Average Error: 47.7 → 16.3
Time: 1.4m
Precision: 64
Internal Precision: 3200
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.46173035710953614:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\sqrt[3]{\frac{i}{n}}} \cdot \left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}}\right)\\ \mathbf{if}\;i \le 0.0003444745819302249:\\ \;\;\;\;\frac{\left(100 \cdot n\right) \cdot \left(1 + i \cdot \frac{1}{2}\right)}{1}\\ \mathbf{if}\;i \le 1.0840697675335321 \cdot 10^{+201}:\\ \;\;\;\;\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}{\frac{1}{n}} \cdot 100\\ \mathbf{if}\;i \le +\infty:\\ \;\;\;\;\left({\left(e^{n}\right)}^{\left(\log i - \log n\right)} - 1\right) \cdot \left(n \cdot \frac{100}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(e^{n}\right)}^{\left(\log i - \log n\right)} - 1\right) \cdot \left(n \cdot \frac{100}{i}\right)\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.7
Target47.1
Herbie16.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -0.46173035710953614

    1. Initial program 28.0

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

    3. Applied add-cube-cbrt28.3

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

    4. Applied add-cube-cbrt28.3

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

    5. Applied times-frac28.3

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

    6. Applied associate-*r*28.3

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

    if -0.46173035710953614 < i < 0.0003444745819302249

    1. Initial program 57.8

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

    2. Taylor expanded around 0 57.5

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

    3. Applied simplify25.6

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

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

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

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

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

    7. Applied times-frac25.6

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

    8. Applied simplify25.6

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

    9. Applied simplify9.2

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

    if 0.0003444745819302249 < i < 1.0840697675335321e+201

    1. Initial program 31.1

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

    3. Applied div-inv31.1

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

    4. Applied associate-/r*31.1

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

    if 1.0840697675335321e+201 < i

    1. Initial program 35.5

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

    2. Taylor expanded around inf 26.2

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

    3. Applied simplify26.5

      \[\leadsto \color{blue}{\left(\frac{100}{i} \cdot n\right) \cdot \left({\left(e^{n}\right)}^{\left(\left(\log i + 0\right) - \log n\right)} - 1\right)}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify16.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;i \le -0.46173035710953614:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\sqrt[3]{\frac{i}{n}}} \cdot \left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}}\right)\\ \mathbf{if}\;i \le 0.0003444745819302249:\\ \;\;\;\;\frac{\left(100 \cdot n\right) \cdot \left(1 + i \cdot \frac{1}{2}\right)}{1}\\ \mathbf{if}\;i \le 1.0840697675335321 \cdot 10^{+201}:\\ \;\;\;\;\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}{\frac{1}{n}} \cdot 100\\ \mathbf{if}\;i \le +\infty:\\ \;\;\;\;\left({\left(e^{n}\right)}^{\left(\log i - \log n\right)} - 1\right) \cdot \left(n \cdot \frac{100}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(e^{n}\right)}^{\left(\log i - \log n\right)} - 1\right) \cdot \left(n \cdot \frac{100}{i}\right)\\ \end{array}}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(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))))