Average Error: 47.2 → 16.4
Time: 2.2m
Precision: 64
Internal Precision: 3392
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.11560161101003551 \lor \neg \left(i \le 6929.885049504173\right):\\ \;\;\;\;100 \cdot \frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - 1}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right)}}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \sqrt[3]{1 + \frac{1}{2} \cdot i}\right) \cdot \left(\sqrt[3]{1 + \frac{1}{2} \cdot i} \cdot \left(100 \cdot \sqrt[3]{1 + \frac{1}{2} \cdot i}\right)\right)\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.2
Target47.0
Herbie16.4
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 2 regimes

  2. if i < -0.11560161101003551 or 6929.885049504173 < i

    1. Initial program 28.7

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

    3. Applied flip3--28.7

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

    4. Applied simplify28.7

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

    5. Applied simplify28.7

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

    if -0.11560161101003551 < i < 6929.885049504173

    1. Initial program 57.4

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

    2. Taylor expanded around 0 57.1

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

    3. Applied simplify25.8

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

    5. Applied *-un-lft-identity25.8

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

    6. Applied times-frac25.8

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

    7. Applied add-cube-cbrt25.8

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

    8. Applied times-frac25.7

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

    9. Applied simplify25.7

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

    10. Applied simplify9.5

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

  4. Applied simplify16.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;i \le -0.11560161101003551 \lor \neg \left(i \le 6929.885049504173\right):\\ \;\;\;\;100 \cdot \frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - 1}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right)}}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \sqrt[3]{1 + \frac{1}{2} \cdot i}\right) \cdot \left(\sqrt[3]{1 + \frac{1}{2} \cdot i} \cdot \left(100 \cdot \sqrt[3]{1 + \frac{1}{2} \cdot i}\right)\right)\\ \end{array}}\]

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed 2018207 
(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))))