Average Error: 42.7 → 25.6
Time: 1.2m
Precision: 64
Internal Precision: 128
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -2.377439836906217 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt[3]{(100 \cdot \left(e^{\log_* (1 + \frac{i}{n}) \cdot n}\right) + -100)_*} \cdot \sqrt[3]{(100 \cdot \left(e^{\log_* (1 + \frac{i}{n}) \cdot n}\right) + -100)_*}}{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \left(\frac{\sqrt[3]{(100 \cdot \left(e^{\log_* (1 + \frac{i}{n}) \cdot n}\right) + -100)_*}}{\sqrt[3]{i}} \cdot n\right)\\ \mathbf{elif}\;i \le -1.4645460330626419 \cdot 10^{-239}:\\ \;\;\;\;\frac{i \cdot (i \cdot \left((i \cdot \frac{50}{3} + 50)_*\right) + 100)_*}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -2.7483213362865855 \cdot 10^{-288}:\\ \;\;\;\;\frac{\sqrt[3]{(100 \cdot \left(e^{\log_* (1 + \frac{i}{n}) \cdot n}\right) + -100)_*} \cdot \sqrt[3]{(100 \cdot \left(e^{\log_* (1 + \frac{i}{n}) \cdot n}\right) + -100)_*}}{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \left(\frac{\sqrt[3]{(100 \cdot \left(e^{\log_* (1 + \frac{i}{n}) \cdot n}\right) + -100)_*}}{\sqrt[3]{i}} \cdot n\right)\\ \mathbf{elif}\;i \le 0.0020594158667992255:\\ \;\;\;\;\frac{i \cdot (i \cdot \left((i \cdot \frac{50}{3} + 50)_*\right) + 100)_*}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(50 \cdot \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) + \left((\frac{50}{3} \cdot \left(\left(n \cdot \left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right)\right) \cdot \log i\right) + \left((100 \cdot \left(n \cdot \log i\right) + \left(\left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right) \cdot 50 + 50 \cdot \left(\log i \cdot \left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot n\right)\right)\right))_*\right))_*\right))_* - \left(\left((\left(\left(\left(n \cdot n\right) \cdot \left(n \cdot \log n\right)\right) \cdot \left(\log i \cdot \log i\right)\right) \cdot \frac{100}{3} + \left(\left(\log i \cdot 50\right) \cdot \left(\left(n \cdot n\right) \cdot \log n\right)\right))_* + \left(\log i \cdot 50\right) \cdot \left(\left(n \cdot n\right) \cdot \log n\right)\right) + (\frac{50}{3} \cdot \left(\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot n\right) \cdot \log n\right) + \left((\left(\left(\left(n \cdot n\right) \cdot \left(n \cdot \log n\right)\right) \cdot \left(\log i \cdot \log i\right)\right) \cdot \frac{50}{3} + \left(100 \cdot \left(n \cdot \log n\right)\right))_*\right))_*\right)}{\frac{i}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original42.7
Target42.2
Herbie25.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -2.377439836906217e-21 or -1.4645460330626419e-239 < i < -2.7483213362865855e-288

    1. Initial program 32.9

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

    2. Simplified32.9

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

    4. Applied add-exp-log32.9

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

    5. Applied pow-exp32.9

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

    6. Simplified14.8

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

    8. Applied associate-/r/15.2

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

    10. Applied add-cube-cbrt15.6

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

    11. Applied add-cube-cbrt15.8

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

    12. Applied times-frac15.9

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

    13. Applied associate-*l*15.9

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

    if -2.377439836906217e-21 < i < -1.4645460330626419e-239 or -2.7483213362865855e-288 < i < 0.0020594158667992255

    1. Initial program 50.6

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

    2. Simplified50.6

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

    3. Taylor expanded around 0 31.7

      \[\leadsto \frac{\color{blue}{100 \cdot i + \left(50 \cdot {i}^{2} + \frac{50}{3} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]

    4. Simplified31.7

      \[\leadsto \frac{\color{blue}{(i \cdot \left((i \cdot \frac{50}{3} + 50)_*\right) + 100)_* \cdot i}}{\frac{i}{n}}\]

    if 0.0020594158667992255 < i

    1. Initial program 31.1

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

    2. Simplified31.0

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

    4. Applied add-exp-log50.3

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

    5. Applied pow-exp50.3

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

    6. Simplified48.9

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

    7. Taylor expanded around 0 21.5

      \[\leadsto \frac{\color{blue}{\left(50 \cdot \left({n}^{2} \cdot {\left(\log n\right)}^{2}\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot {\left(\log i\right)}^{3}\right) + \left(100 \cdot \left(n \cdot \log i\right) + \left(\frac{100}{3} \cdot \left({n}^{3} \cdot \left({\left(\log n\right)}^{2} \cdot \log i\right)\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot \left(\log i \cdot {\left(\log n\right)}^{2}\right)\right) + 50 \cdot \left({n}^{2} \cdot {\left(\log i\right)}^{2}\right)\right)\right)\right)\right)\right) - \left(\frac{100}{3} \cdot \left({n}^{3} \cdot \left({\left(\log i\right)}^{2} \cdot \log n\right)\right) + \left(50 \cdot \left({n}^{2} \cdot \left(\log n \cdot \log i\right)\right) + \left(50 \cdot \left({n}^{2} \cdot \left(\log i \cdot \log n\right)\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot \left(\log n \cdot {\left(\log i\right)}^{2}\right)\right) + 100 \cdot \left(n \cdot \log n\right)\right)\right)\right)\right)\right)}}{\frac{i}{n}}\]

    8. Simplified21.5

      \[\leadsto \frac{\color{blue}{(50 \cdot \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) + \left((\frac{50}{3} \cdot \left(\log i \cdot \left(n \cdot \left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right)\right)\right) + \left((100 \cdot \left(n \cdot \log i\right) + \left(\left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right) \cdot 50 + \left(\log i \cdot \left(n \cdot \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right)\right)\right) \cdot 50\right))_*\right))_*\right))_* - \left(\left((\left(\left(\log i \cdot \log i\right) \cdot \left(\left(n \cdot n\right) \cdot \left(n \cdot \log n\right)\right)\right) \cdot \frac{100}{3} + \left(\left(\left(n \cdot n\right) \cdot \log n\right) \cdot \left(\log i \cdot 50\right)\right))_* + \left(\left(n \cdot n\right) \cdot \log n\right) \cdot \left(\log i \cdot 50\right)\right) + (\frac{50}{3} \cdot \left(\log n \cdot \left(n \cdot \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right)\right)\right) + \left((\left(\left(\log i \cdot \log i\right) \cdot \left(\left(n \cdot n\right) \cdot \left(n \cdot \log n\right)\right)\right) \cdot \frac{50}{3} + \left(100 \cdot \left(n \cdot \log n\right)\right))_*\right))_*\right)}}{\frac{i}{n}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification25.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.377439836906217 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt[3]{(100 \cdot \left(e^{\log_* (1 + \frac{i}{n}) \cdot n}\right) + -100)_*} \cdot \sqrt[3]{(100 \cdot \left(e^{\log_* (1 + \frac{i}{n}) \cdot n}\right) + -100)_*}}{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \left(\frac{\sqrt[3]{(100 \cdot \left(e^{\log_* (1 + \frac{i}{n}) \cdot n}\right) + -100)_*}}{\sqrt[3]{i}} \cdot n\right)\\ \mathbf{elif}\;i \le -1.4645460330626419 \cdot 10^{-239}:\\ \;\;\;\;\frac{i \cdot (i \cdot \left((i \cdot \frac{50}{3} + 50)_*\right) + 100)_*}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -2.7483213362865855 \cdot 10^{-288}:\\ \;\;\;\;\frac{\sqrt[3]{(100 \cdot \left(e^{\log_* (1 + \frac{i}{n}) \cdot n}\right) + -100)_*} \cdot \sqrt[3]{(100 \cdot \left(e^{\log_* (1 + \frac{i}{n}) \cdot n}\right) + -100)_*}}{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \left(\frac{\sqrt[3]{(100 \cdot \left(e^{\log_* (1 + \frac{i}{n}) \cdot n}\right) + -100)_*}}{\sqrt[3]{i}} \cdot n\right)\\ \mathbf{elif}\;i \le 0.0020594158667992255:\\ \;\;\;\;\frac{i \cdot (i \cdot \left((i \cdot \frac{50}{3} + 50)_*\right) + 100)_*}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(50 \cdot \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) + \left((\frac{50}{3} \cdot \left(\left(n \cdot \left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right)\right) \cdot \log i\right) + \left((100 \cdot \left(n \cdot \log i\right) + \left(\left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right) \cdot 50 + 50 \cdot \left(\log i \cdot \left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot n\right)\right)\right))_*\right))_*\right))_* - \left(\left((\left(\left(\left(n \cdot n\right) \cdot \left(n \cdot \log n\right)\right) \cdot \left(\log i \cdot \log i\right)\right) \cdot \frac{100}{3} + \left(\left(\log i \cdot 50\right) \cdot \left(\left(n \cdot n\right) \cdot \log n\right)\right))_* + \left(\log i \cdot 50\right) \cdot \left(\left(n \cdot n\right) \cdot \log n\right)\right) + (\frac{50}{3} \cdot \left(\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot n\right) \cdot \log n\right) + \left((\left(\left(\left(n \cdot n\right) \cdot \left(n \cdot \log n\right)\right) \cdot \left(\log i \cdot \log i\right)\right) \cdot \frac{50}{3} + \left(100 \cdot \left(n \cdot \log n\right)\right))_*\right))_*\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 +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))))