Average Error: 42.7 → 28.9
Time: 4.2m
Precision: 64
Internal Precision: 128
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -2.643994791550655 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sqrt[3]{\log \left(e^{-1 + {\left(\frac{i}{n}\right)}^{n}}\right) \cdot \left(\log \left(e^{-1 + {\left(\frac{i}{n}\right)}^{n}}\right) \cdot \log \left(e^{-1 + {\left(\frac{i}{n}\right)}^{n}}\right)\right)}}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;i \le 2.5132717108214893 \cdot 10^{-17}:\\ \;\;\;\;100 \cdot \frac{\left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right) + i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(\left(\log i \cdot n + \left(\left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right) \cdot \frac{1}{2} + \left(\left(\left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right) \cdot n\right) \cdot \log i\right) \cdot \frac{1}{2}\right)\right) + \left(\frac{1}{6} \cdot \left(\left(\left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right) \cdot n\right) \cdot \log i\right) + \left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right) \cdot \frac{1}{2}\right)\right) - \left(\frac{1}{3} \cdot \left(\left(\left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right) \cdot n\right) \cdot \log n\right) + \left(\left(\log n \cdot n + \left(\frac{1}{6} \cdot \log n\right) \cdot \left(\left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right) \cdot n\right)\right) + \left(\left(\left(\left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right) \cdot n\right) \cdot \log n\right) \cdot \frac{1}{6} + \log i \cdot \left(\left(n \cdot n\right) \cdot \log n\right)\right)\right)\right)}{\frac{i}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original42.7
Target41.9
Herbie28.9
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -2.643994791550655e-18

    1. Initial program 30.6

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

    2. Taylor expanded around inf 62.9

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

    3. Simplified21.5

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

    5. Applied add-log-exp21.5

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

    6. Applied add-log-exp21.5

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

    7. Applied sum-log21.5

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

    8. Simplified21.5

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

    10. Applied add-cbrt-cube21.5

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

    if -2.643994791550655e-18 < i < 2.5132717108214893e-17

    1. Initial program 49.6

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

    2. Taylor expanded around 0 33.4

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

    3. Simplified33.4

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

    if 2.5132717108214893e-17 < i

    1. Initial program 35.2

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

    2. Taylor expanded around inf 31.7

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

    3. Simplified35.3

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

    5. Applied add-log-exp35.8

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

    6. Applied add-log-exp35.8

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

    7. Applied sum-log35.8

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

    8. Simplified35.8

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

    9. Taylor expanded around 0 22.8

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

    10. Simplified22.8

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

  4. Final simplification28.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.643994791550655 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sqrt[3]{\log \left(e^{-1 + {\left(\frac{i}{n}\right)}^{n}}\right) \cdot \left(\log \left(e^{-1 + {\left(\frac{i}{n}\right)}^{n}}\right) \cdot \log \left(e^{-1 + {\left(\frac{i}{n}\right)}^{n}}\right)\right)}}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;i \le 2.5132717108214893 \cdot 10^{-17}:\\ \;\;\;\;100 \cdot \frac{\left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right) + i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(\left(\log i \cdot n + \left(\left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right) \cdot \frac{1}{2} + \left(\left(\left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right) \cdot n\right) \cdot \log i\right) \cdot \frac{1}{2}\right)\right) + \left(\frac{1}{6} \cdot \left(\left(\left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right) \cdot n\right) \cdot \log i\right) + \left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right) \cdot \frac{1}{2}\right)\right) - \left(\frac{1}{3} \cdot \left(\left(\left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right) \cdot n\right) \cdot \log n\right) + \left(\left(\log n \cdot n + \left(\frac{1}{6} \cdot \log n\right) \cdot \left(\left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right) \cdot n\right)\right) + \left(\left(\left(\left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right) \cdot n\right) \cdot \log n\right) \cdot \frac{1}{6} + \log i \cdot \left(\left(n \cdot n\right) \cdot \log n\right)\right)\right)\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 
(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))))