1\right) - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n - \frac{1}{\frac{i}{n}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (i n)
  :name "Compound Interest"

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))