Average Error: 47.3 → 10.8
Time: 57.3s
Precision: 64
Internal Precision: 3136
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;100 \cdot \left(\left(\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot n\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i}\right) \le 2.728305979952768 \cdot 10^{-207}:\\ \;\;\;\;100 \cdot \frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}}\\ \mathbf{elif}\;100 \cdot \left(\left(\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot n\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i}\right) \le 1.748251769543444 \cdot 10^{+163}:\\ \;\;\;\;(\left({\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\frac{n \cdot 100}{i}\right) + \left(\frac{-100}{\frac{i}{n}}\right))_*\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{i}^{n}}{{n}^{n}} - 1}{\frac{i}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.3
Target46.5
Herbie10.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if (* 100 (* (/ (* (cbrt (- (pow (+ 1 (/ i n)) n) 1)) (cbrt (- (pow (+ 1 (/ i n)) n) 1))) i) (* n (cbrt (- (pow (+ 1 (/ i n)) n) 1))))) < 2.728305979952768e-207

    1. Initial program 48.7

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

    3. Applied add-exp-log49.0

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

    4. Applied pow-exp49.0

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

    5. Applied expm1-def42.7

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

    6. Simplified11.7

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

    if 2.728305979952768e-207 < (* 100 (* (/ (* (cbrt (- (pow (+ 1 (/ i n)) n) 1)) (cbrt (- (pow (+ 1 (/ i n)) n) 1))) i) (* n (cbrt (- (pow (+ 1 (/ i n)) n) 1))))) < 1.748251769543444e+163

    1. Initial program 2.1

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

    2. Initial simplification2.2

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

    if 1.748251769543444e+163 < (* 100 (* (/ (* (cbrt (- (pow (+ 1 (/ i n)) n) 1)) (cbrt (- (pow (+ 1 (/ i n)) n) 1))) i) (* n (cbrt (- (pow (+ 1 (/ i n)) n) 1)))))

    1. Initial program 63.0

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.1

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{{i}^{n}}{{n}^{n}} - 1}}{\frac{i}{n}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;100 \cdot \left(\left(\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot n\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i}\right) \le 2.728305979952768 \cdot 10^{-207}:\\ \;\;\;\;100 \cdot \frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}}\\ \mathbf{elif}\;100 \cdot \left(\left(\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot n\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i}\right) \le 1.748251769543444 \cdot 10^{+163}:\\ \;\;\;\;(\left({\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \left(\frac{n \cdot 100}{i}\right) + \left(\frac{-100}{\frac{i}{n}}\right))_*\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{i}^{n}}{{n}^{n}} - 1}{\frac{i}{n}}\\ \end{array}\]

Runtime

Time bar (total: 57.3s)Debug logProfile

herbie shell --seed 2018215 +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))))