Average Error: 47.4 → 17.0
Time: 1.9m
Precision: 64
Internal Precision: 3200
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(1 + \frac{1}{2} \cdot i\right) \le 93.98030941154852:\\ \;\;\;\;\frac{i \cdot \frac{1}{2} + 1}{\frac{1}{n \cdot 100}}\\ \mathbf{if}\;\log \left(1 + \frac{1}{2} \cdot i\right) \le 619.7267873526765:\\ \;\;\;\;\frac{i \cdot \frac{1}{2} + 1}{\log \left(e^{\frac{\frac{1}{n}}{100}}\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.4
Target46.8
Herbie17.0
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if (log (+ 1 (* 1/2 i))) < 93.98030941154852

    1. Initial program 56.8

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

    2. Taylor expanded around 0 56.9

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

    3. Applied simplify27.0

      \[\leadsto \color{blue}{\frac{i \cdot \frac{1}{2} + 1}{\frac{\frac{i}{n}}{100 \cdot i}}}\]
    4. Using strategy rm

    5. Applied clear-num27.0

      \[\leadsto \frac{i \cdot \frac{1}{2} + 1}{\color{blue}{\frac{1}{\frac{100 \cdot i}{\frac{i}{n}}}}}\]

    6. Applied simplify10.6

      \[\leadsto \frac{i \cdot \frac{1}{2} + 1}{\frac{1}{\color{blue}{n \cdot 100}}}\]

    if 93.98030941154852 < (log (+ 1 (* 1/2 i))) < 619.7267873526765

    1. Initial program 30.7

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

    2. Taylor expanded around 0 56.2

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

    3. Applied simplify41.2

      \[\leadsto \color{blue}{\frac{i \cdot \frac{1}{2} + 1}{\frac{\frac{i}{n}}{100 \cdot i}}}\]
    4. Using strategy rm

    5. Applied add-log-exp29.5

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

    6. Applied simplify29.5

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

    if 619.7267873526765 < (log (+ 1 (* 1/2 i)))

    1. Initial program 29.1

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(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))))