Average Error: 47.2 → 11.5
Time: 53.6s
Precision: 64
Internal Precision: 3136
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{i}{n} \le -118194.01059359145:\\ \;\;\;\;\frac{100 \cdot n}{i} \cdot {\left(1 + \frac{i}{n}\right)}^{n} - \frac{100 \cdot n}{i}\\ \mathbf{elif}\;\frac{i}{n} \le 2.3868888510701978 \cdot 10^{-18}:\\ \;\;\;\;\left(n + \sqrt[3]{\left(i \cdot n\right) \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)} \cdot e^{\log \left(\sqrt[3]{\left(i \cdot n\right) \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)} \cdot \sqrt[3]{\left(i \cdot n\right) \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)}\right)}\right) \cdot 100\\ \mathbf{elif}\;\frac{i}{n} \le 3.700705677166386 \cdot 10^{+303}:\\ \;\;\;\;\left(\left(n \cdot {\left(1 + \frac{i}{n}\right)}^{n} - n\right) \cdot \frac{1}{i}\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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
Herbie11.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ i n) < -118194.01059359145

    1. Initial program 0.4

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

    2. Initial simplification0.2

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

    if -118194.01059359145 < (/ i n) < 2.3868888510701978e-18

    1. Initial program 61.5

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

    2. Taylor expanded around 0 24.8

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

    3. Simplified24.7

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

    4. Taylor expanded around inf 8.8

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

    5. Simplified8.7

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

    7. Applied add-cube-cbrt8.7

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

    9. Applied add-exp-log8.7

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

    if 2.3868888510701978e-18 < (/ i n) < 3.700705677166386e+303

    1. Initial program 24.9

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

    3. Applied div-inv24.9

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

    4. Applied *-un-lft-identity24.9

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

    5. Applied times-frac24.9

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

    6. Simplified24.9

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

    if 3.700705677166386e+303 < (/ i n)

    1. Initial program 29.0

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

    2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{0}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{i}{n} \le -118194.01059359145:\\ \;\;\;\;\frac{100 \cdot n}{i} \cdot {\left(1 + \frac{i}{n}\right)}^{n} - \frac{100 \cdot n}{i}\\ \mathbf{elif}\;\frac{i}{n} \le 2.3868888510701978 \cdot 10^{-18}:\\ \;\;\;\;\left(n + \sqrt[3]{\left(i \cdot n\right) \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)} \cdot e^{\log \left(\sqrt[3]{\left(i \cdot n\right) \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)} \cdot \sqrt[3]{\left(i \cdot n\right) \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)}\right)}\right) \cdot 100\\ \mathbf{elif}\;\frac{i}{n} \le 3.700705677166386 \cdot 10^{+303}:\\ \;\;\;\;\left(\left(n \cdot {\left(1 + \frac{i}{n}\right)}^{n} - n\right) \cdot \frac{1}{i}\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Runtime

Time bar (total: 53.6s)Debug logProfile

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))))