Average Error: 42.5 → 16.7
Time: 44.6s
Precision: 64
Internal Precision: 128
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -38724.54068622842:\\ \;\;\;\;\frac{100 \cdot e^{i} - 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -3.5166532436816237 \cdot 10^{-43}:\\ \;\;\;\;\frac{n \cdot i}{\frac{1}{100} \cdot i - \left(i \cdot i\right) \cdot \left(\frac{1}{200} - i \cdot \frac{1}{1200}\right)}\\ \mathbf{elif}\;i \le -8.362702829456008 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{\frac{i}{n \cdot \left(100 \cdot e^{i} - 100\right)}}\\ \mathbf{elif}\;i \le 0.00035475927393523827:\\ \;\;\;\;\frac{n \cdot i}{\frac{1}{100} \cdot i - \left(i \cdot i\right) \cdot \left(\frac{1}{200} - i \cdot \frac{1}{1200}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{50}{3} \cdot \frac{{\left(\log i\right)}^{3} \cdot {n}^{4}}{i} + \left(\frac{\log i \cdot {n}^{2}}{i} \cdot 100 + \left(50 \cdot \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i} + \frac{{n}^{3} \cdot {\left(\log n\right)}^{2}}{i} \cdot 50\right)\right)\right) + \frac{50}{3} \cdot \frac{\left({\left(\log n\right)}^{2} \cdot \log i\right) \cdot {n}^{4}}{i}\right) + \frac{\left({\left(\log n\right)}^{2} \cdot \log i\right) \cdot {n}^{4}}{i} \cdot \frac{100}{3}\right) - \left(\left(\frac{50}{3} \cdot \frac{\left({\left(\log i\right)}^{2} \cdot \log n\right) \cdot {n}^{4}}{i} + \left(\left(\left(50 \cdot \frac{{n}^{3} \cdot \left(\log i \cdot \log n\right)}{i} + 100 \cdot \frac{\log n \cdot {n}^{2}}{i}\right) + \frac{100}{3} \cdot \frac{\left({\left(\log i\right)}^{2} \cdot \log n\right) \cdot {n}^{4}}{i}\right) + 50 \cdot \frac{{n}^{3} \cdot \left(\log i \cdot \log n\right)}{i}\right)\right) + \frac{{\left(\log n\right)}^{3} \cdot {n}^{4}}{i} \cdot \frac{50}{3}\right)\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.5
Target42.3
Herbie16.7
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -38724.54068622842

    1. Initial program 27.5

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

    2. Initial simplification28.0

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

    3. Taylor expanded around -inf 13.2

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

    5. Applied associate-/l*12.4

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

    if -38724.54068622842 < i < -3.5166532436816237e-43 or -8.362702829456008e-70 < i < 0.00035475927393523827

    1. Initial program 50.1

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

    2. Initial simplification50.1

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

    3. Taylor expanded around -inf 49.7

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

    4. Taylor expanded around 0 16.1

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

    5. Simplified16.1

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

    7. Applied associate-/l*16.1

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

    8. Taylor expanded around 0 16.0

      \[\leadsto \frac{i \cdot n}{\color{blue}{\left(\frac{1}{100} \cdot i + \frac{1}{1200} \cdot {i}^{3}\right) - \frac{1}{200} \cdot {i}^{2}}}\]

    9. Simplified16.0

      \[\leadsto \frac{i \cdot n}{\color{blue}{\frac{1}{100} \cdot i - \left(i \cdot i\right) \cdot \left(\frac{1}{200} - \frac{1}{1200} \cdot i\right)}}\]

    if -3.5166532436816237e-43 < i < -8.362702829456008e-70

    1. Initial program 50.9

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

    2. Initial simplification50.9

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

    3. Taylor expanded around -inf 51.2

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

    5. Applied clear-num51.2

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

    if 0.00035475927393523827 < i

    1. Initial program 30.9

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

    2. Initial simplification30.9

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

    3. Taylor expanded around 0 21.1

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

  4. Final simplification16.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -38724.54068622842:\\ \;\;\;\;\frac{100 \cdot e^{i} - 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -3.5166532436816237 \cdot 10^{-43}:\\ \;\;\;\;\frac{n \cdot i}{\frac{1}{100} \cdot i - \left(i \cdot i\right) \cdot \left(\frac{1}{200} - i \cdot \frac{1}{1200}\right)}\\ \mathbf{elif}\;i \le -8.362702829456008 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{\frac{i}{n \cdot \left(100 \cdot e^{i} - 100\right)}}\\ \mathbf{elif}\;i \le 0.00035475927393523827:\\ \;\;\;\;\frac{n \cdot i}{\frac{1}{100} \cdot i - \left(i \cdot i\right) \cdot \left(\frac{1}{200} - i \cdot \frac{1}{1200}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{50}{3} \cdot \frac{{\left(\log i\right)}^{3} \cdot {n}^{4}}{i} + \left(\frac{\log i \cdot {n}^{2}}{i} \cdot 100 + \left(50 \cdot \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i} + \frac{{n}^{3} \cdot {\left(\log n\right)}^{2}}{i} \cdot 50\right)\right)\right) + \frac{50}{3} \cdot \frac{\left({\left(\log n\right)}^{2} \cdot \log i\right) \cdot {n}^{4}}{i}\right) + \frac{\left({\left(\log n\right)}^{2} \cdot \log i\right) \cdot {n}^{4}}{i} \cdot \frac{100}{3}\right) - \left(\left(\frac{50}{3} \cdot \frac{\left({\left(\log i\right)}^{2} \cdot \log n\right) \cdot {n}^{4}}{i} + \left(\left(\left(50 \cdot \frac{{n}^{3} \cdot \left(\log i \cdot \log n\right)}{i} + 100 \cdot \frac{\log n \cdot {n}^{2}}{i}\right) + \frac{100}{3} \cdot \frac{\left({\left(\log i\right)}^{2} \cdot \log n\right) \cdot {n}^{4}}{i}\right) + 50 \cdot \frac{{n}^{3} \cdot \left(\log i \cdot \log n\right)}{i}\right)\right) + \frac{{\left(\log n\right)}^{3} \cdot {n}^{4}}{i} \cdot \frac{50}{3}\right)\\ \end{array}\]

Runtime

Time bar (total: 44.6s)Debug logProfile

herbie shell --seed 2018278 
(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))))