Average Error: 47.1 → 11.1
Time: 3.4m
Precision: 64
Internal Precision: 3200
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -6.086370500870421 \cdot 10^{-15}:\\ \;\;\;\;100 \cdot \frac{e^{n \cdot \log_* (1 + \frac{i}{n})} - 1}{\frac{i}{n}}\\ \mathbf{if}\;i \le 0.14419417012382005:\\ \;\;\;\;n \cdot \left(\frac{100}{1} \cdot (\frac{1}{2} \cdot i + 1)_*\right)\\ \mathbf{if}\;i \le 2.389454476516875 \cdot 10^{+233}:\\ \;\;\;\;\left(\frac{100}{i} \cdot n\right) \cdot \left(\frac{{i}^{n}}{{n}^{n}} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i} \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\right) \cdot 100\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.1
Target46.7
Herbie11.1
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -6.086370500870421e-15

    1. Initial program 28.5

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

    3. Applied add-exp-log28.5

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

    4. Applied pow-exp28.5

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

    5. Applied simplify6.6

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

    if -6.086370500870421e-15 < i < 0.14419417012382005

    1. Initial program 57.4

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

    2. Taylor expanded around 0 57.2

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

    3. Applied simplify26.5

      \[\leadsto \color{blue}{\frac{100}{\frac{i}{n}} \cdot (i \cdot \left(i \cdot \frac{1}{2}\right) + i)_*}\]
    4. Using strategy rm

    5. Applied pow126.5

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

    6. Applied pow126.5

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

    7. Applied pow-prod-down26.5

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

    8. Applied simplify9.1

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

    if 0.14419417012382005 < i < 2.389454476516875e+233

    1. Initial program 32.8

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

    2. Taylor expanded around inf 28.5

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

    3. Applied simplify28.4

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

    if 2.389454476516875e+233 < i

    1. Initial program 30.4

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

    3. Applied div-inv30.4

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

    4. Applied add-cube-cbrt30.4

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

    5. Applied times-frac30.5

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

  4. Applied simplify11.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;i \le -6.086370500870421 \cdot 10^{-15}:\\ \;\;\;\;100 \cdot \frac{e^{n \cdot \log_* (1 + \frac{i}{n})} - 1}{\frac{i}{n}}\\ \mathbf{if}\;i \le 0.14419417012382005:\\ \;\;\;\;n \cdot \left(\frac{100}{1} \cdot (\frac{1}{2} \cdot i + 1)_*\right)\\ \mathbf{if}\;i \le 2.389454476516875 \cdot 10^{+233}:\\ \;\;\;\;\left(\frac{100}{i} \cdot n\right) \cdot \left(\frac{{i}^{n}}{{n}^{n}} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i} \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\right) \cdot 100\\ \end{array}}\]

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' +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))))