Average Error: 47.6 → 17.9
Time: 1.2m
Precision: 64
Internal Precision: 3200
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.098474083814846:\\ \;\;\;\;\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\left({\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)\right) \cdot \frac{i}{n}} \cdot 100\\ \mathbf{if}\;i \le 6.511266593892657 \cdot 10^{-22}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{2} \cdot i\right) \cdot \left(100 \cdot n\right)}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\left({\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)\right) \cdot \frac{i}{n}} \cdot 100\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.6
Target46.5
Herbie17.9
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 2 regimes

  2. if i < -1.098474083814846 or 6.511266593892657e-22 < i

    1. Initial program 31.6

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

    3. Applied flip3--31.6

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

    4. Applied associate-/l/31.6

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

    5. Applied simplify31.6

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

    if -1.098474083814846 < i < 6.511266593892657e-22

    1. Initial program 57.5

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

    2. Taylor expanded around 0 57.3

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

    3. Applied simplify25.8

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

    5. Applied *-un-lft-identity25.8

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

    6. Applied *-un-lft-identity25.8

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

    7. Applied times-frac25.8

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

    8. Applied simplify25.8

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

    9. Applied simplify9.3

      \[\leadsto 1 \cdot \color{blue}{\frac{\left(n \cdot 100\right) \cdot \left(1 + \frac{1}{2} \cdot i\right)}{1}}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify17.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;i \le -1.098474083814846:\\ \;\;\;\;\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\left({\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)\right) \cdot \frac{i}{n}} \cdot 100\\ \mathbf{if}\;i \le 6.511266593892657 \cdot 10^{-22}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{2} \cdot i\right) \cdot \left(100 \cdot n\right)}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\left({\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)\right) \cdot \frac{i}{n}} \cdot 100\\ \end{array}}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(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))))