Average Error: 46.9 → 17.8
Time: 2.3m
Precision: 64
Internal Precision: 3456
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;1 + \frac{1}{2} \cdot i \le 0.91045916734681:\\ \;\;\;\;100 \cdot \frac{\frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - 1}{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}}{\frac{i}{n}}\\ \mathbf{if}\;1 + \frac{1}{2} \cdot i \le 2.0500823968808866 \cdot 10^{+26}:\\ \;\;\;\;\frac{\left(n \cdot 100\right) \cdot \left(1 + \frac{1}{2} \cdot i\right)}{1}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{e^{\frac{\log n - \log i}{n}} - 1}{\frac{i}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original46.9
Target46.9
Herbie17.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if (+ 1 (* i 1/2)) < 0.91045916734681

    1. Initial program 27.6

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

    3. Applied flip3--27.6

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

    4. Applied simplify27.6

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

    5. Applied simplify27.6

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

    if 0.91045916734681 < (+ 1 (* i 1/2)) < 2.0500823968808866e+26

    1. Initial program 56.8

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

    2. Taylor expanded around 0 56.7

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

    3. Applied simplify26.6

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

    5. Applied *-un-lft-identity26.6

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

    6. Applied *-un-lft-identity26.6

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

    7. Applied times-frac26.6

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

    8. Applied simplify26.6

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

    9. Applied simplify10.4

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

    if 2.0500823968808866e+26 < (+ 1 (* i 1/2))

    1. Initial program 30.6

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

    2. Taylor expanded around inf 38.3

      \[\leadsto 100 \cdot \frac{\color{blue}{e^{\frac{\log n - \log i}{n}} - 1}}{\frac{i}{n}}\]
  3. Recombined 3 regimes into one program.

  4. Applied simplify17.8

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;1 + \frac{1}{2} \cdot i \le 0.91045916734681:\\ \;\;\;\;100 \cdot \frac{\frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - 1}{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}}{\frac{i}{n}}\\ \mathbf{if}\;1 + \frac{1}{2} \cdot i \le 2.0500823968808866 \cdot 10^{+26}:\\ \;\;\;\;\frac{\left(n \cdot 100\right) \cdot \left(1 + \frac{1}{2} \cdot i\right)}{1}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{e^{\frac{\log n - \log i}{n}} - 1}{\frac{i}{n}}\\ \end{array}}\]

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' 
(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))))