Average Error: 42.4 → 19.8
Time: 51.0s
Precision: 64
Internal Precision: 128
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.6474128662585373 \cdot 10^{-07}:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.4274539951123314 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \left(\left(\frac{1}{2} + \sqrt[3]{i \cdot \frac{1}{6}} \cdot \left(\sqrt[3]{\sqrt[3]{i \cdot \frac{1}{6}}} \cdot \left(\sqrt[3]{i \cdot \frac{1}{6}} \cdot \left(\sqrt[3]{\sqrt[3]{i \cdot \frac{1}{6}}} \cdot \sqrt[3]{\sqrt[3]{i \cdot \frac{1}{6}}}\right)\right)\right)\right) \cdot \left(n \cdot i\right) + n\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{n} + 1\right)}^{n} \cdot \frac{100 \cdot n}{i} - \frac{100 \cdot n}{i}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.4
Target42.0
Herbie19.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -1.6474128662585373e-07

    1. Initial program 28.4

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

    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. Simplified19.4

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

    if -1.6474128662585373e-07 < i < 1.4274539951123314e+28

    1. Initial program 49.8

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

    3. Applied div-sub49.8

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

    4. Taylor expanded around 0 17.7

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

    5. Simplified17.7

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

    7. Applied add-cube-cbrt17.7

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

    9. Applied add-cube-cbrt17.7

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

    10. Applied associate-*r*17.7

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

    if 1.4274539951123314e+28 < i

    1. Initial program 32.1

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

    2. Initial simplification32.0

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

  4. Final simplification19.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.6474128662585373 \cdot 10^{-07}:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.4274539951123314 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \left(\left(\frac{1}{2} + \sqrt[3]{i \cdot \frac{1}{6}} \cdot \left(\sqrt[3]{\sqrt[3]{i \cdot \frac{1}{6}}} \cdot \left(\sqrt[3]{i \cdot \frac{1}{6}} \cdot \left(\sqrt[3]{\sqrt[3]{i \cdot \frac{1}{6}}} \cdot \sqrt[3]{\sqrt[3]{i \cdot \frac{1}{6}}}\right)\right)\right)\right) \cdot \left(n \cdot i\right) + n\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{n} + 1\right)}^{n} \cdot \frac{100 \cdot n}{i} - \frac{100 \cdot n}{i}\\ \end{array}\]

Runtime

Time bar (total: 51.0s)Debug logProfile

herbie shell --seed 2018296 
(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))))