Average Error: 47.2 → 15.4
Time: 44.3s
Precision: 64
Internal Precision: 3136
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -3683827.786026568:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -3.163487401650747 \cdot 10^{-94}:\\ \;\;\;\;\frac{\left(n \cdot n - \left(\left(n \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) \cdot \left(\left(n \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right) \cdot 100}{n - \left(n \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}\\ \mathbf{elif}\;i \le 2.0374409315644748 \cdot 10^{-19}:\\ \;\;\;\;\left(\sqrt[3]{\left(n \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)} \cdot \left(\sqrt[3]{\left(n \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)} \cdot \sqrt[3]{\left(n \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}\right) + n\right) \cdot 100\\ \mathbf{elif}\;i \le 1.3600583183937914 \cdot 10^{+233}:\\ \;\;\;\;\frac{100 \cdot n}{i} \cdot {\left(1 + \frac{i}{n}\right)}^{n} - \frac{100 \cdot n}{i}\\ \mathbf{elif}\;i \le 3.1352508579636618 \cdot 10^{+280}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.2
Target47.3
Herbie15.4
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 5 regimes

  2. if i < -3683827.786026568 or 3.1352508579636618e+280 < i

    1. Initial program 27.2

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

    2. Taylor expanded around inf 61.5

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

    3. Simplified18.5

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

    if -3683827.786026568 < i < -3.163487401650747e-94

    1. Initial program 51.9

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

    2. Taylor expanded around 0 23.8

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(\frac{1}{2} \cdot {i}^{2} + \frac{1}{6} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]

    3. Simplified23.8

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)}}{\frac{i}{n}}\]

    4. Taylor expanded around inf 20.4

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({i}^{2} \cdot n\right) + \left(n + \frac{1}{2} \cdot \left(i \cdot n\right)\right)\right)}\]

    5. Simplified20.4

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

    7. Applied flip-+27.5

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot n - \left(\left(i \cdot n\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)\right) \cdot \left(\left(i \cdot n\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)\right)}{n - \left(i \cdot n\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)}}\]

    8. Applied associate-*r/27.8

      \[\leadsto \color{blue}{\frac{100 \cdot \left(n \cdot n - \left(\left(i \cdot n\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)\right) \cdot \left(\left(i \cdot n\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)\right)\right)}{n - \left(i \cdot n\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)}}\]

    if -3.163487401650747e-94 < i < 2.0374409315644748e-19

    1. Initial program 58.4

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

    2. Taylor expanded around 0 25.6

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(\frac{1}{2} \cdot {i}^{2} + \frac{1}{6} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]

    3. Simplified25.6

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)}}{\frac{i}{n}}\]

    4. Taylor expanded around inf 7.7

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({i}^{2} \cdot n\right) + \left(n + \frac{1}{2} \cdot \left(i \cdot n\right)\right)\right)}\]

    5. Simplified7.7

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

    7. Applied add-cube-cbrt7.7

      \[\leadsto 100 \cdot \left(n + \color{blue}{\left(\sqrt[3]{\left(i \cdot n\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)} \cdot \sqrt[3]{\left(i \cdot n\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)}\right) \cdot \sqrt[3]{\left(i \cdot n\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)}}\right)\]

    if 2.0374409315644748e-19 < i < 1.3600583183937914e+233

    1. Initial program 33.1

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

    2. Initial simplification33.1

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

    if 1.3600583183937914e+233 < i < 3.1352508579636618e+280

    1. Initial program 30.9

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

    2. Taylor expanded around 0 31.4

      \[\leadsto \color{blue}{0}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification15.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -3683827.786026568:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -3.163487401650747 \cdot 10^{-94}:\\ \;\;\;\;\frac{\left(n \cdot n - \left(\left(n \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) \cdot \left(\left(n \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right) \cdot 100}{n - \left(n \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}\\ \mathbf{elif}\;i \le 2.0374409315644748 \cdot 10^{-19}:\\ \;\;\;\;\left(\sqrt[3]{\left(n \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)} \cdot \left(\sqrt[3]{\left(n \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)} \cdot \sqrt[3]{\left(n \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}\right) + n\right) \cdot 100\\ \mathbf{elif}\;i \le 1.3600583183937914 \cdot 10^{+233}:\\ \;\;\;\;\frac{100 \cdot n}{i} \cdot {\left(1 + \frac{i}{n}\right)}^{n} - \frac{100 \cdot n}{i}\\ \mathbf{elif}\;i \le 3.1352508579636618 \cdot 10^{+280}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \end{array}\]

Runtime

Time bar (total: 44.3s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes28.115.48.419.764.4%
herbie shell --seed 2018274 
(FPCore (i n)
  :name "Compound Interest"

  :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))))