Average Error: 42.4 → 16.2
Time: 34.1s
Precision: 64
Internal Precision: 128
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.00017710199551183856:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot e^{i} - 100\right)}{i}\\ \mathbf{elif}\;i \le 0.0007767270897403532:\\ \;\;\;\;\left(n \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right) + 100 \cdot n\\ \mathbf{elif}\;i \le 2.4023589389181742 \cdot 10^{+246} \lor \neg \left(i \le 9.79079137447263 \cdot 10^{+280}\right):\\ \;\;\;\;\left(\left(\frac{50}{3} \cdot \frac{\left({\left(\log n\right)}^{2} \cdot \log i\right) \cdot {n}^{4}}{i} + \left(\left(\left(\frac{{n}^{3} \cdot {\left(\log i\right)}^{2}}{i} \cdot 50 + 50 \cdot \frac{{\left(\log n\right)}^{2} \cdot {n}^{3}}{i}\right) + 100 \cdot \frac{\log i \cdot {n}^{2}}{i}\right) + \frac{50}{3} \cdot \frac{{\left(\log i\right)}^{3} \cdot {n}^{4}}{i}\right)\right) + \frac{\left({\left(\log n\right)}^{2} \cdot \log i\right) \cdot {n}^{4}}{i} \cdot \frac{100}{3}\right) - \left(\frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} \cdot \frac{50}{3} + \left(\frac{50}{3} \cdot \frac{\left(\log n \cdot {\left(\log i\right)}^{2}\right) \cdot {n}^{4}}{i} + \left(\frac{\left(\log i \cdot \log n\right) \cdot {n}^{3}}{i} \cdot 50 + \left(\left(\frac{\log n \cdot {n}^{2}}{i} \cdot 100 + \frac{\left(\log i \cdot \log n\right) \cdot {n}^{3}}{i} \cdot 50\right) + \frac{100}{3} \cdot \frac{\left(\log n \cdot {\left(\log i\right)}^{2}\right) \cdot {n}^{4}}{i}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{n} + 1\right)}^{n} \cdot \frac{100 \cdot n}{i} - \frac{n}{\frac{i}{100}}\\ \end{array}\]

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
Target42.4
Herbie16.2
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -0.00017710199551183856

    1. Initial program 27.8

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

    2. Initial simplification28.3

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

    3. Taylor expanded around -inf 12.0

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

    4. Taylor expanded around -inf 12.0

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

    if -0.00017710199551183856 < i < 0.0007767270897403532

    1. Initial program 50.2

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

    2. Initial simplification50.2

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

    3. Taylor expanded around -inf 50.2

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

    4. Taylor expanded around 0 16.8

      \[\leadsto \color{blue}{\frac{50}{3} \cdot \left({i}^{2} \cdot n\right) + \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)}\]

    5. Simplified16.8

      \[\leadsto \color{blue}{\left(i \cdot n\right) \cdot \left(50 + \frac{50}{3} \cdot i\right) + 100 \cdot n}\]

    if 0.0007767270897403532 < i < 2.4023589389181742e+246 or 9.79079137447263e+280 < i

    1. Initial program 32.9

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

    2. Initial simplification32.9

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

    3. Taylor expanded around 0 19.6

      \[\leadsto \color{blue}{\left(\frac{100}{3} \cdot \frac{{n}^{4} \cdot \left({\left(\log n\right)}^{2} \cdot \log i\right)}{i} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot \left(\log i \cdot {\left(\log n\right)}^{2}\right)}{i} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log i\right)}^{3}}{i} + \left(100 \cdot \frac{{n}^{2} \cdot \log i}{i} + \left(50 \cdot \frac{{n}^{3} \cdot {\left(\log i\right)}^{2}}{i} + 50 \cdot \frac{{n}^{3} \cdot {\left(\log n\right)}^{2}}{i}\right)\right)\right)\right)\right) - \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot \left(\log n \cdot {\left(\log i\right)}^{2}\right)}{i} + \left(50 \cdot \frac{{n}^{3} \cdot \left(\log n \cdot \log i\right)}{i} + \left(\frac{100}{3} \cdot \frac{{n}^{4} \cdot \left({\left(\log i\right)}^{2} \cdot \log n\right)}{i} + \left(100 \cdot \frac{{n}^{2} \cdot \log n}{i} + 50 \cdot \frac{{n}^{3} \cdot \left(\log i \cdot \log n\right)}{i}\right)\right)\right)\right)\right)}\]

    if 2.4023589389181742e+246 < i < 9.79079137447263e+280

    1. Initial program 26.6

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

    2. Initial simplification26.6

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

    4. Applied associate-/l*26.5

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

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.00017710199551183856:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot e^{i} - 100\right)}{i}\\ \mathbf{elif}\;i \le 0.0007767270897403532:\\ \;\;\;\;\left(n \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right) + 100 \cdot n\\ \mathbf{elif}\;i \le 2.4023589389181742 \cdot 10^{+246} \lor \neg \left(i \le 9.79079137447263 \cdot 10^{+280}\right):\\ \;\;\;\;\left(\left(\frac{50}{3} \cdot \frac{\left({\left(\log n\right)}^{2} \cdot \log i\right) \cdot {n}^{4}}{i} + \left(\left(\left(\frac{{n}^{3} \cdot {\left(\log i\right)}^{2}}{i} \cdot 50 + 50 \cdot \frac{{\left(\log n\right)}^{2} \cdot {n}^{3}}{i}\right) + 100 \cdot \frac{\log i \cdot {n}^{2}}{i}\right) + \frac{50}{3} \cdot \frac{{\left(\log i\right)}^{3} \cdot {n}^{4}}{i}\right)\right) + \frac{\left({\left(\log n\right)}^{2} \cdot \log i\right) \cdot {n}^{4}}{i} \cdot \frac{100}{3}\right) - \left(\frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} \cdot \frac{50}{3} + \left(\frac{50}{3} \cdot \frac{\left(\log n \cdot {\left(\log i\right)}^{2}\right) \cdot {n}^{4}}{i} + \left(\frac{\left(\log i \cdot \log n\right) \cdot {n}^{3}}{i} \cdot 50 + \left(\left(\frac{\log n \cdot {n}^{2}}{i} \cdot 100 + \frac{\left(\log i \cdot \log n\right) \cdot {n}^{3}}{i} \cdot 50\right) + \frac{100}{3} \cdot \frac{\left(\log n \cdot {\left(\log i\right)}^{2}\right) \cdot {n}^{4}}{i}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{n} + 1\right)}^{n} \cdot \frac{100 \cdot n}{i} - \frac{n}{\frac{i}{100}}\\ \end{array}\]

Runtime

Time bar (total: 34.1s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes33.616.24.129.559.1%
herbie shell --seed 2018353 
(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))))