Average Error: 47.2 → 12.7
Time: 37.6s
Precision: 64
Internal Precision: 3136
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -2.4968555287033732 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{\left(e^{i} \cdot 10000\right) \cdot \left(n \cdot e^{i}\right) + n \cdot -10000}{100 + 100 \cdot e^{i}}}{i}\\ \mathbf{elif}\;i \le 0.3817627801330132:\\ \;\;\;\;100 \cdot n + \left(n \cdot i\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)\\ \mathbf{elif}\;i \le 1.173843906973232 \cdot 10^{+214}:\\ \;\;\;\;\left(\left(\left(\left(\left(50 \cdot \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i} + 50 \cdot \frac{{n}^{3} \cdot {\left(\log n\right)}^{2}}{i}\right) + \frac{\log i \cdot {n}^{2}}{i} \cdot 100\right) + \frac{{\left(\log i\right)}^{3} \cdot {n}^{4}}{i} \cdot \frac{50}{3}\right) + \frac{{n}^{4} \cdot \left(\log i \cdot {\left(\log n\right)}^{2}\right)}{i} \cdot \frac{50}{3}\right) + \frac{100}{3} \cdot \frac{{n}^{4} \cdot \left(\log i \cdot {\left(\log n\right)}^{2}\right)}{i}\right) - \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(\left(\left(\frac{\left(\log n \cdot {\left(\log i\right)}^{2}\right) \cdot {n}^{4}}{i} \cdot \frac{100}{3} + \left(50 \cdot \frac{{n}^{3} \cdot \left(\log i \cdot \log n\right)}{i} + 100 \cdot \frac{{n}^{2} \cdot \log n}{i}\right)\right) + 50 \cdot \frac{{n}^{3} \cdot \left(\log i \cdot \log n\right)}{i}\right) + \frac{50}{3} \cdot \frac{\left(\log n \cdot {\left(\log i\right)}^{2}\right) \cdot {n}^{4}}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot n}{i} \cdot {\left(\frac{i}{n} + 1\right)}^{n} - 100 \cdot \frac{n}{i}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.2
Target46.7
Herbie12.7
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -2.4968555287033732e-30

    1. Initial program 30.5

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

    2. Initial simplification30.8

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

    3. Taylor expanded around -inf 16.8

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

    5. Applied flip--16.8

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

    6. Applied associate-*l/17.1

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

    7. Simplified17.1

      \[\leadsto \frac{\frac{\color{blue}{n \cdot -10000 + \left(n \cdot e^{i}\right) \cdot \left(e^{i} \cdot 10000\right)}}{100 \cdot e^{i} + 100}}{i}\]

    if -2.4968555287033732e-30 < i < 0.3817627801330132

    1. Initial program 57.7

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

    2. Initial simplification57.7

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

    3. Taylor expanded around -inf 57.6

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

    4. Taylor expanded around 0 8.7

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

    5. Simplified8.7

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

    if 0.3817627801330132 < i < 1.173843906973232e+214

    1. Initial program 31.3

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

    2. Initial simplification31.3

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

    3. Taylor expanded around 0 14.5

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

    if 1.173843906973232e+214 < i

    1. Initial program 33.7

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

    2. Initial simplification33.7

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

    3. Taylor expanded around inf 34.0

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

  4. Final simplification12.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.4968555287033732 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{\left(e^{i} \cdot 10000\right) \cdot \left(n \cdot e^{i}\right) + n \cdot -10000}{100 + 100 \cdot e^{i}}}{i}\\ \mathbf{elif}\;i \le 0.3817627801330132:\\ \;\;\;\;100 \cdot n + \left(n \cdot i\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)\\ \mathbf{elif}\;i \le 1.173843906973232 \cdot 10^{+214}:\\ \;\;\;\;\left(\left(\left(\left(\left(50 \cdot \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i} + 50 \cdot \frac{{n}^{3} \cdot {\left(\log n\right)}^{2}}{i}\right) + \frac{\log i \cdot {n}^{2}}{i} \cdot 100\right) + \frac{{\left(\log i\right)}^{3} \cdot {n}^{4}}{i} \cdot \frac{50}{3}\right) + \frac{{n}^{4} \cdot \left(\log i \cdot {\left(\log n\right)}^{2}\right)}{i} \cdot \frac{50}{3}\right) + \frac{100}{3} \cdot \frac{{n}^{4} \cdot \left(\log i \cdot {\left(\log n\right)}^{2}\right)}{i}\right) - \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(\left(\left(\frac{\left(\log n \cdot {\left(\log i\right)}^{2}\right) \cdot {n}^{4}}{i} \cdot \frac{100}{3} + \left(50 \cdot \frac{{n}^{3} \cdot \left(\log i \cdot \log n\right)}{i} + 100 \cdot \frac{{n}^{2} \cdot \log n}{i}\right)\right) + 50 \cdot \frac{{n}^{3} \cdot \left(\log i \cdot \log n\right)}{i}\right) + \frac{50}{3} \cdot \frac{\left(\log n \cdot {\left(\log i\right)}^{2}\right) \cdot {n}^{4}}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot n}{i} \cdot {\left(\frac{i}{n} + 1\right)}^{n} - 100 \cdot \frac{n}{i}\\ \end{array}\]

Runtime

Time bar (total: 37.6s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes28.712.74.224.465.4%
herbie shell --seed 2018286 
(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))))