Average Error: 47.4 → 11.4
Time: 54.0s
Precision: 64
Internal Precision: 3392
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -175606663.94769424:\\ \;\;\;\;\frac{100 \cdot e^{i} - 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -3.8745861266623317 \cdot 10^{-187}:\\ \;\;\;\;\frac{\left(n \cdot i\right) \cdot \left(\left(i \cdot i\right) \cdot \frac{50}{3} + \left(50 \cdot i + 100\right)\right)}{i}\\ \mathbf{elif}\;i \le 1.3541607029647162 \cdot 10^{-12}:\\ \;\;\;\;100 \cdot n + \left(n \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\left(n \cdot n\right) \cdot \left(n \cdot 50\right)\right) \cdot \left(\log i \cdot \log i + \log n \cdot \log n\right) + \left(\left(\log n \cdot \log n\right) \cdot \left({n}^{4} \cdot \log i\right)\right) \cdot \frac{50}{3}\right) + \left(\left({n}^{4} \cdot \log i\right) \cdot \left(\frac{100}{3} \cdot \left(\log n \cdot \log n\right)\right) + \left({\left(\log i\right)}^{3} \cdot \left(\frac{50}{3} \cdot {n}^{4}\right) + \left(\left(n \cdot n\right) \cdot 100\right) \cdot \log i\right)\right)\right) - \left(50 \cdot \left(\left(\log i \cdot \log i\right) \cdot \left({n}^{4} \cdot \log n\right)\right) + \left(\left(\left(n \cdot n\right) \cdot \left(n \cdot 50\right) + \left(n \cdot n\right) \cdot \left(n \cdot 50\right)\right) \cdot \left(\log i \cdot \log n\right) + \left({\left(\log n\right)}^{3} \cdot \left(\frac{50}{3} \cdot {n}^{4}\right) + 100 \cdot \left(\left(n \cdot n\right) \cdot \log n\right)\right)\right)\right)}{i}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.4
Target46.8
Herbie11.4
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -175606663.94769424

    1. Initial program 28.1

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

    2. Initial simplification28.5

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

    3. Taylor expanded around -inf 12.5

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

    5. Applied associate-/l*11.9

      \[\leadsto \color{blue}{\frac{100 \cdot e^{i} - 100}{\frac{i}{n}}}\]

    if -175606663.94769424 < i < -3.8745861266623317e-187

    1. Initial program 55.3

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

    2. Initial simplification55.2

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

    3. Taylor expanded around -inf 53.0

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

    4. Taylor expanded around 0 12.5

      \[\leadsto \frac{\color{blue}{50 \cdot \left({i}^{2} \cdot n\right) + \left(\frac{50}{3} \cdot \left({i}^{3} \cdot n\right) + 100 \cdot \left(i \cdot n\right)\right)}}{i}\]

    5. Simplified12.5

      \[\leadsto \frac{\color{blue}{\left(i \cdot n\right) \cdot \left(\left(50 \cdot i + 100\right) + \left(i \cdot i\right) \cdot \frac{50}{3}\right)}}{i}\]

    if -3.8745861266623317e-187 < i < 1.3541607029647162e-12

    1. Initial program 58.4

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

    2. Initial simplification58.4

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

    3. Taylor expanded around -inf 58.6

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

    4. Taylor expanded around 0 7.7

      \[\leadsto \color{blue}{\frac{50}{3} \cdot \left({i}^{2} \cdot n\right) + \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)}\]

    5. Simplified7.7

      \[\leadsto \color{blue}{\left(i \cdot n\right) \cdot \left(50 + \frac{50}{3} \cdot i\right) + 100 \cdot n}\]

    if 1.3541607029647162e-12 < i

    1. Initial program 32.3

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

    2. Initial simplification32.2

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

    3. Taylor expanded around inf 30.1

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

    4. Simplified32.3

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

    6. Applied associate-*l/32.3

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

    7. Applied sub-div32.3

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

    8. Taylor expanded around 0 21.2

      \[\leadsto \frac{\color{blue}{\left(\frac{50}{3} \cdot \left({n}^{4} \cdot \left(\log i \cdot {\left(\log n\right)}^{2}\right)\right) + \left(50 \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right) + \left(50 \cdot \left({n}^{3} \cdot {\left(\log i\right)}^{2}\right) + \left(\frac{100}{3} \cdot \left({n}^{4} \cdot \left({\left(\log n\right)}^{2} \cdot \log i\right)\right) + \left(\frac{50}{3} \cdot \left({n}^{4} \cdot {\left(\log i\right)}^{3}\right) + 100 \cdot \left({n}^{2} \cdot \log i\right)\right)\right)\right)\right)\right) - \left(50 \cdot \left({n}^{3} \cdot \left(\log n \cdot \log i\right)\right) + \left(50 \cdot \left({n}^{3} \cdot \left(\log i \cdot \log n\right)\right) + \left(100 \cdot \left({n}^{2} \cdot \log n\right) + \left(\frac{50}{3} \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{3}\right) + \left(\frac{50}{3} \cdot \left({n}^{4} \cdot \left(\log n \cdot {\left(\log i\right)}^{2}\right)\right) + \frac{100}{3} \cdot \left({n}^{4} \cdot \left({\left(\log i\right)}^{2} \cdot \log n\right)\right)\right)\right)\right)\right)\right)}}{i}\]

    9. Simplified21.2

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{50}{3} \cdot \left(\left({n}^{4} \cdot \log i\right) \cdot \left(\log n \cdot \log n\right)\right) + \left(\left(50 \cdot n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \log n + \log i \cdot \log i\right)\right) + \left(\left({n}^{4} \cdot \log i\right) \cdot \left(\left(\log n \cdot \log n\right) \cdot \frac{100}{3}\right) + \left(\left({n}^{4} \cdot \frac{50}{3}\right) \cdot {\left(\log i\right)}^{3} + \log i \cdot \left(100 \cdot \left(n \cdot n\right)\right)\right)\right)\right) - \left(\left(\left(\log i \cdot \log n\right) \cdot \left(\left(50 \cdot n\right) \cdot \left(n \cdot n\right) + \left(50 \cdot n\right) \cdot \left(n \cdot n\right)\right) + \left({\left(\log n\right)}^{3} \cdot \left({n}^{4} \cdot \frac{50}{3}\right) + \left(\left(n \cdot n\right) \cdot \log n\right) \cdot 100\right)\right) + \left(\left(\log i \cdot \log i\right) \cdot \left({n}^{4} \cdot \log n\right)\right) \cdot 50\right)}}{i}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -175606663.94769424:\\ \;\;\;\;\frac{100 \cdot e^{i} - 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -3.8745861266623317 \cdot 10^{-187}:\\ \;\;\;\;\frac{\left(n \cdot i\right) \cdot \left(\left(i \cdot i\right) \cdot \frac{50}{3} + \left(50 \cdot i + 100\right)\right)}{i}\\ \mathbf{elif}\;i \le 1.3541607029647162 \cdot 10^{-12}:\\ \;\;\;\;100 \cdot n + \left(n \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\left(n \cdot n\right) \cdot \left(n \cdot 50\right)\right) \cdot \left(\log i \cdot \log i + \log n \cdot \log n\right) + \left(\left(\log n \cdot \log n\right) \cdot \left({n}^{4} \cdot \log i\right)\right) \cdot \frac{50}{3}\right) + \left(\left({n}^{4} \cdot \log i\right) \cdot \left(\frac{100}{3} \cdot \left(\log n \cdot \log n\right)\right) + \left({\left(\log i\right)}^{3} \cdot \left(\frac{50}{3} \cdot {n}^{4}\right) + \left(\left(n \cdot n\right) \cdot 100\right) \cdot \log i\right)\right)\right) - \left(50 \cdot \left(\left(\log i \cdot \log i\right) \cdot \left({n}^{4} \cdot \log n\right)\right) + \left(\left(\left(n \cdot n\right) \cdot \left(n \cdot 50\right) + \left(n \cdot n\right) \cdot \left(n \cdot 50\right)\right) \cdot \left(\log i \cdot \log n\right) + \left({\left(\log n\right)}^{3} \cdot \left(\frac{50}{3} \cdot {n}^{4}\right) + 100 \cdot \left(\left(n \cdot n\right) \cdot \log n\right)\right)\right)\right)}{i}\\ \end{array}\]

Runtime

Time bar (total: 54.0s)Debug logProfile

herbie shell --seed 2018277 
(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))))