Average Error: 47.3 → 19.1
Time: 2.2m
Precision: 64
Internal Precision: 2880
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left({\left(1 + i \cdot \frac{1}{2}\right)}^{\frac{1}{3}} \cdot e^{\log \left(100 \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right)}\right) \cdot \left(n \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right) \le -3.834816283885112 \cdot 10^{+297}:\\ \;\;\;\;\frac{\log \left(e^{{\left(\frac{i}{n} + 1\right)}^{n} - 1}\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{if}\;\left({\left(1 + i \cdot \frac{1}{2}\right)}^{\frac{1}{3}} \cdot e^{\log \left(100 \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right)}\right) \cdot \left(n \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right) \le -2.524949745714951 \cdot 10^{-223}:\\ \;\;\;\;\left({\left(1 + i \cdot \frac{1}{2}\right)}^{\frac{1}{3}} \cdot n\right) \cdot \left(\left(100 \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right) \cdot {\left(1 + i \cdot \frac{1}{2}\right)}^{\frac{1}{3}}\right)\\ \mathbf{if}\;\left({\left(1 + i \cdot \frac{1}{2}\right)}^{\frac{1}{3}} \cdot e^{\log \left(100 \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right)}\right) \cdot \left(n \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right) \le 2.1930611554132226 \cdot 10^{-256}:\\ \;\;\;\;\frac{\log \left(e^{{\left(\frac{i}{n} + 1\right)}^{n} - 1}\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{if}\;\left({\left(1 + i \cdot \frac{1}{2}\right)}^{\frac{1}{3}} \cdot e^{\log \left(100 \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right)}\right) \cdot \left(n \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right) \le 1.10824615154099 \cdot 10^{+31} \lor \left({\left(1 + i \cdot \frac{1}{2}\right)}^{\frac{1}{3}} \cdot e^{\log \left(100 \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right)}\right) \cdot \left(n \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right) \le 1.4614977427470927 \cdot 10^{+286}:\\ \;\;\;\;\left({\left(1 + i \cdot \frac{1}{2}\right)}^{\frac{1}{3}} \cdot n\right) \cdot \left(\left(100 \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right) \cdot {\left(1 + i \cdot \frac{1}{2}\right)}^{\frac{1}{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.3
Target46.5
Herbie19.1
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if (* (* (exp (log (* (cbrt (+ 1 (* 1/2 i))) 100))) (pow (+ 1 (* 1/2 i)) 1/3)) (* (cbrt (+ 1 (* 1/2 i))) n)) < -3.834816283885112e+297 or -2.524949745714951e-223 < (* (* (exp (log (* (cbrt (+ 1 (* 1/2 i))) 100))) (pow (+ 1 (* 1/2 i)) 1/3)) (* (cbrt (+ 1 (* 1/2 i))) n)) < 2.1930611554132226e-256

    1. Initial program 19.3

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

    3. Applied add-log-exp19.3

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

    if -3.834816283885112e+297 < (* (* (exp (log (* (cbrt (+ 1 (* 1/2 i))) 100))) (pow (+ 1 (* 1/2 i)) 1/3)) (* (cbrt (+ 1 (* 1/2 i))) n)) < -2.524949745714951e-223 or 1.10824615154099e+31 < (* (* (exp (log (* (cbrt (+ 1 (* 1/2 i))) 100))) (pow (+ 1 (* 1/2 i)) 1/3)) (* (cbrt (+ 1 (* 1/2 i))) n)) < 1.4614977427470927e+286

    1. Initial program 58.3

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

    2. Taylor expanded around 0 60.7

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

    3. Applied simplify27.4

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

    5. Applied *-un-lft-identity27.4

      \[\leadsto \frac{i \cdot \frac{1}{2} + 1}{\frac{\color{blue}{1 \cdot \frac{i}{n}}}{100 \cdot i}}\]

    6. Applied times-frac27.3

      \[\leadsto \frac{i \cdot \frac{1}{2} + 1}{\color{blue}{\frac{1}{100} \cdot \frac{\frac{i}{n}}{i}}}\]

    7. Applied add-cube-cbrt27.4

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

    8. Applied times-frac27.3

      \[\leadsto \color{blue}{\frac{\sqrt[3]{i \cdot \frac{1}{2} + 1} \cdot \sqrt[3]{i \cdot \frac{1}{2} + 1}}{\frac{1}{100}} \cdot \frac{\sqrt[3]{i \cdot \frac{1}{2} + 1}}{\frac{\frac{i}{n}}{i}}}\]

    9. Applied simplify27.3

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

    10. Applied simplify10.6

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

    12. Applied pow1/310.6

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

    14. Applied pow1/310.6

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

    if 2.1930611554132226e-256 < (* (* (exp (log (* (cbrt (+ 1 (* 1/2 i))) 100))) (pow (+ 1 (* 1/2 i)) 1/3)) (* (cbrt (+ 1 (* 1/2 i))) n)) < 1.10824615154099e+31

    1. Initial program 53.8

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

    2. Taylor expanded around 0 55.4

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

    3. Applied simplify19.6

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

    5. Applied *-un-lft-identity19.6

      \[\leadsto \frac{i \cdot \frac{1}{2} + 1}{\frac{\color{blue}{1 \cdot \frac{i}{n}}}{100 \cdot i}}\]

    6. Applied times-frac19.6

      \[\leadsto \frac{i \cdot \frac{1}{2} + 1}{\color{blue}{\frac{1}{100} \cdot \frac{\frac{i}{n}}{i}}}\]

    7. Applied add-cube-cbrt19.7

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

    8. Applied times-frac19.5

      \[\leadsto \color{blue}{\frac{\sqrt[3]{i \cdot \frac{1}{2} + 1} \cdot \sqrt[3]{i \cdot \frac{1}{2} + 1}}{\frac{1}{100}} \cdot \frac{\sqrt[3]{i \cdot \frac{1}{2} + 1}}{\frac{\frac{i}{n}}{i}}}\]

    9. Applied simplify19.5

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

    10. Applied simplify27.6

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

    12. Applied pow1/327.6

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

    14. Applied pow1/327.6

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

    if 1.4614977427470927e+286 < (* (* (exp (log (* (cbrt (+ 1 (* 1/2 i))) 100))) (pow (+ 1 (* 1/2 i)) 1/3)) (* (cbrt (+ 1 (* 1/2 i))) n))

    1. Initial program 29.7

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

    3. Applied associate-*r/29.7

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

  4. Applied simplify19.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left({\left(1 + i \cdot \frac{1}{2}\right)}^{\frac{1}{3}} \cdot e^{\log \left(100 \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right)}\right) \cdot \left(n \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right) \le -3.834816283885112 \cdot 10^{+297}:\\ \;\;\;\;\frac{\log \left(e^{{\left(\frac{i}{n} + 1\right)}^{n} - 1}\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{if}\;\left({\left(1 + i \cdot \frac{1}{2}\right)}^{\frac{1}{3}} \cdot e^{\log \left(100 \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right)}\right) \cdot \left(n \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right) \le -2.524949745714951 \cdot 10^{-223}:\\ \;\;\;\;\left({\left(1 + i \cdot \frac{1}{2}\right)}^{\frac{1}{3}} \cdot n\right) \cdot \left(\left(100 \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right) \cdot {\left(1 + i \cdot \frac{1}{2}\right)}^{\frac{1}{3}}\right)\\ \mathbf{if}\;\left({\left(1 + i \cdot \frac{1}{2}\right)}^{\frac{1}{3}} \cdot e^{\log \left(100 \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right)}\right) \cdot \left(n \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right) \le 2.1930611554132226 \cdot 10^{-256}:\\ \;\;\;\;\frac{\log \left(e^{{\left(\frac{i}{n} + 1\right)}^{n} - 1}\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{if}\;\left({\left(1 + i \cdot \frac{1}{2}\right)}^{\frac{1}{3}} \cdot e^{\log \left(100 \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right)}\right) \cdot \left(n \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right) \le 1.10824615154099 \cdot 10^{+31} \lor \left({\left(1 + i \cdot \frac{1}{2}\right)}^{\frac{1}{3}} \cdot e^{\log \left(100 \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right)}\right) \cdot \left(n \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right) \le 1.4614977427470927 \cdot 10^{+286}:\\ \;\;\;\;\left({\left(1 + i \cdot \frac{1}{2}\right)}^{\frac{1}{3}} \cdot n\right) \cdot \left(\left(100 \cdot \sqrt[3]{1 + i \cdot \frac{1}{2}}\right) \cdot {\left(1 + i \cdot \frac{1}{2}\right)}^{\frac{1}{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \end{array}}\]

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed 2018199 
(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))))