Average Error: 47.3 → 5.9
Time: 48.5s
Precision: 64
Internal Precision: 3392
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.291037782358374 \cdot 10^{-53}:\\ \;\;\;\;\left((e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot 100\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;i \le 2.7369923000369392 \cdot 10^{-77}:\\ \;\;\;\;\frac{1}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*} \cdot 100\\ \mathbf{elif}\;i \le 4.3239306309604786 \cdot 10^{+82}:\\ \;\;\;\;\left((e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot 100\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;i \le 8.833898493107465 \cdot 10^{+221}:\\ \;\;\;\;\left(\sqrt[3]{\frac{1}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*} \cdot 100} \cdot \sqrt[3]{\frac{1}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*} \cdot 100}\right) \cdot \sqrt[3]{\frac{1}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*} \cdot 100}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} - 1} \cdot n\right) \cdot \frac{\sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} - 1} \cdot \sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} - 1}}{i}\right) \cdot 100\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.3
Target46.8
Herbie5.9
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -1.291037782358374e-53 or 2.7369923000369392e-77 < i < 4.3239306309604786e+82

    1. Initial program 35.4

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

    3. Applied add-exp-log35.6

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

    4. Applied pow-exp35.6

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

    5. Applied expm1-def27.4

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

    6. Simplified1.9

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

    8. Applied div-inv2.3

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

    9. Applied associate-*r*2.3

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

    10. Simplified2.6

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

    if -1.291037782358374e-53 < i < 2.7369923000369392e-77

    1. Initial program 58.7

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

    3. Applied add-exp-log58.7

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

    4. Applied pow-exp58.7

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

    5. Applied expm1-def54.4

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

    6. Simplified19.1

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

    8. Applied clear-num19.4

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

    9. Taylor expanded around 0 8.3

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

    10. Simplified3.4

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

    if 4.3239306309604786e+82 < i < 8.833898493107465e+221

    1. Initial program 32.6

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

    3. Applied add-exp-log48.9

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

    4. Applied pow-exp48.9

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

    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. Simplified36.7

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

    8. Applied clear-num37.2

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

    9. Taylor expanded around 0 46.9

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

    10. Simplified26.7

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

    12. Applied add-cube-cbrt26.7

      \[\leadsto \color{blue}{\left(\sqrt[3]{100 \cdot \frac{1}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*}} \cdot \sqrt[3]{100 \cdot \frac{1}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*}}\right) \cdot \sqrt[3]{100 \cdot \frac{1}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*}}}\]

    if 8.833898493107465e+221 < i

    1. Initial program 34.9

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

    3. Applied div-inv34.9

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

    4. Applied add-cube-cbrt34.9

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right) \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}}{i \cdot \frac{1}{n}}\]

    5. Applied times-frac34.9

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i} \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\right)}\]

    6. Simplified34.9

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.291037782358374 \cdot 10^{-53}:\\ \;\;\;\;\left((e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot 100\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;i \le 2.7369923000369392 \cdot 10^{-77}:\\ \;\;\;\;\frac{1}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*} \cdot 100\\ \mathbf{elif}\;i \le 4.3239306309604786 \cdot 10^{+82}:\\ \;\;\;\;\left((e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot 100\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;i \le 8.833898493107465 \cdot 10^{+221}:\\ \;\;\;\;\left(\sqrt[3]{\frac{1}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*} \cdot 100} \cdot \sqrt[3]{\frac{1}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*} \cdot 100}\right) \cdot \sqrt[3]{\frac{1}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*} \cdot 100}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} - 1} \cdot n\right) \cdot \frac{\sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} - 1} \cdot \sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} - 1}}{i}\right) \cdot 100\\ \end{array}\]

Runtime

Time bar (total: 48.5s)Debug logProfile

herbie shell --seed 2018225 +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))))