Average Error: 47.1 → 14.8
Time: 40.2s
Precision: 64
Internal Precision: 3136
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -809395029.0043625:\\ \;\;\;\;\frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{100 \cdot n}}\\ \mathbf{elif}\;i \le 6.201896158457 \cdot 10^{-20}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}} \cdot \sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}}\right)} \cdot \left(\left(n \cdot \left(i \cdot 100\right)\right) \cdot \sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}}\right) + 100 \cdot n\\ \mathbf{elif}\;i \le 1.4823630125186987 \cdot 10^{+235}:\\ \;\;\;\;\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - 1}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right) + {\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{i}{100 \cdot n}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.1
Target47.0
Herbie14.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -809395029.0043625

    1. Initial program 27.7

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

    2. Initial simplification27.8

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

    3. Taylor expanded around inf 62.9

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

    4. Simplified18.3

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

    if -809395029.0043625 < i < 6.201896158457e-20

    1. Initial program 57.5

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

    2. Initial simplification57.5

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

    3. Taylor expanded around 0 26.5

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

    4. Simplified26.5

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

    6. Applied *-un-lft-identity26.5

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

    7. Applied *-un-lft-identity26.5

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

    8. Applied times-frac26.5

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

    9. Simplified26.5

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

    10. Simplified9.6

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

    12. Applied add-cube-cbrt9.6

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

    13. Applied associate-*l*9.6

      \[\leadsto 1 \cdot \left(\color{blue}{\left(\sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}} \cdot \sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}}\right) \cdot \left(\sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}} \cdot \left(\left(i \cdot 100\right) \cdot n\right)\right)} + n \cdot 100\right)\]
    14. Using strategy rm

    15. Applied add-exp-log9.6

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

    if 6.201896158457e-20 < i < 1.4823630125186987e+235

    1. Initial program 31.9

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

    2. Initial simplification31.9

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

    4. Applied flip3--32.0

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

    5. Applied associate-/l/32.0

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

    6. Simplified32.0

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

    if 1.4823630125186987e+235 < i

    1. Initial program 32.9

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

    2. Initial simplification32.9

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

    3. Taylor expanded around 0 28.8

      \[\leadsto \color{blue}{0}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification14.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -809395029.0043625:\\ \;\;\;\;\frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{100 \cdot n}}\\ \mathbf{elif}\;i \le 6.201896158457 \cdot 10^{-20}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}} \cdot \sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}}\right)} \cdot \left(\left(n \cdot \left(i \cdot 100\right)\right) \cdot \sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}}\right) + 100 \cdot n\\ \mathbf{elif}\;i \le 1.4823630125186987 \cdot 10^{+235}:\\ \;\;\;\;\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - 1}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right) + {\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{i}{100 \cdot n}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Runtime

Time bar (total: 40.2s)Debug logProfile

herbie shell --seed 2018220 
(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))))