Average Error: 47.2 → 23.7
Time: 47.8s
Precision: 64
Internal Precision: 3136
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -3.531906327313068 \cdot 10^{-27}:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 3.1321616989668866 \cdot 10^{-12}:\\ \;\;\;\;\frac{\left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right) + i}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;i \le 2.3822219758927888 \cdot 10^{+215}:\\ \;\;\;\;\left(\left(\left(\left(\left(\log n \cdot \log n\right) \cdot \left(\log i \cdot {n}^{3}\right)\right) \cdot \frac{1}{2} + \left(\log i \cdot n + \left(\log i \cdot \log i\right) \cdot \left(\left(n \cdot n\right) \cdot \frac{1}{2}\right)\right)\right) + \left({\left(\log i\right)}^{3} \cdot \left(\left(\frac{1}{6} \cdot n\right) \cdot \left(n \cdot n\right)\right) + \left(\left(n \cdot n\right) \cdot \frac{1}{2}\right) \cdot \left(\log n \cdot \log n\right)\right)\right) - \left(\left(\left(\log n \cdot n + \left(\left(\frac{1}{6} \cdot n\right) \cdot \left(n \cdot n\right)\right) \cdot {\left(\log n\right)}^{3}\right) + \left(\left(\frac{1}{6} \cdot n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \left(\log i \cdot \log i\right)\right)\right) + \left(\log n \cdot \left(\log i \cdot \left(n \cdot n\right)\right) + \left(\log n \cdot \left(\log i \cdot \log i\right)\right) \cdot \left(\left(n \cdot n\right) \cdot \left(n \cdot \frac{1}{3}\right)\right)\right)\right)\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \end{array}\]

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
Target46.6
Herbie23.7
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -3.531906327313068e-27 or 2.3822219758927888e+215 < i

    1. Initial program 30.8

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

    2. Taylor expanded around inf 56.8

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

    3. Simplified23.3

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

    4. Taylor expanded around inf 56.8

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

    5. Simplified23.3

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

    if -3.531906327313068e-27 < i < 3.1321616989668866e-12

    1. Initial program 57.8

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

    2. Taylor expanded around 0 24.7

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

    3. Simplified24.7

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

    if 3.1321616989668866e-12 < i < 2.3822219758927888e+215

    1. Initial program 34.4

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

    2. Taylor expanded around inf 32.7

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

    3. Simplified34.5

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

    4. Taylor expanded around inf 32.7

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

    5. Simplified34.5

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

    6. Taylor expanded around 0 19.4

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

    7. Simplified19.4

      \[\leadsto \frac{100}{\frac{i}{n}} \cdot \color{blue}{\left(\left(\left(\left(\left(\frac{1}{6} \cdot n\right) \cdot \left(n \cdot n\right)\right) \cdot {\left(\log i\right)}^{3} + \left(\left(n \cdot n\right) \cdot \frac{1}{2}\right) \cdot \left(\log n \cdot \log n\right)\right) + \left(\left(\left(\log i \cdot {n}^{3}\right) \cdot \left(\log n \cdot \log n\right)\right) \cdot \frac{1}{2} + \left(\left(\log i \cdot \log i\right) \cdot \left(\left(n \cdot n\right) \cdot \frac{1}{2}\right) + n \cdot \log i\right)\right)\right) - \left(\left(\left(\left(\frac{1}{3} \cdot n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \left(\log i \cdot \log i\right)\right) + \left(\left(\log i \cdot \left(n \cdot n\right)\right) \cdot \log n\right) \cdot 1\right) + \left(\left({\left(\log n\right)}^{3} \cdot \left(\left(\frac{1}{6} \cdot n\right) \cdot \left(n \cdot n\right)\right) + n \cdot \log n\right) + \left(\left(\frac{1}{6} \cdot n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \left(\log i \cdot \log i\right)\right)\right)\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification23.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -3.531906327313068 \cdot 10^{-27}:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 3.1321616989668866 \cdot 10^{-12}:\\ \;\;\;\;\frac{\left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right) + i}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;i \le 2.3822219758927888 \cdot 10^{+215}:\\ \;\;\;\;\left(\left(\left(\left(\left(\log n \cdot \log n\right) \cdot \left(\log i \cdot {n}^{3}\right)\right) \cdot \frac{1}{2} + \left(\log i \cdot n + \left(\log i \cdot \log i\right) \cdot \left(\left(n \cdot n\right) \cdot \frac{1}{2}\right)\right)\right) + \left({\left(\log i\right)}^{3} \cdot \left(\left(\frac{1}{6} \cdot n\right) \cdot \left(n \cdot n\right)\right) + \left(\left(n \cdot n\right) \cdot \frac{1}{2}\right) \cdot \left(\log n \cdot \log n\right)\right)\right) - \left(\left(\left(\log n \cdot n + \left(\left(\frac{1}{6} \cdot n\right) \cdot \left(n \cdot n\right)\right) \cdot {\left(\log n\right)}^{3}\right) + \left(\left(\frac{1}{6} \cdot n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \left(\log i \cdot \log i\right)\right)\right) + \left(\log n \cdot \left(\log i \cdot \left(n \cdot n\right)\right) + \left(\log n \cdot \left(\log i \cdot \log i\right)\right) \cdot \left(\left(n \cdot n\right) \cdot \left(n \cdot \frac{1}{3}\right)\right)\right)\right)\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \end{array}\]

Runtime

Time bar (total: 47.8s)Debug logProfile

herbie shell --seed 2018254 
(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))))