Average Error: 42.4 → 18.8
Time: 44.3s
Precision: 64
Internal Precision: 128
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -0.5589490298612874:\\ \;\;\;\;100 \cdot \frac{(e^{n \cdot (e^{\log_* (1 + \log_* (1 + \frac{i}{n}))} - 1)^*} - 1)^*}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 3.718399696914079 \cdot 10^{-236}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.4
Target42.1
Herbie18.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if n < -0.5589490298612874

    1. Initial program 45.1

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

    3. Applied add-exp-log45.1

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

    4. Applied expm1-def45.1

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

    5. Simplified24.9

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

    7. Applied expm1-log1p-u24.9

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

    8. Applied log1p-expm124.9

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

    10. Applied expm1-log1p-u24.9

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

    if -0.5589490298612874 < n < 3.718399696914079e-236

    1. Initial program 18.7

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

    3. Applied add-exp-log18.7

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

    4. Applied expm1-def18.7

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

    5. Simplified21.7

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

    6. Taylor expanded around 0 15.5

      \[\leadsto \color{blue}{0}\]

    if 3.718399696914079e-236 < n

    1. Initial program 56.2

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

    3. Applied add-exp-log56.2

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

    4. Applied expm1-def56.2

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

    5. Simplified16.1

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

    7. Applied expm1-log1p-u16.1

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

    8. Applied log1p-expm116.1

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

  4. Final simplification18.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -0.5589490298612874:\\ \;\;\;\;100 \cdot \frac{(e^{n \cdot (e^{\log_* (1 + \log_* (1 + \frac{i}{n}))} - 1)^*} - 1)^*}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 3.718399696914079 \cdot 10^{-236}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}}\\ \end{array}\]

Runtime

Time bar (total: 44.3s)Debug logProfile

herbie shell --seed 2018349 +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))))