Average Error: 43.0 → 13.5
Time: 24.7s
Precision: 64
Internal Precision: 128
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -106.96805828974858:\\ \;\;\;\;\frac{1}{\frac{1}{(\left(e^{\log_* (1 + \frac{i}{n}) \cdot n}\right) \cdot 100 + -100)_*} \cdot \frac{i}{n}}\\ \mathbf{elif}\;i \le 2.428070897256001 \cdot 10^{+105}:\\ \;\;\;\;\frac{1}{(\left(\frac{\frac{1}{200}}{n} - \frac{1}{200}\right) \cdot \left(\frac{i}{n}\right) + \left(\frac{1}{\frac{n}{\frac{1}{100}}}\right))_*}\\ \mathbf{elif}\;i \le 1.8625811257567917 \cdot 10^{+273}:\\ \;\;\;\;\frac{\log \left(e^{(\left({\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100 + -100)_*}\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{(\left(\left(\sqrt[3]{\frac{\frac{1}{200}}{n}} \cdot \sqrt[3]{\frac{\frac{1}{200}}{n}}\right) \cdot \sqrt[3]{\frac{\frac{1}{200}}{n}} - \frac{1}{200}\right) \cdot \left(\frac{i}{n}\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original43.0
Target42.9
Herbie13.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -106.96805828974858

    1. Initial program 28.0

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

    2. Simplified28.0

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

    4. Applied add-exp-log28.1

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

    5. Applied pow-exp28.1

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

    6. Simplified5.4

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

    8. Applied *-un-lft-identity5.4

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

    9. Applied associate-/l*5.5

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

    11. Applied div-inv5.5

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

    if -106.96805828974858 < i < 2.428070897256001e+105

    1. Initial program 49.5

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

    2. Simplified49.5

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

    4. Applied add-exp-log50.0

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

    5. Applied pow-exp50.0

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

    6. Simplified48.9

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

    8. Applied *-un-lft-identity48.9

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

    9. Applied associate-/l*48.9

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

    10. Taylor expanded around 0 15.5

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{200} \cdot \frac{i}{{n}^{2}} + \frac{1}{100} \cdot \frac{1}{n}\right) - \frac{1}{200} \cdot \frac{i}{n}}}\]

    11. Simplified13.6

      \[\leadsto \frac{1}{\color{blue}{(\left(\frac{\frac{1}{200}}{n} - \frac{1}{200}\right) \cdot \left(\frac{i}{n}\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}}\]
    12. Using strategy rm

    13. Applied clear-num13.5

      \[\leadsto \frac{1}{(\left(\frac{\frac{1}{200}}{n} - \frac{1}{200}\right) \cdot \left(\frac{i}{n}\right) + \color{blue}{\left(\frac{1}{\frac{n}{\frac{1}{100}}}\right)})_*}\]

    if 2.428070897256001e+105 < i < 1.8625811257567917e+273

    1. Initial program 32.1

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

    2. Simplified32.1

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

    4. Applied add-log-exp32.5

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

    if 1.8625811257567917e+273 < i

    1. Initial program 30.7

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

    2. Simplified30.6

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

    4. Applied add-exp-log61.3

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

    5. Applied pow-exp61.4

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

    6. Simplified61.4

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

    8. Applied *-un-lft-identity61.4

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

    9. Applied associate-/l*61.4

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

    10. Taylor expanded around 0 57.1

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{200} \cdot \frac{i}{{n}^{2}} + \frac{1}{100} \cdot \frac{1}{n}\right) - \frac{1}{200} \cdot \frac{i}{n}}}\]

    11. Simplified27.3

      \[\leadsto \frac{1}{\color{blue}{(\left(\frac{\frac{1}{200}}{n} - \frac{1}{200}\right) \cdot \left(\frac{i}{n}\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}}\]
    12. Using strategy rm

    13. Applied add-cube-cbrt27.3

      \[\leadsto \frac{1}{(\left(\color{blue}{\left(\sqrt[3]{\frac{\frac{1}{200}}{n}} \cdot \sqrt[3]{\frac{\frac{1}{200}}{n}}\right) \cdot \sqrt[3]{\frac{\frac{1}{200}}{n}}} - \frac{1}{200}\right) \cdot \left(\frac{i}{n}\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification13.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -106.96805828974858:\\ \;\;\;\;\frac{1}{\frac{1}{(\left(e^{\log_* (1 + \frac{i}{n}) \cdot n}\right) \cdot 100 + -100)_*} \cdot \frac{i}{n}}\\ \mathbf{elif}\;i \le 2.428070897256001 \cdot 10^{+105}:\\ \;\;\;\;\frac{1}{(\left(\frac{\frac{1}{200}}{n} - \frac{1}{200}\right) \cdot \left(\frac{i}{n}\right) + \left(\frac{1}{\frac{n}{\frac{1}{100}}}\right))_*}\\ \mathbf{elif}\;i \le 1.8625811257567917 \cdot 10^{+273}:\\ \;\;\;\;\frac{\log \left(e^{(\left({\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100 + -100)_*}\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{(\left(\left(\sqrt[3]{\frac{\frac{1}{200}}{n}} \cdot \sqrt[3]{\frac{\frac{1}{200}}{n}}\right) \cdot \sqrt[3]{\frac{\frac{1}{200}}{n}} - \frac{1}{200}\right) \cdot \left(\frac{i}{n}\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019068 +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))))