Average Error: 47.0 → 16.6
Time: 2.8m
Precision: 64
Internal Precision: 3456
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l}
\mathbf{if}\;i \le -2.4581221205466595 \cdot 10^{-10}:\\
\;\;\;\;100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i}{n} \cdot \left(\left({\left(\frac{i}{n} + 1\right)}^{n} + 1\right) + {\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right)}\\
\mathbf{if}\;i \le -5.844914035232024 \cdot 10^{-224}:\\
\;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \frac{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot \left(i \cdot i\right) + i}{\frac{1}{n}}\right)\\
\mathbf{if}\;i \le 4.70133616777004 \cdot 10^{-216}:\\
\;\;\;\;100 \cdot \left(\frac{\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot \left(i \cdot i\right) + i}{i} \cdot n\right)\\
\mathbf{if}\;i \le 2.292576075217319 \cdot 10^{-18}:\\
\;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \frac{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot \left(i \cdot i\right) + i}{\frac{1}{n}}\right)\\
\mathbf{else}:\\
\;\;\;\;\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\\
\end{array}\]
Target
| Original | 47.0 |
|---|
| Target | 47.1 |
|---|
| Herbie | 16.6 |
|---|
\[100 \cdot \frac{e^{n \cdot \begin{array}{l}
\mathbf{if}\;1 + \frac{i}{n} = 1:\\
\;\;\;\;\frac{i}{n}\\
\mathbf{else}:\\
\;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\
\end{array}} - 1}{\frac{i}{n}}\]
Derivation
- Split input into 4 regimes
if i < -2.4581221205466595e-10
Initial program 27.7
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
- Using strategy
rm Applied flip3--27.7
\[\leadsto 100 \cdot \frac{\color{blue}{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}{\frac{i}{n}}\]
Applied associate-/l/27.7
\[\leadsto 100 \cdot \color{blue}{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i}{n} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)\right)}}\]
Applied simplify27.7
\[\leadsto 100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\color{blue}{\frac{i}{n} \cdot \left(\left({\left(\frac{i}{n} + 1\right)}^{n} + 1\right) + {\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right)}}\]
if -2.4581221205466595e-10 < i < -5.844914035232024e-224 or 4.70133616777004e-216 < i < 2.292576075217319e-18
Initial program 57.0
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
Taylor expanded around 0 23.6
\[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{2} \cdot {i}^{2} + \left(\frac{1}{6} \cdot {i}^{3} + i\right)}}{\frac{i}{n}}\]
- Using strategy
rm Applied div-inv23.7
\[\leadsto 100 \cdot \frac{\frac{1}{2} \cdot {i}^{2} + \left(\frac{1}{6} \cdot {i}^{3} + i\right)}{\color{blue}{i \cdot \frac{1}{n}}}\]
Applied *-un-lft-identity23.7
\[\leadsto 100 \cdot \frac{\color{blue}{1 \cdot \left(\frac{1}{2} \cdot {i}^{2} + \left(\frac{1}{6} \cdot {i}^{3} + i\right)\right)}}{i \cdot \frac{1}{n}}\]
Applied times-frac10.4
\[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{i} \cdot \frac{\frac{1}{2} \cdot {i}^{2} + \left(\frac{1}{6} \cdot {i}^{3} + i\right)}{\frac{1}{n}}\right)}\]
Applied simplify10.4
\[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \color{blue}{\frac{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot \left(i \cdot i\right) + i}{\frac{1}{n}}}\right)\]
if -5.844914035232024e-224 < i < 4.70133616777004e-216
Initial program 60.6
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
Taylor expanded around 0 28.2
\[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{2} \cdot {i}^{2} + \left(\frac{1}{6} \cdot {i}^{3} + i\right)}}{\frac{i}{n}}\]
- Using strategy
rm Applied associate-/r/2.8
\[\leadsto 100 \cdot \color{blue}{\left(\frac{\frac{1}{2} \cdot {i}^{2} + \left(\frac{1}{6} \cdot {i}^{3} + i\right)}{i} \cdot n\right)}\]
Applied simplify2.8
\[\leadsto 100 \cdot \left(\color{blue}{\frac{\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot \left(i \cdot i\right) + i}{i}} \cdot n\right)\]
if 2.292576075217319e-18 < i
Initial program 33.1
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
- Using strategy
rm Applied div-inv33.1
\[\leadsto 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\color{blue}{i \cdot \frac{1}{n}}}\]
Applied add-cube-cbrt33.2
\[\leadsto 100 \cdot \frac{\color{blue}{\left(\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right) \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}}{i \cdot \frac{1}{n}}\]
Applied times-frac33.1
\[\leadsto 100 \cdot \color{blue}{\left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i} \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\right)}\]
Applied associate-*r*33.1
\[\leadsto \color{blue}{\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}}\]
- Recombined 4 regimes into one program.
Runtime
herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)'
(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))))
