Average Error: 52.5 → 10.1
Time: 6.7m
Precision: 64
Internal Precision: 3456
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l}
\mathbf{if}\;i \le -0.0016917120535581073:\\
\;\;\;\;\left(\sqrt[3]{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \cdot \sqrt[3]{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}}\right) \cdot \sqrt[3]{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}}\\
\mathbf{if}\;i \le 0.001497256652680892:\\
\;\;\;\;n \cdot \left(\left(\frac{50}{3} \cdot i\right) \cdot i + \left(100 + 50 \cdot i\right)\right)\\
\mathbf{if}\;i \le 3.509098710768375 \cdot 10^{+188}:\\
\;\;\;\;e^{\log \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)}\\
\mathbf{if}\;i \le 3.076781514658324 \cdot 10^{+222}:\\
\;\;\;\;\frac{100}{\frac{i}{n}} \cdot \left(\frac{\frac{1}{2}}{i \cdot i} + \left(i + \frac{\frac{\frac{1}{6}}{i}}{i \cdot i}\right)\right)\\
\mathbf{if}\;i \le 3.156165085498486 \cdot 10^{+258}:\\
\;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{n}{i}\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{100}{\frac{i}{n}} \cdot \left(\frac{\frac{1}{2}}{i \cdot i} + \left(i + \frac{\frac{\frac{1}{6}}{i}}{i \cdot i}\right)\right)\\
\end{array}\]
Target
| Original | 52.5 |
|---|
| Target | 51.7 |
|---|
| Herbie | 10.1 |
|---|
\[100 \cdot \frac{e^{n \cdot \begin{array}{l}
\mathbf{if}\;1 + \frac{i}{n} = 1:\\
\;\;\;\;\frac{i}{n}\\
\mathbf{else}:\\
\;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\
\end{array}} - 1}{\frac{i}{n}}\]
Derivation
- Split input into 5 regimes
if i < -0.0016917120535581073
Initial program 27.1
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
- Using strategy
rm Applied add-cube-cbrt27.4
\[\leadsto \color{blue}{\left(\sqrt[3]{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \cdot \sqrt[3]{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}}\right) \cdot \sqrt[3]{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}}}\]
if -0.0016917120535581073 < i < 0.001497256652680892
Initial program 61.7
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
Taylor expanded around 0 14.1
\[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{2} \cdot {i}^{2} + \left(\frac{1}{6} \cdot {i}^{3} + i\right)}}{\frac{i}{n}}\]
Taylor expanded around 0 0.0
\[\leadsto \color{blue}{100 \cdot n + \left(\frac{50}{3} \cdot \left(n \cdot {i}^{2}\right) + 50 \cdot \left(n \cdot i\right)\right)}\]
Applied simplify0.0
\[\leadsto \color{blue}{n \cdot \left(\left(\frac{50}{3} \cdot i\right) \cdot i + \left(100 + 50 \cdot i\right)\right)}\]
if 0.001497256652680892 < i < 3.509098710768375e+188
Initial program 30.1
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
- Using strategy
rm Applied add-exp-log30.8
\[\leadsto \color{blue}{e^{\log \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)}}\]
if 3.509098710768375e+188 < i < 3.076781514658324e+222 or 3.156165085498486e+258 < i
Initial program 61.8
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
Taylor expanded around 0 61.7
\[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{2} \cdot {i}^{2} + \left(\frac{1}{6} \cdot {i}^{3} + i\right)}}{\frac{i}{n}}\]
Taylor expanded around inf 23.4
\[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{6} \cdot \frac{1}{{i}^{3}} + \left(i + \frac{1}{2} \cdot \frac{1}{{i}^{2}}\right)}}{\frac{i}{n}}\]
Applied simplify23.6
\[\leadsto \color{blue}{\frac{100}{\frac{i}{n}} \cdot \left(\frac{\frac{1}{2}}{i \cdot i} + \left(i + \frac{\frac{\frac{1}{6}}{i}}{i \cdot i}\right)\right)}\]
if 3.076781514658324e+222 < i < 3.156165085498486e+258
Initial program 31.0
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
- Using strategy
rm Applied div-sub31.1
\[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)}\]
Applied simplify33.0
\[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right)\]
- Recombined 5 regimes into one program.
- Removed slow
pow expressions.
Runtime
herbie shell --seed '#(1062930989 876886121 3990119081 3032829768 3060892583 1929069376)'
(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))))
