Average Error: 47.5 → 16.7
Time: 2.6m
Precision: 64
Internal Precision: 3392
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt[3]{1 + \frac{1}{2} \cdot i} \cdot 100\right) + \log \left(\sqrt[3]{1 + \frac{1}{2} \cdot i}\right)} \cdot \left(\sqrt[3]{1 + \frac{1}{2} \cdot i} \cdot n\right) \le -1.7749380109575957 \cdot 10^{+308}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{if}\;e^{\log \left(\sqrt[3]{1 + \frac{1}{2} \cdot i} \cdot 100\right) + \log \left(\sqrt[3]{1 + \frac{1}{2} \cdot i}\right)} \cdot \left(\sqrt[3]{1 + \frac{1}{2} \cdot i} \cdot n\right) \le -8.819795090297178 \cdot 10^{-190}:\\ \;\;\;\;\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)\\ \mathbf{if}\;e^{\log \left(\sqrt[3]{1 + \frac{1}{2} \cdot i} \cdot 100\right) + \log \left(\sqrt[3]{1 + \frac{1}{2} \cdot i}\right)} \cdot \left(\sqrt[3]{1 + \frac{1}{2} \cdot i} \cdot n\right) \le 2.860878887320379 \cdot 10^{-287}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{if}\;e^{\log \left(\sqrt[3]{1 + \frac{1}{2} \cdot i} \cdot 100\right) + \log \left(\sqrt[3]{1 + \frac{1}{2} \cdot i}\right)} \cdot \left(\sqrt[3]{1 + \frac{1}{2} \cdot i} \cdot n\right) \le 2.5004732237782533 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{1 + \frac{1}{2} \cdot i}{\sqrt[3]{\frac{\frac{1}{100}}{n}} \cdot \sqrt[3]{\frac{\frac{1}{100}}{n}}}}{\sqrt[3]{\frac{\frac{i}{n}}{100 \cdot i}}}\\ \mathbf{if}\;e^{\log \left(\sqrt[3]{1 + \frac{1}{2} \cdot i} \cdot 100\right) + \log \left(\sqrt[3]{1 + \frac{1}{2} \cdot i}\right)} \cdot \left(\sqrt[3]{1 + \frac{1}{2} \cdot i} \cdot n\right) \le 3.5565440192703386 \cdot 10^{+306}:\\ \;\;\;\;\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)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - 1}{\left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right) + {\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n}}}{\frac{i}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.5
Target46.9
Herbie16.7
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if (* (exp (+ (log (* (cbrt (+ 1 (* 1/2 i))) 100)) (log (cbrt (+ 1 (* 1/2 i)))))) (* (cbrt (+ 1 (* 1/2 i))) n)) < -1.7749380109575957e+308 or -8.819795090297178e-190 < (* (exp (+ (log (* (cbrt (+ 1 (* 1/2 i))) 100)) (log (cbrt (+ 1 (* 1/2 i)))))) (* (cbrt (+ 1 (* 1/2 i))) n)) < 2.860878887320379e-287

    1. Initial program 18.9

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

    3. Applied div-inv18.9

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

    4. Applied *-un-lft-identity18.9

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

    5. Applied times-frac18.9

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

    6. Applied associate-*r*18.9

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

    7. Applied simplify18.9

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

    if -1.7749380109575957e+308 < (* (exp (+ (log (* (cbrt (+ 1 (* 1/2 i))) 100)) (log (cbrt (+ 1 (* 1/2 i)))))) (* (cbrt (+ 1 (* 1/2 i))) n)) < -8.819795090297178e-190 or 2.5004732237782533e+58 < (* (exp (+ (log (* (cbrt (+ 1 (* 1/2 i))) 100)) (log (cbrt (+ 1 (* 1/2 i)))))) (* (cbrt (+ 1 (* 1/2 i))) n)) < 3.5565440192703386e+306

    1. Initial program 58.7

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

    2. Taylor expanded around 0 60.9

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

    3. Applied simplify28.4

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

    5. Applied *-un-lft-identity28.4

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

    6. Applied times-frac28.4

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

    7. Applied add-cube-cbrt28.4

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

    8. Applied times-frac28.3

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

    9. Applied simplify28.3

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

    10. Applied simplify9.0

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

    11. Taylor expanded around 0 9.0

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

    if 2.860878887320379e-287 < (* (exp (+ (log (* (cbrt (+ 1 (* 1/2 i))) 100)) (log (cbrt (+ 1 (* 1/2 i)))))) (* (cbrt (+ 1 (* 1/2 i))) n)) < 2.5004732237782533e+58

    1. Initial program 52.5

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

    2. Taylor expanded around 0 53.9

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

    3. Applied simplify20.1

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

    5. Applied add-cube-cbrt20.6

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

    6. Applied associate-/r*20.6

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

    7. Applied simplify20.5

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

    if 3.5565440192703386e+306 < (* (exp (+ (log (* (cbrt (+ 1 (* 1/2 i))) 100)) (log (cbrt (+ 1 (* 1/2 i)))))) (* (cbrt (+ 1 (* 1/2 i))) n))

    1. Initial program 28.1

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

    3. Applied flip3--28.1

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

    4. Applied simplify28.1

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

    5. Applied simplify28.1

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

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018178 
(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))))