Average Error: 47.4 → 18.2
Time: 2.0m
Precision: 64
Internal Precision: 3200
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(i \cdot 50 + 100\right) \le 199.3041047917235:\\ \;\;\;\;\left(i \cdot 50 + 100\right) \cdot n\\ \mathbf{if}\;\log \left(i \cdot 50 + 100\right) \le 657.442695206473:\\ \;\;\;\;\frac{i}{\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}} \cdot \frac{i \cdot 50 + 100}{\sqrt[3]{\frac{i}{n}}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.4
Target46.8
Herbie18.2
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if (log (+ 100 (* 50 i))) < 199.3041047917235

    1. Initial program 56.2

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

    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 simplify27.5

      \[\leadsto \color{blue}{\frac{i \cdot \left(i \cdot 50 + 100\right)}{\frac{i}{n}}}\]
    4. Using strategy rm

    5. Applied div-inv27.6

      \[\leadsto \frac{i \cdot \left(i \cdot 50 + 100\right)}{\color{blue}{i \cdot \frac{1}{n}}}\]

    6. Applied times-frac11.8

      \[\leadsto \color{blue}{\frac{i}{i} \cdot \frac{i \cdot 50 + 100}{\frac{1}{n}}}\]

    7. Applied simplify11.8

      \[\leadsto \color{blue}{1} \cdot \frac{i \cdot 50 + 100}{\frac{1}{n}}\]

    8. Applied simplify11.6

      \[\leadsto 1 \cdot \color{blue}{\left(\left(100 + 50 \cdot i\right) \cdot n\right)}\]

    if 199.3041047917235 < (log (+ 100 (* 50 i))) < 657.442695206473

    1. Initial program 31.0

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

    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 simplify57.0

      \[\leadsto \color{blue}{\frac{i \cdot \left(i \cdot 50 + 100\right)}{\frac{i}{n}}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt57.0

      \[\leadsto \frac{i \cdot \left(i \cdot 50 + 100\right)}{\color{blue}{\left(\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}\right) \cdot \sqrt[3]{\frac{i}{n}}}}\]

    6. Applied times-frac38.4

      \[\leadsto \color{blue}{\frac{i}{\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}} \cdot \frac{i \cdot 50 + 100}{\sqrt[3]{\frac{i}{n}}}}\]

    if 657.442695206473 < (log (+ 100 (* 50 i)))

    1. Initial program 29.0

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

  4. Applied simplify18.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\log \left(i \cdot 50 + 100\right) \le 199.3041047917235:\\ \;\;\;\;\left(i \cdot 50 + 100\right) \cdot n\\ \mathbf{if}\;\log \left(i \cdot 50 + 100\right) \le 657.442695206473:\\ \;\;\;\;\frac{i}{\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}} \cdot \frac{i \cdot 50 + 100}{\sqrt[3]{\frac{i}{n}}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}\\ \end{array}}\]

Runtime

Time bar (total: 2.0m)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))))