Average Error: 47.8 → 17.0
Time: 3.0m
Precision: 64
Internal Precision: 3392
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.2203645458010985:\\ \;\;\;\;\left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\sqrt{\frac{i}{n}}} \cdot 100\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\sqrt{\frac{i}{n}}}\\ \mathbf{if}\;i \le 9.361359717821282 \cdot 10^{+23}:\\ \;\;\;\;\left(1 + \frac{1}{2} \cdot i\right) \cdot \left(100 \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{2} \cdot i}{\log \left(e^{\frac{\frac{1}{n}}{100}}\right)}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.8
Target47.1
Herbie17.0
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -1.2203645458010985

    1. Initial program 28.6

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

    3. Applied add-sqr-sqrt29.8

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

    4. Applied add-cube-cbrt29.8

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

    5. Applied times-frac29.8

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

    6. Applied associate-*r*29.8

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

    if -1.2203645458010985 < i < 9.361359717821282e+23

    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 simplify26.2

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

    5. Applied *-un-lft-identity26.2

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

    6. Applied *-un-lft-identity26.2

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

    7. Applied times-frac26.2

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

    8. Applied simplify26.2

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

    9. Applied simplify10.0

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

    if 9.361359717821282e+23 < i

    1. Initial program 32.0

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

    2. Taylor expanded around 0 56.4

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

    3. Applied simplify39.8

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

    5. Applied add-log-exp30.8

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

    6. Applied simplify30.6

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

  4. Applied simplify17.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;i \le -1.2203645458010985:\\ \;\;\;\;\left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\sqrt{\frac{i}{n}}} \cdot 100\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\sqrt{\frac{i}{n}}}\\ \mathbf{if}\;i \le 9.361359717821282 \cdot 10^{+23}:\\ \;\;\;\;\left(1 + \frac{1}{2} \cdot i\right) \cdot \left(100 \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{2} \cdot i}{\log \left(e^{\frac{\frac{1}{n}}{100}}\right)}\\ \end{array}}\]

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed '#(1072330854 3074818769 591214268 3603999196 3863745332 3332387116)' 
(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))))