Average Error: 42.5 → 30.1
Time: 35.6s
Precision: 64
Internal Precision: 128
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.11652804108123575:\\ \;\;\;\;\left(-1 + {\left(\frac{i}{n}\right)}^{n}\right) \cdot \frac{1}{\frac{i}{100 \cdot n}}\\ \mathbf{elif}\;i \le -2.8091648519346336 \cdot 10^{-110}:\\ \;\;\;\;\frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;i \le -5.027682451313799 \cdot 10^{-244}:\\ \;\;\;\;\sqrt[3]{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \cdot \left(\sqrt[3]{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \cdot \sqrt[3]{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100}\right)\\ \mathbf{elif}\;i \le 0.034213549460077985:\\ \;\;\;\;\frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{1}{2} \cdot \left(\left(\log n \cdot \log n\right) \cdot \left(\log i \cdot {n}^{3}\right)\right) + \left(\left(\frac{1}{2} \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot \log i\right) + \log i \cdot n\right)\right) + \left({\left(\log i\right)}^{3} \cdot \left(\left(n \cdot n\right) \cdot \left(n \cdot \frac{1}{6}\right)\right) + \left(\log n \cdot \log n\right) \cdot \left(\frac{1}{2} \cdot \left(n \cdot n\right)\right)\right)\right) - \left(\left(\left(\left(\log i \cdot \log i\right) \cdot \log n\right) \cdot \left(\left(n \cdot n\right) \cdot \left(n \cdot \frac{1}{6}\right)\right) + \left(\log n \cdot n + \left(\left(n \cdot n\right) \cdot \left(n \cdot \frac{1}{6}\right)\right) \cdot {\left(\log n\right)}^{3}\right)\right) + \left(\log n \cdot \left(\left(n \cdot n\right) \cdot \log i\right) + \left(\left(\log i \cdot \log i\right) \cdot \log n\right) \cdot \left(\left(\frac{1}{3} \cdot n\right) \cdot \left(n \cdot n\right)\right)\right)\right)\right) \cdot \frac{100}{\frac{i}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.5
Target42.2
Herbie30.1
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -0.11652804108123575

    1. Initial program 28.3

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

    2. Taylor expanded around inf 62.9

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

    3. Simplified18.7

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

    5. Applied clear-num18.7

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

    7. Applied associate-/l/19.1

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

    if -0.11652804108123575 < i < -2.8091648519346336e-110 or -5.027682451313799e-244 < i < 0.034213549460077985

    1. Initial program 50.2

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

    2. Taylor expanded around 0 32.5

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

    3. Simplified32.5

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

    if -2.8091648519346336e-110 < i < -5.027682451313799e-244

    1. Initial program 48.3

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

    3. Applied add-cube-cbrt48.3

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

    if 0.034213549460077985 < i

    1. Initial program 32.0

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

    2. Taylor expanded around inf 31.1

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

    3. Simplified32.1

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

    4. Taylor expanded around 0 21.4

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

    5. Simplified21.4

      \[\leadsto \frac{100}{\frac{i}{n}} \cdot \color{blue}{\left(\left(\left(\left(\left(\frac{1}{6} \cdot n\right) \cdot \left(n \cdot n\right)\right) \cdot {\left(\log i\right)}^{3} + \left(\left(n \cdot n\right) \cdot \frac{1}{2}\right) \cdot \left(\log n \cdot \log n\right)\right) + \left(\left(\left(\log i \cdot {n}^{3}\right) \cdot \left(\log n \cdot \log n\right)\right) \cdot \frac{1}{2} + \left(\left(\log i \cdot \log i\right) \cdot \left(\left(n \cdot n\right) \cdot \frac{1}{2}\right) + n \cdot \log i\right)\right)\right) - \left(\left(\left(\left(\frac{1}{3} \cdot n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \left(\log i \cdot \log i\right)\right) + \left(\left(\log i \cdot \left(n \cdot n\right)\right) \cdot \log n\right) \cdot 1\right) + \left(\left({\left(\log n\right)}^{3} \cdot \left(\left(\frac{1}{6} \cdot n\right) \cdot \left(n \cdot n\right)\right) + n \cdot \log n\right) + \left(\left(\frac{1}{6} \cdot n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \left(\log i \cdot \log i\right)\right)\right)\right)\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification30.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.11652804108123575:\\ \;\;\;\;\left(-1 + {\left(\frac{i}{n}\right)}^{n}\right) \cdot \frac{1}{\frac{i}{100 \cdot n}}\\ \mathbf{elif}\;i \le -2.8091648519346336 \cdot 10^{-110}:\\ \;\;\;\;\frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;i \le -5.027682451313799 \cdot 10^{-244}:\\ \;\;\;\;\sqrt[3]{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \cdot \left(\sqrt[3]{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \cdot \sqrt[3]{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100}\right)\\ \mathbf{elif}\;i \le 0.034213549460077985:\\ \;\;\;\;\frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{1}{2} \cdot \left(\left(\log n \cdot \log n\right) \cdot \left(\log i \cdot {n}^{3}\right)\right) + \left(\left(\frac{1}{2} \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot \log i\right) + \log i \cdot n\right)\right) + \left({\left(\log i\right)}^{3} \cdot \left(\left(n \cdot n\right) \cdot \left(n \cdot \frac{1}{6}\right)\right) + \left(\log n \cdot \log n\right) \cdot \left(\frac{1}{2} \cdot \left(n \cdot n\right)\right)\right)\right) - \left(\left(\left(\left(\log i \cdot \log i\right) \cdot \log n\right) \cdot \left(\left(n \cdot n\right) \cdot \left(n \cdot \frac{1}{6}\right)\right) + \left(\log n \cdot n + \left(\left(n \cdot n\right) \cdot \left(n \cdot \frac{1}{6}\right)\right) \cdot {\left(\log n\right)}^{3}\right)\right) + \left(\log n \cdot \left(\left(n \cdot n\right) \cdot \log i\right) + \left(\left(\log i \cdot \log i\right) \cdot \log n\right) \cdot \left(\left(\frac{1}{3} \cdot n\right) \cdot \left(n \cdot n\right)\right)\right)\right)\right) \cdot \frac{100}{\frac{i}{n}}\\ \end{array}\]

Runtime

Time bar (total: 35.6s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes43.030.116.226.848.1%
herbie shell --seed 2018355 
(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))))