Average Error: 47.1 → 14.2
Time: 2.5m
Precision: 64
Internal Precision: 3136
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;100 \cdot \left(e^{\left(\frac{1}{2} \cdot i + \left(\frac{1}{24} \cdot {i}^{2} + \left(\frac{1}{36} \cdot {i}^{4} + \left(\frac{1}{648} \cdot {i}^{6} + \frac{1}{72} \cdot {i}^{5}\right)\right)\right)\right) - \frac{1}{24} \cdot {i}^{3}} \cdot n\right) \le -1.7828617811379277 \cdot 10^{+308}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{n}{i}\right)\\ \mathbf{if}\;100 \cdot \left(e^{\left(\frac{1}{2} \cdot i + \left(\frac{1}{24} \cdot {i}^{2} + \left(\frac{1}{36} \cdot {i}^{4} + \left(\frac{1}{648} \cdot {i}^{6} + \frac{1}{72} \cdot {i}^{5}\right)\right)\right)\right) - \frac{1}{24} \cdot {i}^{3}} \cdot n\right) \le -7.557055664580623 \cdot 10^{-180}:\\ \;\;\;\;{\left(e^{\frac{1}{24} \cdot i}\right)}^{\left(i - i \cdot i\right)} \cdot \left(\sqrt{e^{i}} \cdot \left(100 \cdot n\right)\right)\\ \mathbf{if}\;100 \cdot \left(e^{\left(\frac{1}{2} \cdot i + \left(\frac{1}{24} \cdot {i}^{2} + \left(\frac{1}{36} \cdot {i}^{4} + \left(\frac{1}{648} \cdot {i}^{6} + \frac{1}{72} \cdot {i}^{5}\right)\right)\right)\right) - \frac{1}{24} \cdot {i}^{3}} \cdot n\right) \le 3.4826183084898404 \cdot 10^{-243}:\\ \;\;\;\;\left(\frac{100}{i} \cdot n\right) \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)\\ \mathbf{if}\;100 \cdot \left(e^{\left(\frac{1}{2} \cdot i + \left(\frac{1}{24} \cdot {i}^{2} + \left(\frac{1}{36} \cdot {i}^{4} + \left(\frac{1}{648} \cdot {i}^{6} + \frac{1}{72} \cdot {i}^{5}\right)\right)\right)\right) - \frac{1}{24} \cdot {i}^{3}} \cdot n\right) \le 1.8473422884204718 \cdot 10^{+298}:\\ \;\;\;\;100 \cdot \left(e^{\left(\frac{1}{2} \cdot i + \left(\frac{1}{24} \cdot {i}^{2} + \left(\frac{1}{36} \cdot {i}^{4} + \left(\frac{1}{648} \cdot {i}^{6} + \frac{1}{72} \cdot {i}^{5}\right)\right)\right)\right) - \frac{1}{24} \cdot {i}^{3}} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{100 \cdot n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

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Results

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Target

Original47.1
Target46.7
Herbie14.2
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 5 regimes

  2. if (* 100 (* (exp (- (+ (* 1/2 i) (+ (* 1/24 (pow i 2)) (+ (* 1/36 (pow i 4)) (+ (* 1/648 (pow i 6)) (* 1/72 (pow i 5)))))) (* 1/24 (pow i 3)))) n)) < -1.7828617811379277e+308

    1. Initial program 35.4

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

    3. Applied div-sub35.5

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

    4. Applied simplify38.4

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

    if -1.7828617811379277e+308 < (* 100 (* (exp (- (+ (* 1/2 i) (+ (* 1/24 (pow i 2)) (+ (* 1/36 (pow i 4)) (+ (* 1/648 (pow i 6)) (* 1/72 (pow i 5)))))) (* 1/24 (pow i 3)))) n)) < -7.557055664580623e-180

    1. Initial program 62.2

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

    2. Taylor expanded around 0 22.2

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(\frac{1}{2} \cdot {i}^{2} + \frac{1}{6} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]
    3. Using strategy rm

    4. Applied associate-/r/4.7

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{i + \left(\frac{1}{2} \cdot {i}^{2} + \frac{1}{6} \cdot {i}^{3}\right)}{i} \cdot n\right)}\]
    5. Using strategy rm

    6. Applied add-exp-log4.7

      \[\leadsto 100 \cdot \left(\color{blue}{e^{\log \left(\frac{i + \left(\frac{1}{2} \cdot {i}^{2} + \frac{1}{6} \cdot {i}^{3}\right)}{i}\right)}} \cdot n\right)\]

    7. Taylor expanded around 0 4.7

      \[\leadsto 100 \cdot \left(e^{\color{blue}{\left(\frac{1}{2} \cdot i + \frac{1}{24} \cdot {i}^{2}\right) - \frac{1}{24} \cdot {i}^{3}}} \cdot n\right)\]

    8. Applied simplify4.7

      \[\leadsto \color{blue}{{\left(e^{\frac{1}{24} \cdot i}\right)}^{\left(i - i \cdot i\right)} \cdot \left(\sqrt{e^{i}} \cdot \left(100 \cdot n\right)\right)}\]

    if -7.557055664580623e-180 < (* 100 (* (exp (- (+ (* 1/2 i) (+ (* 1/24 (pow i 2)) (+ (* 1/36 (pow i 4)) (+ (* 1/648 (pow i 6)) (* 1/72 (pow i 5)))))) (* 1/24 (pow i 3)))) n)) < 3.4826183084898404e-243

    1. Initial program 26.6

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

    2. Taylor expanded around inf 48.5

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

    3. Applied simplify27.1

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

    if 3.4826183084898404e-243 < (* 100 (* (exp (- (+ (* 1/2 i) (+ (* 1/24 (pow i 2)) (+ (* 1/36 (pow i 4)) (+ (* 1/648 (pow i 6)) (* 1/72 (pow i 5)))))) (* 1/24 (pow i 3)))) n)) < 1.8473422884204718e+298

    1. Initial program 59.7

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

    2. Taylor expanded around 0 23.7

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(\frac{1}{2} \cdot {i}^{2} + \frac{1}{6} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]
    3. Using strategy rm

    4. Applied associate-/r/7.0

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{i + \left(\frac{1}{2} \cdot {i}^{2} + \frac{1}{6} \cdot {i}^{3}\right)}{i} \cdot n\right)}\]
    5. Using strategy rm

    6. Applied add-exp-log7.0

      \[\leadsto 100 \cdot \left(\color{blue}{e^{\log \left(\frac{i + \left(\frac{1}{2} \cdot {i}^{2} + \frac{1}{6} \cdot {i}^{3}\right)}{i}\right)}} \cdot n\right)\]

    7. Taylor expanded around inf 7.0

      \[\leadsto 100 \cdot \left(e^{\color{blue}{\left(\frac{1}{2} \cdot i + \left(\frac{1}{24} \cdot {i}^{2} + \left(\frac{1}{36} \cdot {i}^{4} + \left(\frac{1}{648} \cdot {i}^{6} + \frac{1}{72} \cdot {i}^{5}\right)\right)\right)\right) - \frac{1}{24} \cdot {i}^{3}}} \cdot n\right)\]

    if 1.8473422884204718e+298 < (* 100 (* (exp (- (+ (* 1/2 i) (+ (* 1/24 (pow i 2)) (+ (* 1/36 (pow i 4)) (+ (* 1/648 (pow i 6)) (* 1/72 (pow i 5)))))) (* 1/24 (pow i 3)))) n))

    1. Initial program 29.1

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

    2. Taylor expanded around inf 49.8

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

    3. Applied simplify22.6

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018296 
(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))))