Average Error: 47.2 → 14.1
Time: 30.3s
Precision: 64
Internal Precision: 3136
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 5.077754959869468 \cdot 10^{+188}:\\ \;\;\;\;100 \cdot \frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot (\left({\left(\frac{i}{n}\right)}^{n}\right) \cdot \left(\frac{n}{i}\right) + \left(\frac{-n}{i}\right))_*\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.2
Target46.7
Herbie14.1
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 2 regimes

  2. if i < 5.077754959869468e+188

    1. Initial program 48.2

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

    3. Applied add-exp-log48.2

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

    4. Applied expm1-def48.2

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

    5. Simplified12.8

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

    7. Applied *-commutative12.8

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

    if 5.077754959869468e+188 < i

    1. Initial program 33.4

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

    2. Taylor expanded around inf 28.6

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

    3. Simplified33.4

      \[\leadsto 100 \cdot \color{blue}{(\left({\left(\frac{i}{n}\right)}^{n}\right) \cdot \left(\frac{n}{i}\right) + \left(\frac{-n}{i}\right))_*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification14.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 5.077754959869468 \cdot 10^{+188}:\\ \;\;\;\;100 \cdot \frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot (\left({\left(\frac{i}{n}\right)}^{n}\right) \cdot \left(\frac{n}{i}\right) + \left(\frac{-n}{i}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 30.3s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes15.614.19.66.024.8%
herbie shell --seed 2018286 +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))))