Compound Interest

Average Error: 47.6 → 16.8

Time: 15.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -1.80321636632118687:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 7030377895.16914845:\\ \;\;\;\;100 \cdot \left(\left(\left(0.5 \cdot i + \left(\frac{\log 1 \cdot n}{i} + 1\right)\right) - 0.5 \cdot \left(i \cdot \log 1\right)\right) \cdot n\right)\\ \mathbf{elif}\;i \le 1.31897584370110781 \cdot 10^{260}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 4.058977410036035 \cdot 10^{302}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original	47.6
Target	47.5
Herbie	16.8

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

Derivation

1. Split input into 4 regimes

2. if i < -1.80321636632118687

   1. Initial program 28.0

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

   3. Applied flip-- 28.0

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

   4. Simplified 28.0

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

   if -1.80321636632118687 < i < 7030377895.16914845

   1. Initial program 58.2

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

   2. Taylor expanded around 0 27.2

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{i}{n}}\]
   3. Using strategy rm

   4. Applied associate-/r/ 9.8

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)}\]

   5. Taylor expanded around 0 9.8

      \[\leadsto 100 \cdot \left(\color{blue}{\left(\left(1 + \left(0.5 \cdot i + \frac{\log 1 \cdot n}{i}\right)\right) - 0.5 \cdot \left(i \cdot \log 1\right)\right)} \cdot n\right)\]

   6. Simplified 9.8

      \[\leadsto 100 \cdot \left(\color{blue}{\left(\left(0.5 \cdot i + \left(\frac{\log 1 \cdot n}{i} + 1\right)\right) - 0.5 \cdot \left(i \cdot \log 1\right)\right)} \cdot n\right)\]

   if 7030377895.16914845 < i < 1.31897584370110781e260 or 4.058977410036035e302 < i

   1. Initial program 30.6

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

   3. Applied div-inv 30.6

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

   4. Applied *-un-lft-identity 30.6

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

   5. Applied times-frac 30.6

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

   6. Applied associate-*r* 30.6

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

   7. Simplified 30.6

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

   if 1.31897584370110781e260 < i < 4.058977410036035e302

   1. Initial program 32.6

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

   2. Taylor expanded around 0 31.8

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

4. Final simplification 16.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.80321636632118687:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 7030377895.16914845:\\ \;\;\;\;100 \cdot \left(\left(\left(0.5 \cdot i + \left(\frac{\log 1 \cdot n}{i} + 1\right)\right) - 0.5 \cdot \left(i \cdot \log 1\right)\right) \cdot n\right)\\ \mathbf{elif}\;i \le 1.31897584370110781 \cdot 10^{260}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 4.058977410036035 \cdot 10^{302}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020174 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))