Average Error: 47.5 → 10.7
Time: 9.6m
Precision: 64
Internal Precision: 3392
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\frac{1}{2} \cdot i + 1)_* \le 0.9999999879064155:\\ \;\;\;\;\frac{e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{if}\;(\frac{1}{2} \cdot i + 1)_* \le 42954578076.05249:\\ \;\;\;\;\left((\frac{1}{2} \cdot i + 1)_* \cdot 100\right) \cdot n\\ \mathbf{if}\;(\frac{1}{2} \cdot i + 1)_* \le 3.6661174704394984 \cdot 10^{+204}:\\ \;\;\;\;\left(n \cdot \frac{100}{i}\right) \cdot \left(\frac{{i}^{n}}{{n}^{n}} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{1}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.5
Target46.9
Herbie10.7
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if (fma 1/2 i 1) < 0.9999999879064155

    1. Initial program 28.8

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

    3. Applied add-exp-log28.8

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

    4. Applied pow-exp28.8

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

    5. Applied simplify5.9

      \[\leadsto 100 \cdot \frac{e^{\color{blue}{n \cdot \log_* (1 + \frac{i}{n})}} - 1}{\frac{i}{n}}\]

    if 0.9999999879064155 < (fma 1/2 i 1) < 42954578076.05249

    1. Initial program 57.8

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

    2. Taylor expanded around 0 57.7

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

    3. Applied simplify26.5

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

    5. Applied pow126.5

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

    6. Applied pow126.5

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

    7. Applied pow-prod-down26.5

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

    8. Applied simplify9.1

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

    if 42954578076.05249 < (fma 1/2 i 1) < 3.6661174704394984e+204

    1. Initial program 32.7

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

    2. Taylor expanded around inf 25.7

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

    3. Applied simplify25.7

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

    if 3.6661174704394984e+204 < (fma 1/2 i 1)

    1. Initial program 30.1

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

    3. Applied div-inv30.2

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

    4. Applied *-un-lft-identity30.2

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

    5. Applied times-frac30.2

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

    6. Applied associate-*r*30.2

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

    7. Applied simplify30.1

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

  4. Applied simplify10.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;(\frac{1}{2} \cdot i + 1)_* \le 0.9999999879064155:\\ \;\;\;\;\frac{e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{if}\;(\frac{1}{2} \cdot i + 1)_* \le 42954578076.05249:\\ \;\;\;\;\left((\frac{1}{2} \cdot i + 1)_* \cdot 100\right) \cdot n\\ \mathbf{if}\;(\frac{1}{2} \cdot i + 1)_* \le 3.6661174704394984 \cdot 10^{+204}:\\ \;\;\;\;\left(n \cdot \frac{100}{i}\right) \cdot \left(\frac{{i}^{n}}{{n}^{n}} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{1}{n}}\\ \end{array}}\]

Runtime

Time bar (total: 9.6m)Debug logProfile

herbie shell --seed 2018178 +o rules:numerics
(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))))