Average Error: 47.3 → 6.6
Time: 42.4s
Precision: 64
Internal Precision: 3136
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.7361288934095412 \cdot 10^{-39}:\\ \;\;\;\;\frac{100}{\frac{i}{n} \cdot \frac{1}{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.3
Target47.3
Herbie6.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 2 regimes

  2. if i < -1.7361288934095412e-39

    1. Initial program 31.3

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

    3. Applied add-exp-log31.3

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

    4. Applied expm1-def31.3

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

    5. Simplified0.5

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

    7. Applied associate-*r/0.4

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

    9. Applied associate-/l*0.6

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

    11. Applied div-inv0.6

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

    if -1.7361288934095412e-39 < i

    1. Initial program 53.0

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

    3. Applied add-exp-log53.0

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

    4. Applied expm1-def53.0

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

    5. Simplified21.2

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

    7. Applied associate-*r/21.2

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

    9. Applied associate-/l*21.5

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

    10. Taylor expanded around 0 15.6

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

    11. Simplified8.7

      \[\leadsto \frac{100}{\color{blue}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.7361288934095412 \cdot 10^{-39}:\\ \;\;\;\;\frac{100}{\frac{i}{n} \cdot \frac{1}{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 42.4s)Debug logProfile

herbie shell --seed 2018255 +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))))