Average Error: 46.9 → 16.6
Time: 1.9m
Precision: 64
Internal Precision: 3200
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.48234294450906656:\\ \;\;\;\;\left(100 \cdot \left(\sqrt[3]{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \cdot \sqrt[3]{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}}\right)\right) \cdot \sqrt[3]{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}}\\ \mathbf{if}\;i \le 6.611841488837997 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt[3]{i \cdot \frac{1}{2} + 1} \cdot \left(\left(1 + \frac{1}{6} \cdot i\right) - \frac{1}{36} \cdot {i}^{2}\right)}{\frac{1}{100}} \cdot \left(\left(\left(1 + \frac{1}{6} \cdot i\right) - \frac{1}{36} \cdot {i}^{2}\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\left(\frac{i}{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right) + \frac{i}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original46.9
Target46.8
Herbie16.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -0.48234294450906656

    1. Initial program 26.3

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

    3. Applied add-cube-cbrt26.5

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

    4. Applied associate-*r*26.5

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

    if -0.48234294450906656 < i < 6.611841488837997e-10

    1. Initial program 57.7

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

    2. Taylor expanded around 0 57.4

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

    3. Applied simplify25.1

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

    5. Applied *-un-lft-identity25.1

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

    6. Applied times-frac25.1

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

    7. Applied add-cube-cbrt25.1

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{i \cdot \frac{1}{2} + 1} \cdot \sqrt[3]{i \cdot \frac{1}{2} + 1}\right) \cdot \sqrt[3]{i \cdot \frac{1}{2} + 1}}}{\frac{1}{100} \cdot \frac{\frac{i}{n}}{i}}\]

    8. Applied times-frac25.0

      \[\leadsto \color{blue}{\frac{\sqrt[3]{i \cdot \frac{1}{2} + 1} \cdot \sqrt[3]{i \cdot \frac{1}{2} + 1}}{\frac{1}{100}} \cdot \frac{\sqrt[3]{i \cdot \frac{1}{2} + 1}}{\frac{\frac{i}{n}}{i}}}\]

    9. Applied simplify9.2

      \[\leadsto \frac{\sqrt[3]{i \cdot \frac{1}{2} + 1} \cdot \sqrt[3]{i \cdot \frac{1}{2} + 1}}{\frac{1}{100}} \cdot \color{blue}{\left(\sqrt[3]{1 + \frac{1}{2} \cdot i} \cdot n\right)}\]

    10. Taylor expanded around 0 9.2

      \[\leadsto \frac{\sqrt[3]{i \cdot \frac{1}{2} + 1} \cdot \color{blue}{\left(\left(1 + \frac{1}{6} \cdot i\right) - \frac{1}{36} \cdot {i}^{2}\right)}}{\frac{1}{100}} \cdot \left(\sqrt[3]{1 + \frac{1}{2} \cdot i} \cdot n\right)\]

    11. Taylor expanded around 0 9.2

      \[\leadsto \frac{\sqrt[3]{i \cdot \frac{1}{2} + 1} \cdot \left(\left(1 + \frac{1}{6} \cdot i\right) - \frac{1}{36} \cdot {i}^{2}\right)}{\frac{1}{100}} \cdot \left(\color{blue}{\left(\left(1 + \frac{1}{6} \cdot i\right) - \frac{1}{36} \cdot {i}^{2}\right)} \cdot n\right)\]

    if 6.611841488837997e-10 < i

    1. Initial program 32.7

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

    3. Applied flip3--32.7

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

    4. Applied associate-/l/32.7

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

    5. Applied simplify32.7

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' 
(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))))