Average Error: 47.6 → 16.2
Time: 52.6s
Precision: 64
Internal Precision: 3136
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -2.0143314523031926 \cdot 10^{+20}:\\ \;\;\;\;\frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{100 \cdot n}}\\ \mathbf{elif}\;i \le 7.638535803835229 \cdot 10^{+95}:\\ \;\;\;\;100 \cdot n + \left(n \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \frac{100}{i}\right) \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.6
Target47.5
Herbie16.2
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -2.0143314523031926e+20

    1. Initial program 26.9

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

    2. Initial simplification27.0

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

    3. Taylor expanded around inf 62.9

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

    4. Simplified18.5

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

    if -2.0143314523031926e+20 < i < 7.638535803835229e+95

    1. Initial program 56.3

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

    2. Initial simplification56.3

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

    3. Taylor expanded around 0 28.4

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

    4. Simplified28.4

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

    5. Taylor expanded around inf 13.4

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

    6. Simplified13.4

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

    if 7.638535803835229e+95 < i

    1. Initial program 30.7

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

    2. Initial simplification30.7

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

    3. Taylor expanded around inf 29.9

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

    4. Simplified30.7

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

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.0143314523031926 \cdot 10^{+20}:\\ \;\;\;\;\frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{100 \cdot n}}\\ \mathbf{elif}\;i \le 7.638535803835229 \cdot 10^{+95}:\\ \;\;\;\;100 \cdot n + \left(n \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \frac{100}{i}\right) \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)\\ \end{array}\]

Runtime

Time bar (total: 52.6s)Debug logProfile

herbie shell --seed 2018227 
(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))))