Average Error: 47.5 → 11.2
Time: 2.0m
Precision: 64
Internal Precision: 3200
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -2.9366393669284026 \cdot 10^{-24}:\\ \;\;\;\;100 \cdot \frac{e^{n \cdot \log_* (1 + \frac{i}{n})} - 1}{\frac{i}{n}}\\ \mathbf{if}\;i \le 8.339067327181505:\\ \;\;\;\;n \cdot \left(100 \cdot (\left((\frac{1}{6} \cdot i + \frac{1}{2})_*\right) \cdot i + 1)_*\right)\\ \mathbf{if}\;i \le 1.5957855862421325 \cdot 10^{+150}:\\ \;\;\;\;\frac{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot n}{i} \cdot \left(\frac{{i}^{n}}{{n}^{n}} - 1\right)\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.5
Target47.2
Herbie11.2
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -2.9366393669284026e-24

    1. Initial program 30.4

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

    3. Applied add-exp-log30.5

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

    4. Applied pow-exp30.5

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

    5. Applied simplify7.9

      \[\leadsto 100 \cdot \frac{e^{\color{blue}{n \cdot \log_* (1 + \frac{i}{n})}} - 1}{\frac{i}{n}}\]

    if -2.9366393669284026e-24 < i < 8.339067327181505

    1. Initial program 57.9

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

    2. Taylor expanded around 0 24.9

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

    3. Applied simplify26.4

      \[\leadsto \color{blue}{\left(n \cdot \frac{100}{i}\right) \cdot (\left(i \cdot i\right) \cdot \left((i \cdot \frac{1}{6} + \frac{1}{2})_*\right) + i)_*}\]
    4. Using strategy rm

    5. Applied associate-*l*9.0

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

    6. Applied simplify8.7

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

    if 8.339067327181505 < i < 1.5957855862421325e+150

    1. Initial program 30.0

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

    3. Applied associate-*r/30.0

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

    if 1.5957855862421325e+150 < i

    1. Initial program 31.0

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

    2. Taylor expanded around inf 28.5

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

    3. Applied simplify28.5

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

  4. Applied simplify11.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;i \le -2.9366393669284026 \cdot 10^{-24}:\\ \;\;\;\;100 \cdot \frac{e^{n \cdot \log_* (1 + \frac{i}{n})} - 1}{\frac{i}{n}}\\ \mathbf{if}\;i \le 8.339067327181505:\\ \;\;\;\;n \cdot \left(100 \cdot (\left((\frac{1}{6} \cdot i + \frac{1}{2})_*\right) \cdot i + 1)_*\right)\\ \mathbf{if}\;i \le 1.5957855862421325 \cdot 10^{+150}:\\ \;\;\;\;\frac{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot n}{i} \cdot \left(\frac{{i}^{n}}{{n}^{n}} - 1\right)\\ \end{array}}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' +o rules:numerics
(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))))