Average Error: 47.1 → 5.7
Time: 47.3s
Precision: 64
Internal Precision: 3392
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -2.5927250546386884 \cdot 10^{-118}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\frac{i}{n}}{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}}\\ \mathbf{elif}\;i \le 1.305700592175497 \cdot 10^{-123} \lor \neg \left(i \le 9.576667012352247 \cdot 10^{+129}\right):\\ \;\;\;\;\frac{1}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left((e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot \frac{1}{\frac{i}{n}}\right)\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.1
Target46.7
Herbie5.7
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -2.5927250546386884e-118

    1. Initial program 36.3

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

    3. Applied add-exp-log36.3

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

    4. Applied pow-exp36.3

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

    5. Applied expm1-def29.8

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

    6. Simplified1.8

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

    8. Applied clear-num2.0

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

    if -2.5927250546386884e-118 < i < 1.305700592175497e-123 or 9.576667012352247e+129 < i

    1. Initial program 54.4

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

    3. Applied add-exp-log58.5

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

    4. Applied pow-exp58.5

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

    5. Applied expm1-def55.0

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

    6. Simplified26.2

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

    8. Applied clear-num26.6

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

    9. Taylor expanded around 0 17.1

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

    10. Simplified6.7

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

    if 1.305700592175497e-123 < i < 9.576667012352247e+129

    1. Initial program 47.6

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

    3. Applied add-exp-log49.7

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

    4. Applied pow-exp49.7

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

    5. Applied expm1-def36.7

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

    6. Simplified9.3

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

    8. Applied clear-num9.7

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

    10. Applied associate-/r/10.2

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

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.5927250546386884 \cdot 10^{-118}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\frac{i}{n}}{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}}\\ \mathbf{elif}\;i \le 1.305700592175497 \cdot 10^{-123} \lor \neg \left(i \le 9.576667012352247 \cdot 10^{+129}\right):\\ \;\;\;\;\frac{1}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left((e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot \frac{1}{\frac{i}{n}}\right)\\ \end{array}\]

Runtime

Time bar (total: 47.3s)Debug logProfile

herbie shell --seed 2018258 +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))))