Average Error: 42.7 → 31.4
Time: 29.5s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -2.56140166535216897 \cdot 10^{135}:\\ \;\;\;\;\frac{100 \cdot \frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.3992561866449662 \cdot 10^{-10}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.51591290926460688 \cdot 10^{-160}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 6.02622510223326963 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 8532543483832934860000:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 7.39555913873958208 \cdot 10^{219}:\\ \;\;\;\;\frac{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -2.56140166535216897 \cdot 10^{135}:\\
\;\;\;\;\frac{100 \cdot \frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\

\mathbf{elif}\;i \le -1.3992561866449662 \cdot 10^{-10}:\\
\;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 2.51591290926460688 \cdot 10^{-160}:\\
\;\;\;\;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)}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 6.02622510223326963 \cdot 10^{-125}:\\
\;\;\;\;\frac{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}}{\frac{1}{n}}\\

\mathbf{elif}\;i \le 8532543483832934860000:\\
\;\;\;\;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)}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 7.39555913873958208 \cdot 10^{219}:\\
\;\;\;\;\frac{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}}{\frac{1}{n}}\\

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

\end{array}
double f(double i, double n) {
        double r267532 = 100.0;
        double r267533 = 1.0;
        double r267534 = i;
        double r267535 = n;
        double r267536 = r267534 / r267535;
        double r267537 = r267533 + r267536;
        double r267538 = pow(r267537, r267535);
        double r267539 = r267538 - r267533;
        double r267540 = r267539 / r267536;
        double r267541 = r267532 * r267540;
        return r267541;
}
↓
double f(double i, double n) {
        double r267542 = i;
        double r267543 = -2.561401665352169e+135;
        bool r267544 = r267542 <= r267543;
        double r267545 = 100.0;
        double r267546 = 1.0;
        double r267547 = n;
        double r267548 = r267542 / r267547;
        double r267549 = r267546 + r267548;
        double r267550 = 2.0;
        double r267551 = r267550 * r267547;
        double r267552 = pow(r267549, r267551);
        double r267553 = r267546 * r267546;
        double r267554 = r267552 - r267553;
        double r267555 = exp(r267554);
        double r267556 = log(r267555);
        double r267557 = pow(r267549, r267547);
        double r267558 = r267557 + r267546;
        double r267559 = r267556 / r267558;
        double r267560 = r267545 * r267559;
        double r267561 = r267560 / r267548;
        double r267562 = -1.3992561866449662e-10;
        bool r267563 = r267542 <= r267562;
        double r267564 = pow(r267548, r267547);
        double r267565 = r267564 - r267546;
        double r267566 = r267545 * r267565;
        double r267567 = r267566 / r267548;
        double r267568 = 2.515912909264607e-160;
        bool r267569 = r267542 <= r267568;
        double r267570 = r267546 * r267542;
        double r267571 = 0.5;
        double r267572 = pow(r267542, r267550);
        double r267573 = r267571 * r267572;
        double r267574 = log(r267546);
        double r267575 = r267574 * r267547;
        double r267576 = r267573 + r267575;
        double r267577 = r267570 + r267576;
        double r267578 = r267572 * r267574;
        double r267579 = r267571 * r267578;
        double r267580 = r267577 - r267579;
        double r267581 = r267580 / r267548;
        double r267582 = r267545 * r267581;
        double r267583 = 6.02622510223327e-125;
        bool r267584 = r267542 <= r267583;
        double r267585 = r267557 - r267546;
        double r267586 = r267545 * r267585;
        double r267587 = r267586 / r267542;
        double r267588 = 1.0;
        double r267589 = r267588 / r267547;
        double r267590 = r267587 / r267589;
        double r267591 = 8.532543483832935e+21;
        bool r267592 = r267542 <= r267591;
        double r267593 = 7.395559138739582e+219;
        bool r267594 = r267542 <= r267593;
        double r267595 = r267575 + r267588;
        double r267596 = r267570 + r267595;
        double r267597 = r267596 - r267546;
        double r267598 = r267597 / r267548;
        double r267599 = r267545 * r267598;
        double r267600 = r267594 ? r267590 : r267599;
        double r267601 = r267592 ? r267582 : r267600;
        double r267602 = r267584 ? r267590 : r267601;
        double r267603 = r267569 ? r267582 : r267602;
        double r267604 = r267563 ? r267567 : r267603;
        double r267605 = r267544 ? r267561 : r267604;
        return r267605;
}

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
Herbie31.4
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 5 regimes

  2. if i < -2.561401665352169e+135

    1. Initial program 15.4

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

    3. Applied associate-*r/15.4

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

    5. Applied flip--15.4

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

    6. Simplified15.4

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

    8. Applied add-log-exp15.4

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

    9. Applied neg-log15.4

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

    10. Applied add-log-exp15.4

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

    11. Applied sum-log15.4

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

    12. Simplified15.4

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

    if -2.561401665352169e+135 < i < -1.3992561866449662e-10

    1. Initial program 41.3

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

    3. Applied associate-*r/41.3

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

    4. Taylor expanded around inf 64.0

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

    5. Simplified27.6

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

    if -1.3992561866449662e-10 < i < 2.515912909264607e-160 or 6.02622510223327e-125 < i < 8.532543483832935e+21

    1. Initial program 49.9

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

    2. Taylor expanded around 0 33.7

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

    if 2.515912909264607e-160 < i < 6.02622510223327e-125 or 8.532543483832935e+21 < i < 7.395559138739582e+219

    1. Initial program 38.2

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

    3. Applied associate-*r/38.1

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

    5. Applied div-inv38.2

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

    6. Applied associate-/r*38.1

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

    if 7.395559138739582e+219 < i

    1. Initial program 30.6

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

    2. Taylor expanded around 0 34.7

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

  4. Final simplification31.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.56140166535216897 \cdot 10^{135}:\\ \;\;\;\;\frac{100 \cdot \frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.3992561866449662 \cdot 10^{-10}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.51591290926460688 \cdot 10^{-160}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 6.02622510223326963 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 8532543483832934860000:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 7.39555913873958208 \cdot 10^{219}:\\ \;\;\;\;\frac{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))))