Average Error: 46.5 → 14.8
Time: 1.1m
Precision: 64
Internal Precision: 2624
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -2.5462627312609303 \cdot 10^{-05}:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 5.470276661507296:\\ \;\;\;\;\left(\left(n \cdot \left(\sqrt[3]{i} + \left(\sqrt[3]{{i}^{4}} \cdot \frac{1}{6} + \frac{1}{36} \cdot \sqrt[3]{{i}^{7}}\right)\right)\right) \cdot \left(\frac{1}{3} \cdot \sqrt[3]{i \cdot i} + \left(\sqrt[3]{{i}^{5}} \cdot \frac{1}{12} + \sqrt[3]{\frac{1}{i}}\right)\right)\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot n}{i} \cdot {\left(\frac{i}{n} + 1\right)}^{n} - \frac{100 \cdot n}{i}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original46.5
Target46.5
Herbie14.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -2.5462627312609303e-05

    1. Initial program 27.8

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

    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. Simplified18.2

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

    if -2.5462627312609303e-05 < i < 5.470276661507296

    1. Initial program 57.1

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

    2. Taylor expanded around 0 26.0

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

    3. Simplified26.0

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

    5. Applied div-inv26.1

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

    6. Applied add-cube-cbrt26.7

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

    7. Applied times-frac10.7

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

    8. Simplified10.7

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

    9. Taylor expanded around 0 37.7

      \[\leadsto 100 \cdot \left(\color{blue}{\left(\frac{1}{3} \cdot {\left({i}^{2}\right)}^{\frac{1}{3}} + \left({\left(\frac{1}{i}\right)}^{\frac{1}{3}} + \frac{1}{12} \cdot {\left({i}^{5}\right)}^{\frac{1}{3}}\right)\right)} \cdot \left(n \cdot \sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)}\right)\right)\]

    10. Simplified10.2

      \[\leadsto 100 \cdot \left(\color{blue}{\left(\sqrt[3]{i \cdot i} \cdot \frac{1}{3} + \left(\frac{1}{12} \cdot \sqrt[3]{{i}^{5}} + \sqrt[3]{\frac{1}{i}}\right)\right)} \cdot \left(n \cdot \sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)}\right)\right)\]

    11. Taylor expanded around 0 37.7

      \[\leadsto 100 \cdot \left(\left(\sqrt[3]{i \cdot i} \cdot \frac{1}{3} + \left(\frac{1}{12} \cdot \sqrt[3]{{i}^{5}} + \sqrt[3]{\frac{1}{i}}\right)\right) \cdot \left(n \cdot \color{blue}{\left({i}^{\frac{1}{3}} + \left(\frac{1}{6} \cdot {\left({i}^{4}\right)}^{\frac{1}{3}} + \frac{1}{36} \cdot {\left({i}^{7}\right)}^{\frac{1}{3}}\right)\right)}\right)\right)\]

    12. Simplified10.2

      \[\leadsto 100 \cdot \left(\left(\sqrt[3]{i \cdot i} \cdot \frac{1}{3} + \left(\frac{1}{12} \cdot \sqrt[3]{{i}^{5}} + \sqrt[3]{\frac{1}{i}}\right)\right) \cdot \left(n \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot \sqrt[3]{{i}^{4}} + \sqrt[3]{{i}^{7}} \cdot \frac{1}{36}\right) + \sqrt[3]{i}\right)}\right)\right)\]

    if 5.470276661507296 < i

    1. Initial program 29.7

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

    2. Initial simplification29.7

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

  4. Final simplification14.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.5462627312609303 \cdot 10^{-05}:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 5.470276661507296:\\ \;\;\;\;\left(\left(n \cdot \left(\sqrt[3]{i} + \left(\sqrt[3]{{i}^{4}} \cdot \frac{1}{6} + \frac{1}{36} \cdot \sqrt[3]{{i}^{7}}\right)\right)\right) \cdot \left(\frac{1}{3} \cdot \sqrt[3]{i \cdot i} + \left(\sqrt[3]{{i}^{5}} \cdot \frac{1}{12} + \sqrt[3]{\frac{1}{i}}\right)\right)\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot n}{i} \cdot {\left(\frac{i}{n} + 1\right)}^{n} - \frac{100 \cdot n}{i}\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.514.88.421.169.8%
herbie shell --seed 2018263 
(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))))