Average Error: 47.6 → 14.6
Time: 42.9s
Precision: 64
Internal Precision: 3136
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -3.661329260570601 \cdot 10^{-27}:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -8.544469710762926 \cdot 10^{-128}:\\ \;\;\;\;\left(\frac{1}{i} \cdot 100\right) \cdot \frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 2.5465214486200763 \cdot 10^{-09}:\\ \;\;\;\;\left(n \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right) + 100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{n}\right)}^{\left(\left(i \cdot i\right) \cdot \frac{50}{3} + \left(50 \cdot i + 100\right)\right)}\right)\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.6
Target47.1
Herbie14.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -3.661329260570601e-27

    1. Initial program 30.6

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

    2. Taylor expanded around inf 62.8

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

    3. Simplified21.7

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

    if -3.661329260570601e-27 < i < -8.544469710762926e-128

    1. Initial program 54.4

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

    2. Taylor expanded around 0 21.5

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

    3. Simplified21.5

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

    5. Applied div-inv21.6

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

    6. Applied *-un-lft-identity21.6

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

    7. Applied times-frac10.6

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

    8. Applied associate-*r*10.7

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

    if -8.544469710762926e-128 < i < 2.5465214486200763e-09

    1. Initial program 58.6

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

    2. Taylor expanded around 0 26.5

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

    3. Simplified26.5

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

    4. Taylor expanded around -inf 7.2

      \[\leadsto \color{blue}{\frac{50}{3} \cdot \left({i}^{2} \cdot n\right) + \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)}\]

    5. Simplified7.2

      \[\leadsto \color{blue}{\left(i \cdot n\right) \cdot \left(50 + \frac{50}{3} \cdot i\right) + 100 \cdot n}\]

    if 2.5465214486200763e-09 < i

    1. Initial program 33.3

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

    2. Taylor expanded around 0 58.1

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

    3. Simplified58.1

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

    4. Taylor expanded around -inf 60.2

      \[\leadsto \color{blue}{\frac{50}{3} \cdot \left({i}^{2} \cdot n\right) + \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)}\]

    5. Simplified60.2

      \[\leadsto \color{blue}{\left(i \cdot n\right) \cdot \left(50 + \frac{50}{3} \cdot i\right) + 100 \cdot n}\]
    6. Using strategy rm

    7. Applied add-log-exp62.4

      \[\leadsto \left(i \cdot n\right) \cdot \left(50 + \frac{50}{3} \cdot i\right) + \color{blue}{\log \left(e^{100 \cdot n}\right)}\]

    8. Applied add-log-exp54.5

      \[\leadsto \color{blue}{\log \left(e^{\left(i \cdot n\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)}\right)} + \log \left(e^{100 \cdot n}\right)\]

    9. Applied sum-log54.5

      \[\leadsto \color{blue}{\log \left(e^{\left(i \cdot n\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)} \cdot e^{100 \cdot n}\right)}\]

    10. Simplified31.3

      \[\leadsto \log \color{blue}{\left({\left(e^{n}\right)}^{\left(\left(i \cdot i\right) \cdot \frac{50}{3} + \left(50 \cdot i + 100\right)\right)}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification14.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -3.661329260570601 \cdot 10^{-27}:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -8.544469710762926 \cdot 10^{-128}:\\ \;\;\;\;\left(\frac{1}{i} \cdot 100\right) \cdot \frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 2.5465214486200763 \cdot 10^{-09}:\\ \;\;\;\;\left(n \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right) + 100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{n}\right)}^{\left(\left(i \cdot i\right) \cdot \frac{50}{3} + \left(50 \cdot i + 100\right)\right)}\right)\\ \end{array}\]

Runtime

Time bar (total: 42.9s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes28.014.68.419.768.6%
herbie shell --seed 2018290 
(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))))