Average Error: 47.6 → 16.9
Time: 2.3m
Precision: 64
Internal Precision: 3392
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.6411582150710616 \lor \neg \left(i \le 2.427056218200348\right):\\ \;\;\;\;\frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \sqrt[3]{1 + \frac{1}{2} \cdot i}\right) \cdot \left(\left(100 + \frac{100}{3} \cdot i\right) - \frac{25}{9} \cdot {i}^{2}\right)\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.6
Target47.6
Herbie16.9
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 2 regimes

  2. if i < -0.6411582150710616 or 2.427056218200348 < i

    1. Initial program 29.8

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

    3. Applied add-log-exp29.9

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

    if -0.6411582150710616 < i < 2.427056218200348

    1. Initial program 57.5

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

    2. Taylor expanded around 0 57.1

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

    3. Applied simplify25.8

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

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

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

    6. Applied times-frac25.8

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

    7. Applied add-cube-cbrt25.8

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

    8. Applied times-frac25.7

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

    9. Applied simplify25.7

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

    10. Applied simplify9.7

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

    11. Taylor expanded around 0 9.7

      \[\leadsto \color{blue}{\left(\left(100 + \frac{100}{3} \cdot i\right) - \frac{25}{9} \cdot {i}^{2}\right)} \cdot \left(\sqrt[3]{1 + \frac{1}{2} \cdot i} \cdot n\right)\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify16.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;i \le -0.6411582150710616 \lor \neg \left(i \le 2.427056218200348\right):\\ \;\;\;\;\frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \sqrt[3]{1 + \frac{1}{2} \cdot i}\right) \cdot \left(\left(100 + \frac{100}{3} \cdot i\right) - \frac{25}{9} \cdot {i}^{2}\right)\\ \end{array}}\]

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' 
(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))))