Average Error: 42.7 → 31.1
Time: 1.1m
Precision: 64
Internal Precision: 128
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -2.649631116597885 \cdot 10^{+140}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100} \cdot \left(\sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100} \cdot \sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100}\right) + -100}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -3.431241481757734 \cdot 10^{+58}:\\ \;\;\;\;100 \cdot \frac{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -3.596398395624021 \cdot 10^{+37}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100} \cdot \left(\sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100} \cdot \sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100}\right) + -100}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -3576926.32044652:\\ \;\;\;\;100 \cdot \frac{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 3.5640030898971636 \cdot 10^{-65}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \le 2.4071012156782965 \cdot 10^{+187}:\\ \;\;\;\;100 \cdot \frac{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{i}{n}\right)}^{n}}{\frac{\frac{i}{100}}{n}} - \frac{100 \cdot n}{i}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.7
Target41.6
Herbie31.1
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if n < -2.649631116597885e+140 or -3.431241481757734e+58 < n < -3.596398395624021e+37

    1. Initial program 50.0

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

    3. Applied associate-*r/50.0

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

    5. Applied sub-neg50.0

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

    6. Applied distribute-lft-in50.0

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

    7. Simplified50.0

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

    9. Applied add-cube-cbrt47.9

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

    if -2.649631116597885e+140 < n < -3.431241481757734e+58 or -3.596398395624021e+37 < n < -3576926.32044652 or 3.5640030898971636e-65 < n < 2.4071012156782965e+187

    1. Initial program 51.0

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

    2. Taylor expanded around 0 29.1

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

    3. Simplified29.1

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

    if -3576926.32044652 < n < 3.5640030898971636e-65

    1. Initial program 28.6

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

    2. Taylor expanded around 0 21.9

      \[\leadsto \color{blue}{0}\]

    if 2.4071012156782965e+187 < n

    1. Initial program 59.7

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

    3. Applied associate-*r/59.7

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

    5. Applied sub-neg59.7

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

    6. Applied distribute-lft-in59.7

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

    7. Simplified59.7

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

    8. Taylor expanded around inf 61.4

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

    9. Simplified43.1

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

  4. Final simplification31.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.649631116597885 \cdot 10^{+140}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100} \cdot \left(\sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100} \cdot \sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100}\right) + -100}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -3.431241481757734 \cdot 10^{+58}:\\ \;\;\;\;100 \cdot \frac{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -3.596398395624021 \cdot 10^{+37}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100} \cdot \left(\sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100} \cdot \sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100}\right) + -100}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -3576926.32044652:\\ \;\;\;\;100 \cdot \frac{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 3.5640030898971636 \cdot 10^{-65}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \le 2.4071012156782965 \cdot 10^{+187}:\\ \;\;\;\;100 \cdot \frac{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{i}{n}\right)}^{n}}{\frac{\frac{i}{100}}{n}} - \frac{100 \cdot n}{i}\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 
(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))))