Average Error: 47.2 → 13.5
Time: 53.1s
Precision: 64
Internal Precision: 3136
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.5827350381820495:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.4050748044378363 \cdot 10^{-199}:\\ \;\;\;\;100 \cdot \left(\frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}{\frac{1}{n}} \cdot \frac{1}{i}\right)\\ \mathbf{elif}\;i \le 111.11466664412667:\\ \;\;\;\;100 \cdot \left({\left(e^{\sqrt[3]{\log \left(\frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}{i}\right)} \cdot \sqrt[3]{\log \left(\frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}{i}\right)}}\right)}^{\left(\sqrt[3]{\log \left(\frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}{i}\right)}\right)} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot 0\\ \end{array}\]

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
Target46.7
Herbie13.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -1.5827350381820495

    1. Initial program 27.9

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

    2. Taylor expanded around inf 62.8

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

    3. Simplified17.1

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

    if -1.5827350381820495 < i < -1.4050748044378363e-199

    1. Initial program 55.3

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

    2. Taylor expanded around 0 24.3

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

    3. Simplified24.3

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

    5. Applied div-inv24.4

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

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

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

    7. Applied times-frac11.6

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

    if -1.4050748044378363e-199 < i < 111.11466664412667

    1. Initial program 58.4

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

    2. Taylor expanded around 0 25.7

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

    3. Simplified25.7

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

    5. Applied associate-/r/7.8

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

    7. Applied add-exp-log7.8

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

    9. Applied add-cube-cbrt7.8

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

    10. Applied exp-prod7.8

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

    if 111.11466664412667 < 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 29.3

      \[\leadsto 100 \cdot \color{blue}{0}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification13.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.5827350381820495:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.4050748044378363 \cdot 10^{-199}:\\ \;\;\;\;100 \cdot \left(\frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}{\frac{1}{n}} \cdot \frac{1}{i}\right)\\ \mathbf{elif}\;i \le 111.11466664412667:\\ \;\;\;\;100 \cdot \left({\left(e^{\sqrt[3]{\log \left(\frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}{i}\right)} \cdot \sqrt[3]{\log \left(\frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}{i}\right)}}\right)}^{\left(\sqrt[3]{\log \left(\frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}{i}\right)}\right)} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot 0\\ \end{array}\]

Runtime

Time bar (total: 53.1s)Debug logProfile

herbie shell --seed 2018219 
(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))))