Average Error: 47.2 → 11.1
Time: 1.4m
Precision: 64
Internal Precision: 3392
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -4.311627950501261 \cdot 10^{-19}:\\ \;\;\;\;100 \cdot \frac{e^{n \cdot \log_* (1 + \frac{i}{n})} - 1}{\frac{i}{n}}\\ \mathbf{if}\;i \le 0.0047815638446876965:\\ \;\;\;\;n \cdot \left(\frac{i}{i} \cdot (50 \cdot i + 100)_*\right)\\ \mathbf{if}\;i \le 1.6123540503597488 \cdot 10^{+153}:\\ \;\;\;\;100 \cdot \frac{(e^{n \cdot \log \left(1 + \frac{i}{n}\right)} - 1)^*}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{{i}^{n}}{{n}^{n}}\right) \cdot 100 + -100)_*}{\frac{i}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.2
Target46.7
Herbie11.1
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -4.311627950501261e-19

    1. Initial program 29.6

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

    3. Applied add-exp-log29.7

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

    4. Applied pow-exp29.7

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

    5. Applied simplify7.2

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

    if -4.311627950501261e-19 < i < 0.0047815638446876965

    1. Initial program 57.8

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

    2. Taylor expanded around 0 57.6

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

    3. Applied simplify25.7

      \[\leadsto \color{blue}{\frac{(50 \cdot \left(i \cdot i\right) + \left(100 \cdot i\right))_*}{\frac{i}{n}}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity25.7

      \[\leadsto \frac{(50 \cdot \left(i \cdot i\right) + \left(100 \cdot i\right))_*}{\color{blue}{1 \cdot \frac{i}{n}}}\]

    6. Applied *-un-lft-identity25.7

      \[\leadsto \frac{\color{blue}{1 \cdot (50 \cdot \left(i \cdot i\right) + \left(100 \cdot i\right))_*}}{1 \cdot \frac{i}{n}}\]

    7. Applied times-frac25.7

      \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{(50 \cdot \left(i \cdot i\right) + \left(100 \cdot i\right))_*}{\frac{i}{n}}}\]

    8. Applied simplify25.7

      \[\leadsto \color{blue}{1} \cdot \frac{(50 \cdot \left(i \cdot i\right) + \left(100 \cdot i\right))_*}{\frac{i}{n}}\]

    9. Applied simplify9.2

      \[\leadsto 1 \cdot \color{blue}{\left(\left(\frac{i}{i} \cdot (50 \cdot i + 100)_*\right) \cdot n\right)}\]

    if 0.0047815638446876965 < i < 1.6123540503597488e+153

    1. Initial program 32.1

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

    3. Applied add-exp-log42.6

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

    4. Applied pow-exp42.6

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

    5. Applied expm1-def25.8

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

    if 1.6123540503597488e+153 < i

    1. Initial program 31.4

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

    2. Taylor expanded around inf 28.4

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

    3. Applied simplify28.4

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

  4. Applied simplify11.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;i \le -4.311627950501261 \cdot 10^{-19}:\\ \;\;\;\;100 \cdot \frac{e^{n \cdot \log_* (1 + \frac{i}{n})} - 1}{\frac{i}{n}}\\ \mathbf{if}\;i \le 0.0047815638446876965:\\ \;\;\;\;n \cdot \left(\frac{i}{i} \cdot (50 \cdot i + 100)_*\right)\\ \mathbf{if}\;i \le 1.6123540503597488 \cdot 10^{+153}:\\ \;\;\;\;100 \cdot \frac{(e^{n \cdot \log \left(1 + \frac{i}{n}\right)} - 1)^*}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{{i}^{n}}{{n}^{n}}\right) \cdot 100 + -100)_*}{\frac{i}{n}}\\ \end{array}}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +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))))