Average Error: 46.9 → 14.1
Time: 44.9s
Precision: 64
Internal Precision: 3136
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -8.320224335699487:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.7853496252095305 \cdot 10^{-05}:\\ \;\;\;\;100 \cdot \left(\left(\left(\left(\sqrt[3]{\sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}}} \cdot \sqrt[3]{\sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}}}\right) \cdot \left(\sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}} \cdot \sqrt[3]{\sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}}}\right)\right) \cdot \left(n \cdot i\right)\right) \cdot \sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}} + n\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original46.9
Target46.2
Herbie14.1
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -8.320224335699487

    1. Initial program 27.9

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

    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. Simplified18.1

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

    if -8.320224335699487 < i < 1.7853496252095305e-05

    1. Initial program 57.5

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

    2. Taylor expanded around 0 25.0

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

    3. Simplified25.0

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

    4. Taylor expanded around 0 9.1

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

    5. Simplified9.1

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

    7. Applied add-cube-cbrt9.1

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

    8. Applied associate-*r*9.1

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

    10. Applied add-cube-cbrt9.1

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

    11. Applied associate-*l*9.1

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

    if 1.7853496252095305e-05 < i

    1. Initial program 32.4

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

    2. Taylor expanded around 0 29.7

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

  4. Final simplification14.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -8.320224335699487:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.7853496252095305 \cdot 10^{-05}:\\ \;\;\;\;100 \cdot \left(\left(\left(\left(\sqrt[3]{\sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}}} \cdot \sqrt[3]{\sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}}}\right) \cdot \left(\sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}} \cdot \sqrt[3]{\sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}}}\right)\right) \cdot \left(n \cdot i\right)\right) \cdot \sqrt[3]{i \cdot \frac{1}{6} + \frac{1}{2}} + n\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Runtime

Time bar (total: 44.9s)Debug logProfile

herbie shell --seed 2018257 
(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))))