Average Error: 47.3 → 18.1
Time: 1.8m
Precision: 64
Internal Precision: 3136
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.2885864716259372 \cdot 10^{+23} \lor \neg \left(i \le 6.156007441366224 \cdot 10^{-43}\right):\\ \;\;\;\;\left(\left(\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right) \cdot 100\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n + \frac{\left(n \cdot i\right) \cdot \left(\left(i \cdot \frac{1}{6}\right) \cdot \left(i \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \frac{1}{2}\right)}{\log \left(e^{i \cdot \frac{1}{6}}\right) - \frac{1}{2}}\right) \cdot 100\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.3
Target46.5
Herbie18.1
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 2 regimes

  2. if i < -1.2885864716259372e+23 or 6.156007441366224e-43 < i

    1. Initial program 30.8

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

    3. Applied *-un-lft-identity30.8

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

    4. Applied add-cube-cbrt30.8

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

    5. Applied times-frac30.8

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

    6. Applied associate-*r*30.8

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

    if -1.2885864716259372e+23 < i < 6.156007441366224e-43

    1. Initial program 57.6

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

    2. Taylor expanded around 0 27.0

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

    3. Simplified27.0

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

    5. Applied *-un-lft-identity27.0

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

    6. Applied *-un-lft-identity27.0

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

    7. Applied times-frac27.0

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

    8. Simplified27.0

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

    9. Simplified10.1

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

    11. Applied flip-+10.1

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

    12. Applied associate-*r/10.1

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

    14. Applied add-log-exp10.1

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

  4. Final simplification18.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.2885864716259372 \cdot 10^{+23} \lor \neg \left(i \le 6.156007441366224 \cdot 10^{-43}\right):\\ \;\;\;\;\left(\left(\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right) \cdot 100\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n + \frac{\left(n \cdot i\right) \cdot \left(\left(i \cdot \frac{1}{6}\right) \cdot \left(i \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \frac{1}{2}\right)}{\log \left(e^{i \cdot \frac{1}{6}}\right) - \frac{1}{2}}\right) \cdot 100\\ \end{array}\]

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed 2018215 
(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))))