Average Error: 42.9 → 30.4
Time: 37.8s
Precision: 64
Internal Precision: 128
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -9.591285212666758 \cdot 10^{+30}:\\ \;\;\;\;\left(n \cdot {\left(\frac{i}{n} + 1\right)}^{n} - n\right) \cdot \left(100 \cdot \frac{1}{i}\right)\\ \mathbf{elif}\;n \le -2.0213172308126555:\\ \;\;\;\;100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 3.15974925982966 \cdot 10^{-310}:\\ \;\;\;\;\left(n \cdot {\left(\frac{i}{n} + 1\right)}^{n} - n\right) \cdot \left(100 \cdot \frac{1}{i}\right)\\ \mathbf{elif}\;n \le 7.604186655321221 \cdot 10^{-112}:\\ \;\;\;\;\frac{100}{\frac{i}{n}} \cdot \left(\left(\left(\frac{1}{2} \cdot \left(\left({n}^{3} \cdot \log i\right) \cdot \left(\log n \cdot \log n\right)\right) + \left(\left(\log i \cdot \log i\right) \cdot \left(\frac{1}{2} \cdot \left(n \cdot n\right)\right) + n \cdot \log i\right)\right) + \left(\left(\frac{1}{2} \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \log n\right) + {\left(\log i\right)}^{3} \cdot \left(\left(n \cdot \frac{1}{6}\right) \cdot \left(n \cdot n\right)\right)\right)\right) - \left(\left(\left(\left(n \cdot \frac{1}{6}\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \left(\log i \cdot \log i\right)\right) + \left({\left(\log n\right)}^{3} \cdot \left(\left(n \cdot \frac{1}{6}\right) \cdot \left(n \cdot n\right)\right) + \log n \cdot n\right)\right) + \left(\left(\left(n \cdot n\right) \cdot \log i\right) \cdot \log n + \left(\left(n \cdot n\right) \cdot \left(n \cdot \frac{1}{3}\right)\right) \cdot \left(\log n \cdot \left(\log i \cdot \log i\right)\right)\right)\right)\right)\\ \mathbf{elif}\;n \le 2.668078310905783 \cdot 10^{-56}:\\ \;\;\;\;100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 0.016254391867187298:\\ \;\;\;\;\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 n\right)}^{2}}{i} \cdot 50 + 50 \cdot \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i}\right) + \frac{\log i \cdot {n}^{2}}{i} \cdot 100\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(\left(\frac{50}{3} \cdot \frac{\left({\left(\log i\right)}^{2} \cdot \log n\right) \cdot {n}^{4}}{i} + \left(50 \cdot \frac{{n}^{3} \cdot \left(\log n \cdot \log i\right)}{i} + \left(\frac{100}{3} \cdot \frac{\left({\left(\log i\right)}^{2} \cdot \log n\right) \cdot {n}^{4}}{i} + \left(\frac{{n}^{2} \cdot \log n}{i} \cdot 100 + 50 \cdot \frac{{n}^{3} \cdot \left(\log n \cdot \log i\right)}{i}\right)\right)\right)\right) + \frac{{\left(\log n\right)}^{3} \cdot {n}^{4}}{i} \cdot \frac{50}{3}\right)\\ \mathbf{elif}\;n \le 1.3508939593683805 \cdot 10^{+246}:\\ \;\;\;\;100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot n\right) \cdot \frac{100}{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

Original42.9
Target42.6
Herbie30.4
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 5 regimes

  2. if n < -9.591285212666758e+30 or -2.0213172308126555 < n < 3.15974925982966e-310

    1. Initial program 32.1

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

    3. Applied div-inv32.1

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

    4. Applied *-un-lft-identity32.1

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

    5. Applied times-frac32.2

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

    6. Applied associate-*r*32.3

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

    7. Simplified32.3

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

    if -9.591285212666758e+30 < n < -2.0213172308126555 or 7.604186655321221e-112 < n < 2.668078310905783e-56 or 0.016254391867187298 < n < 1.3508939593683805e+246

    1. Initial program 56.4

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

    2. Taylor expanded around 0 29.6

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

    3. Simplified29.6

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

    if 3.15974925982966e-310 < n < 7.604186655321221e-112

    1. Initial program 45.4

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

    2. Taylor expanded around inf 26.0

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

    3. Simplified45.6

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

    4. Taylor expanded around 0 17.2

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

    5. Simplified17.2

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

    if 2.668078310905783e-56 < n < 0.016254391867187298

    1. Initial program 55.6

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

    2. Taylor expanded around inf 53.8

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

    3. Simplified55.8

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

    4. Taylor expanded around 0 39.5

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

    if 1.3508939593683805e+246 < n

    1. Initial program 58.9

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

    2. Taylor expanded around inf 61.1

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

    3. Simplified41.6

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

    5. Applied associate-/r/41.6

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

    6. Applied associate-*l*41.6

      \[\leadsto \color{blue}{\frac{100}{i} \cdot \left(n \cdot \left({\left(\frac{i}{n}\right)}^{n} + -1\right)\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification30.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -9.591285212666758 \cdot 10^{+30}:\\ \;\;\;\;\left(n \cdot {\left(\frac{i}{n} + 1\right)}^{n} - n\right) \cdot \left(100 \cdot \frac{1}{i}\right)\\ \mathbf{elif}\;n \le -2.0213172308126555:\\ \;\;\;\;100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 3.15974925982966 \cdot 10^{-310}:\\ \;\;\;\;\left(n \cdot {\left(\frac{i}{n} + 1\right)}^{n} - n\right) \cdot \left(100 \cdot \frac{1}{i}\right)\\ \mathbf{elif}\;n \le 7.604186655321221 \cdot 10^{-112}:\\ \;\;\;\;\frac{100}{\frac{i}{n}} \cdot \left(\left(\left(\frac{1}{2} \cdot \left(\left({n}^{3} \cdot \log i\right) \cdot \left(\log n \cdot \log n\right)\right) + \left(\left(\log i \cdot \log i\right) \cdot \left(\frac{1}{2} \cdot \left(n \cdot n\right)\right) + n \cdot \log i\right)\right) + \left(\left(\frac{1}{2} \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \log n\right) + {\left(\log i\right)}^{3} \cdot \left(\left(n \cdot \frac{1}{6}\right) \cdot \left(n \cdot n\right)\right)\right)\right) - \left(\left(\left(\left(n \cdot \frac{1}{6}\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \left(\log i \cdot \log i\right)\right) + \left({\left(\log n\right)}^{3} \cdot \left(\left(n \cdot \frac{1}{6}\right) \cdot \left(n \cdot n\right)\right) + \log n \cdot n\right)\right) + \left(\left(\left(n \cdot n\right) \cdot \log i\right) \cdot \log n + \left(\left(n \cdot n\right) \cdot \left(n \cdot \frac{1}{3}\right)\right) \cdot \left(\log n \cdot \left(\log i \cdot \log i\right)\right)\right)\right)\right)\\ \mathbf{elif}\;n \le 2.668078310905783 \cdot 10^{-56}:\\ \;\;\;\;100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 0.016254391867187298:\\ \;\;\;\;\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 n\right)}^{2}}{i} \cdot 50 + 50 \cdot \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i}\right) + \frac{\log i \cdot {n}^{2}}{i} \cdot 100\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(\left(\frac{50}{3} \cdot \frac{\left({\left(\log i\right)}^{2} \cdot \log n\right) \cdot {n}^{4}}{i} + \left(50 \cdot \frac{{n}^{3} \cdot \left(\log n \cdot \log i\right)}{i} + \left(\frac{100}{3} \cdot \frac{\left({\left(\log i\right)}^{2} \cdot \log n\right) \cdot {n}^{4}}{i} + \left(\frac{{n}^{2} \cdot \log n}{i} \cdot 100 + 50 \cdot \frac{{n}^{3} \cdot \left(\log n \cdot \log i\right)}{i}\right)\right)\right)\right) + \frac{{\left(\log n\right)}^{3} \cdot {n}^{4}}{i} \cdot \frac{50}{3}\right)\\ \mathbf{elif}\;n \le 1.3508939593683805 \cdot 10^{+246}:\\ \;\;\;\;100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot n\right) \cdot \frac{100}{i}\\ \end{array}\]

Reproduce

herbie shell --seed 2019021 
(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))))