Average Error: 47.3 → 11.2
Time: 9.9m
Precision: 64
Internal Precision: 3392
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -2.9371381809061995 \cdot 10^{-18}:\\ \;\;\;\;100 \cdot \frac{e^{n \cdot \log_* (1 + \frac{i}{n})} - 1}{\frac{i}{n}}\\ \mathbf{if}\;i \le -7.295074826611423 \cdot 10^{-213}:\\ \;\;\;\;\frac{100}{i} \cdot \left(n \cdot (i \cdot \left(\frac{1}{2} \cdot i\right) + i)_*\right)\\ \mathbf{if}\;i \le 7.090344427060975:\\ \;\;\;\;\left(100 \cdot (\frac{1}{2} \cdot i + 1)_*\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{n}{i}\right) \cdot 100\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.3
Target46.6
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 < -2.9371381809061995e-18

    1. Initial program 29.4

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

    3. Applied add-exp-log29.4

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

    4. Applied pow-exp29.4

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

    5. Applied simplify7.1

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

    if -2.9371381809061995e-18 < i < -7.295074826611423e-213

    1. Initial program 55.6

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

    2. Taylor expanded around 0 55.6

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

    3. Applied simplify24.9

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

    5. Applied associate-/r/24.9

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

    6. Applied associate-*l*10.6

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

    if -7.295074826611423e-213 < i < 7.090344427060975

    1. Initial program 58.9

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

    2. Taylor expanded around 0 58.6

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

    3. Applied simplify26.9

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

    5. Applied pow126.9

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

    6. Applied pow126.9

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

    7. Applied pow-prod-down26.9

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

    8. Applied simplify7.0

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

    if 7.090344427060975 < i

    1. Initial program 31.3

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

    3. Applied div-sub31.3

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

    4. Applied simplify33.6

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

  4. Applied simplify11.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;i \le -2.9371381809061995 \cdot 10^{-18}:\\ \;\;\;\;100 \cdot \frac{e^{n \cdot \log_* (1 + \frac{i}{n})} - 1}{\frac{i}{n}}\\ \mathbf{if}\;i \le -7.295074826611423 \cdot 10^{-213}:\\ \;\;\;\;\frac{100}{i} \cdot \left(n \cdot (i \cdot \left(\frac{1}{2} \cdot i\right) + i)_*\right)\\ \mathbf{if}\;i \le 7.090344427060975:\\ \;\;\;\;\left(100 \cdot (\frac{1}{2} \cdot i + 1)_*\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{n}{i}\right) \cdot 100\\ \end{array}}\]

Runtime

Time bar (total: 9.9m)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' +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))))