Average Error: 47.5 → 17.5
Time: 2.2m
Precision: 64
Internal Precision: 3392
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2} \cdot i + 1} \cdot \left(\left(100 + i \cdot \frac{100}{3}\right) - \frac{25}{9} \cdot \left(i \cdot i\right)\right) \le -3.5244619032097613 \cdot 10^{+190}:\\ \;\;\;\;100 \cdot \left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i} \cdot \left(n \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right)\right)\\ \mathbf{if}\;\sqrt[3]{\frac{1}{2} \cdot i + 1} \cdot \left(\left(100 + i \cdot \frac{100}{3}\right) - \frac{25}{9} \cdot \left(i \cdot i\right)\right) \le -4.418830388816365 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{1 + \frac{1}{2} \cdot i}{\sqrt[3]{\frac{\frac{1}{100}}{n}} \cdot \sqrt[3]{\frac{\frac{1}{100}}{n}}}}{\sqrt[3]{\frac{\frac{i}{n}}{100 \cdot i}}}\\ \mathbf{if}\;\sqrt[3]{\frac{1}{2} \cdot i + 1} \cdot \left(\left(100 + i \cdot \frac{100}{3}\right) - \frac{25}{9} \cdot \left(i \cdot i\right)\right) \le -1.34040328536697 \cdot 10^{+64}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{if}\;\sqrt[3]{\frac{1}{2} \cdot i + 1} \cdot \left(\left(100 + i \cdot \frac{100}{3}\right) - \frac{25}{9} \cdot \left(i \cdot i\right)\right) \le 3.0894318730283632 \cdot 10^{+44}:\\ \;\;\;\;\left(\left(100 + \frac{100}{3} \cdot i\right) - \frac{25}{9} \cdot {i}^{2}\right) \cdot \left(\sqrt[3]{1 + \frac{1}{2} \cdot i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.5
Target47.2
Herbie17.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 5 regimes

  2. if (* (cbrt (+ (* 1/2 i) 1)) (- (+ 100 (* i 100/3)) (* 25/9 (* i i)))) < -3.5244619032097613e+190

    1. Initial program 31.0

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

    3. Applied div-inv31.0

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

    4. Applied add-cube-cbrt31.0

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

    5. Applied times-frac31.1

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

    6. Applied simplify31.0

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

    if -3.5244619032097613e+190 < (* (cbrt (+ (* 1/2 i) 1)) (- (+ 100 (* i 100/3)) (* 25/9 (* i i)))) < -4.418830388816365e+93

    1. Initial program 30.9

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

    2. Taylor expanded around 0 50.5

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

    3. Applied simplify50.5

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

    5. Applied add-cube-cbrt50.5

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

    6. Applied associate-/r*50.5

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

    7. Applied simplify50.5

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

    if -4.418830388816365e+93 < (* (cbrt (+ (* 1/2 i) 1)) (- (+ 100 (* i 100/3)) (* 25/9 (* i i)))) < -1.34040328536697e+64

    1. Initial program 36.1

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

    3. Applied associate-*r/36.1

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

    if -1.34040328536697e+64 < (* (cbrt (+ (* 1/2 i) 1)) (- (+ 100 (* i 100/3)) (* 25/9 (* i i)))) < 3.0894318730283632e+44

    1. Initial program 56.9

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

    2. Taylor expanded around 0 57.2

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

    3. Applied simplify27.1

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

    5. Applied *-un-lft-identity27.1

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

    6. Applied times-frac27.2

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

    7. Applied add-cube-cbrt27.2

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

    8. Applied times-frac27.1

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

    9. Applied simplify27.1

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

    10. Applied simplify11.7

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

    11. Taylor expanded around 0 11.7

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

    if 3.0894318730283632e+44 < (* (cbrt (+ (* 1/2 i) 1)) (- (+ 100 (* i 100/3)) (* 25/9 (* i i))))

    1. Initial program 27.1

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

    3. Applied associate-*r/27.1

      \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}}\]
  3. Recombined 5 regimes into one program.

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed 2018206 
(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))))