Average Error: 42.7 → 25.0
Time: 1.5m
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -6.819767655434368 \cdot 10^{-06}:\\ \;\;\;\;\frac{1}{i} \cdot \left((100 \cdot \left(e^{n \cdot \log_* (1 + \frac{i}{n})}\right) + -100)_* \cdot n\right)\\ \mathbf{elif}\;i \le 0.06844872388115948:\\ \;\;\;\;\frac{i \cdot (i \cdot \left((i \cdot \frac{50}{3} + 50)_*\right) + 100)_*}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{(50 \cdot \left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right) + \left((\frac{50}{3} \cdot \left(\left(\left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right) \cdot n\right) \cdot \log i\right) + \left((100 \cdot \left(n \cdot \log i\right) + \left(50 \cdot \left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right) + \left(\log i \cdot \left(n \cdot \left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right)\right)\right) \cdot 50\right))_*\right))_*\right))_* - \left(\left(\left(\log i \cdot 50\right) \cdot \left(\log n \cdot \left(n \cdot n\right)\right) + (\left(\left(\log i \cdot \log i\right) \cdot \left(\left(n \cdot n\right) \cdot \left(\log n \cdot n\right)\right)\right) \cdot \frac{100}{3} + \left(\left(\log i \cdot 50\right) \cdot \left(\log n \cdot \left(n \cdot n\right)\right)\right))_*\right) + (\frac{50}{3} \cdot \left(\log n \cdot \left(n \cdot \left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right)\right)\right) + \left((\left(\left(\log i \cdot \log i\right) \cdot \left(\left(n \cdot n\right) \cdot \left(\log n \cdot n\right)\right)\right) \cdot \frac{50}{3} + \left(\left(\log n \cdot n\right) \cdot 100\right))_*\right))_*\right)}{i}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original42.7
Target42.5
Herbie25.0
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -6.819767655434368e-06

    1. Initial program 28.2

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

    2. Simplified28.1

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

    4. Applied add-exp-log28.2

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

    5. Applied pow-exp28.2

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

    6. Simplified5.2

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

    8. Applied insert-posit166.1

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

    10. Applied div-inv6.2

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

    11. Applied *-un-lft-identity6.2

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

    12. Applied times-frac7.4

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

    13. Simplified6.4

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

    if -6.819767655434368e-06 < i < 0.06844872388115948

    1. Initial program 50.3

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

    2. Simplified50.3

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

    3. Taylor expanded around 0 32.7

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

    4. Simplified32.7

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

    if 0.06844872388115948 < i

    1. Initial program 31.1

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

    2. Simplified31.1

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

    4. Applied add-exp-log50.3

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

    5. Applied pow-exp50.3

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

    6. Simplified49.6

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

    8. Applied insert-posit1631.8

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

    10. Applied associate-/r/31.8

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

    11. Taylor expanded around 0 20.5

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

    12. Simplified20.5

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

  4. Final simplification25.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -6.819767655434368 \cdot 10^{-06}:\\ \;\;\;\;\frac{1}{i} \cdot \left((100 \cdot \left(e^{n \cdot \log_* (1 + \frac{i}{n})}\right) + -100)_* \cdot n\right)\\ \mathbf{elif}\;i \le 0.06844872388115948:\\ \;\;\;\;\frac{i \cdot (i \cdot \left((i \cdot \frac{50}{3} + 50)_*\right) + 100)_*}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{(50 \cdot \left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right) + \left((\frac{50}{3} \cdot \left(\left(\left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right) \cdot n\right) \cdot \log i\right) + \left((100 \cdot \left(n \cdot \log i\right) + \left(50 \cdot \left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right) + \left(\log i \cdot \left(n \cdot \left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right)\right)\right) \cdot 50\right))_*\right))_*\right))_* - \left(\left(\left(\log i \cdot 50\right) \cdot \left(\log n \cdot \left(n \cdot n\right)\right) + (\left(\left(\log i \cdot \log i\right) \cdot \left(\left(n \cdot n\right) \cdot \left(\log n \cdot n\right)\right)\right) \cdot \frac{100}{3} + \left(\left(\log i \cdot 50\right) \cdot \left(\log n \cdot \left(n \cdot n\right)\right)\right))_*\right) + (\frac{50}{3} \cdot \left(\log n \cdot \left(n \cdot \left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right)\right)\right) + \left((\left(\left(\log i \cdot \log i\right) \cdot \left(\left(n \cdot n\right) \cdot \left(\log n \cdot n\right)\right)\right) \cdot \frac{50}{3} + \left(\left(\log n \cdot n\right) \cdot 100\right))_*\right))_*\right)}{i}\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 +o rules:numerics
(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))))