Average Error: 42.6 → 18.8
Time: 22.8s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -120.575514312032225916482275351881980896:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.483222825543963383682498158929342207957 \cdot 10^{-10}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -120.575514312032225916482275351881980896:\\
\;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 2.483222825543963383682498158929342207957 \cdot 10^{-10}:\\
\;\;\;\;\frac{100}{i} \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{1}{n}}\\

\mathbf{else}:\\
\;\;\;\;\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\\

\end{array}
double f(double i, double n) {
        double r114533 = 100.0;
        double r114534 = 1.0;
        double r114535 = i;
        double r114536 = n;
        double r114537 = r114535 / r114536;
        double r114538 = r114534 + r114537;
        double r114539 = pow(r114538, r114536);
        double r114540 = r114539 - r114534;
        double r114541 = r114540 / r114537;
        double r114542 = r114533 * r114541;
        return r114542;
}
↓
double f(double i, double n) {
        double r114543 = i;
        double r114544 = -120.57551431203223;
        bool r114545 = r114543 <= r114544;
        double r114546 = 100.0;
        double r114547 = n;
        double r114548 = r114543 / r114547;
        double r114549 = pow(r114548, r114547);
        double r114550 = 1.0;
        double r114551 = r114549 - r114550;
        double r114552 = r114551 / r114548;
        double r114553 = r114546 * r114552;
        double r114554 = 2.4832228255439634e-10;
        bool r114555 = r114543 <= r114554;
        double r114556 = r114546 / r114543;
        double r114557 = r114550 * r114543;
        double r114558 = 0.5;
        double r114559 = 2.0;
        double r114560 = pow(r114543, r114559);
        double r114561 = r114558 * r114560;
        double r114562 = log(r114550);
        double r114563 = r114562 * r114547;
        double r114564 = r114561 + r114563;
        double r114565 = r114557 + r114564;
        double r114566 = r114560 * r114562;
        double r114567 = r114558 * r114566;
        double r114568 = r114565 - r114567;
        double r114569 = 1.0;
        double r114570 = r114569 / r114547;
        double r114571 = r114568 / r114570;
        double r114572 = r114556 * r114571;
        double r114573 = r114550 + r114548;
        double r114574 = pow(r114573, r114547);
        double r114575 = r114574 - r114550;
        double r114576 = cbrt(r114575);
        double r114577 = r114576 * r114576;
        double r114578 = r114577 / r114543;
        double r114579 = r114546 * r114578;
        double r114580 = r114576 / r114570;
        double r114581 = r114579 * r114580;
        double r114582 = r114555 ? r114572 : r114581;
        double r114583 = r114545 ? r114553 : r114582;
        return r114583;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.6
Target42.7
Herbie18.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -120.57551431203223

    1. Initial program 27.5

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

    2. Taylor expanded around inf 64.0

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

    3. Simplified17.7

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

    if -120.57551431203223 < i < 2.4832228255439634e-10

    1. Initial program 50.6

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

    3. Applied div-inv50.6

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

    4. Applied *-un-lft-identity50.6

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

    5. Applied times-frac50.3

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

    6. Applied associate-*r*50.4

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

    7. Simplified50.4

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

    8. Taylor expanded around 0 16.0

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

    if 2.4832228255439634e-10 < i

    1. Initial program 32.3

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

    3. Applied div-inv32.3

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

    4. Applied add-cube-cbrt32.3

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

    5. Applied times-frac32.3

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

    6. Applied associate-*r*32.3

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

  4. Final simplification18.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -120.575514312032225916482275351881980896:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.483222825543963383682498158929342207957 \cdot 10^{-10}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(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))))