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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)} \cdot \left(n \cdot 100\right)\right) \cdot \frac{\left(\sqrt[3]{i} + \left(\sqrt[3]{{i}^{4}} \cdot \frac{1}{6} + \sqrt[3]{{i}^{7}} \cdot \frac{1}{36}\right)\right) \cdot \sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)}}{i} \le -1.770573189712183 \cdot 10^{+308}:\\ \;\;\;\;\left(\left(n \cdot 100\right) \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\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}\\ \mathbf{elif}\;\left(\sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)} \cdot \left(n \cdot 100\right)\right) \cdot \frac{\left(\sqrt[3]{i} + \left(\sqrt[3]{{i}^{4}} \cdot \frac{1}{6} + \sqrt[3]{{i}^{7}} \cdot \frac{1}{36}\right)\right) \cdot \sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)}}{i} \le -4.115528159098888 \cdot 10^{-225}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(\left(i \cdot 100\right) \cdot n\right)\\ \mathbf{elif}\;\left(\sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)} \cdot \left(n \cdot 100\right)\right) \cdot \frac{\left(\sqrt[3]{i} + \left(\sqrt[3]{{i}^{4}} \cdot \frac{1}{6} + \sqrt[3]{{i}^{7}} \cdot \frac{1}{36}\right)\right) \cdot \sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)}}{i} \le 3.704217697331561 \cdot 10^{-170}:\\ \;\;\;\;\frac{1 + {\left(\sqrt{1 + \frac{i}{n}}\right)}^{n}}{\sqrt[3]{\frac{i}{n \cdot 100}} \cdot \sqrt[3]{\frac{i}{n \cdot 100}}} \cdot \frac{{\left(\sqrt{1 + \frac{i}{n}}\right)}^{n} - 1}{\sqrt[3]{\frac{i}{n \cdot 100}}}\\ \mathbf{elif}\;\left(\sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)} \cdot \left(n \cdot 100\right)\right) \cdot \frac{\left(\sqrt[3]{i} + \left(\sqrt[3]{{i}^{4}} \cdot \frac{1}{6} + \sqrt[3]{{i}^{7}} \cdot \frac{1}{36}\right)\right) \cdot \sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)}}{i} \le 1.7815974963192015 \cdot 10^{+308}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(\left(i \cdot 100\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n \cdot 100}}\\ \end{array}\]

Original	47.1
Target	46.8
Herbie	14.3

Derivation

Split input into 4 regimes
if (* (/ (* (cbrt (+ i (* (* i i) (+ (* 1/6 i) 1/2)))) (+ (+ (* 1/6 (cbrt (pow i 4))) (* (cbrt (pow i 7)) 1/36)) (cbrt i))) i) (* (cbrt (+ i (* (+ (* i 1/6) 1/2) (* i i)))) (* 100 n))) < -1.770573189712183e+308
1. Initial program 13.3
  \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
2. Initial simplification13.3
  \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{100 \cdot n}}\]
3. Using strategy rm
4. Applied div-inv13.4
  \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\color{blue}{i \cdot \frac{1}{100 \cdot n}}}\]
5. Applied add-cube-cbrt13.4
  \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right) \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}}{i \cdot \frac{1}{100 \cdot n}}\]
6. Applied times-frac13.6
  \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i} \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{100 \cdot n}}}\]
7. Simplified13.5
  \[\leadsto \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i} \cdot \color{blue}{\left(\left(n \cdot 100\right) \cdot \sqrt[3]{{\left(\frac{i}{n} + 1\right)}^{n} - 1}\right)}\]
if -1.770573189712183e+308 < (* (/ (* (cbrt (+ i (* (* i i) (+ (* 1/6 i) 1/2)))) (+ (+ (* 1/6 (cbrt (pow i 4))) (* (cbrt (pow i 7)) 1/36)) (cbrt i))) i) (* (cbrt (+ i (* (+ (* i 1/6) 1/2) (* i i)))) (* 100 n))) < -4.115528159098888e-225 or 3.704217697331561e-170 < (* (/ (* (cbrt (+ i (* (* i i) (+ (* 1/6 i) 1/2)))) (+ (+ (* 1/6 (cbrt (pow i 4))) (* (cbrt (pow i 7)) 1/36)) (cbrt i))) i) (* (cbrt (+ i (* (+ (* i 1/6) 1/2) (* i i)))) (* 100 n))) < 1.7815974963192015e+308
1. Initial program 60.1
  \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
2. Initial simplification60.1
  \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{100 \cdot n}}\]
3. Taylor expanded around 0 25.5
  \[\leadsto \frac{\color{blue}{i + \left(\frac{1}{2} \cdot {i}^{2} + \frac{1}{6} \cdot {i}^{3}\right)}}{\frac{i}{100 \cdot n}}\]
4. Simplified25.5
  \[\leadsto \frac{\color{blue}{i + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)}}{\frac{i}{100 \cdot n}}\]
5. Using strategy rm
6. Applied *-un-lft-identity25.5
  \[\leadsto \frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)}{\color{blue}{1 \cdot \frac{i}{100 \cdot n}}}\]
7. Applied *-un-lft-identity25.5
  \[\leadsto \frac{\color{blue}{1 \cdot \left(i + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)\right)}}{1 \cdot \frac{i}{100 \cdot n}}\]
8. Applied times-frac25.5
  \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)}{\frac{i}{100 \cdot n}}}\]
9. Simplified25.5
  \[\leadsto \color{blue}{1} \cdot \frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)}{\frac{i}{100 \cdot n}}\]
10. Simplified8.3
  \[\leadsto 1 \cdot \color{blue}{\left(\left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(\left(i \cdot 100\right) \cdot n\right) + n \cdot 100\right)}\]
if -4.115528159098888e-225 < (* (/ (* (cbrt (+ i (* (* i i) (+ (* 1/6 i) 1/2)))) (+ (+ (* 1/6 (cbrt (pow i 4))) (* (cbrt (pow i 7)) 1/36)) (cbrt i))) i) (* (cbrt (+ i (* (+ (* i 1/6) 1/2) (* i i)))) (* 100 n))) < 3.704217697331561e-170
1. Initial program 28.7
  \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
2. Initial simplification28.8
  \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{100 \cdot n}}\]
3. Using strategy rm
4. Applied add-cube-cbrt28.8
  \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\color{blue}{\left(\sqrt[3]{\frac{i}{100 \cdot n}} \cdot \sqrt[3]{\frac{i}{100 \cdot n}}\right) \cdot \sqrt[3]{\frac{i}{100 \cdot n}}}}\]
5. Applied add-sqr-sqrt28.8
  \[\leadsto \frac{{\color{blue}{\left(\sqrt{1 + \frac{i}{n}} \cdot \sqrt{1 + \frac{i}{n}}\right)}}^{n} - 1}{\left(\sqrt[3]{\frac{i}{100 \cdot n}} \cdot \sqrt[3]{\frac{i}{100 \cdot n}}\right) \cdot \sqrt[3]{\frac{i}{100 \cdot n}}}\]
6. Applied unpow-prod-down28.8
  \[\leadsto \frac{\color{blue}{{\left(\sqrt{1 + \frac{i}{n}}\right)}^{n} \cdot {\left(\sqrt{1 + \frac{i}{n}}\right)}^{n}} - 1}{\left(\sqrt[3]{\frac{i}{100 \cdot n}} \cdot \sqrt[3]{\frac{i}{100 \cdot n}}\right) \cdot \sqrt[3]{\frac{i}{100 \cdot n}}}\]
7. Applied difference-of-sqr-128.8
  \[\leadsto \frac{\color{blue}{\left({\left(\sqrt{1 + \frac{i}{n}}\right)}^{n} + 1\right) \cdot \left({\left(\sqrt{1 + \frac{i}{n}}\right)}^{n} - 1\right)}}{\left(\sqrt[3]{\frac{i}{100 \cdot n}} \cdot \sqrt[3]{\frac{i}{100 \cdot n}}\right) \cdot \sqrt[3]{\frac{i}{100 \cdot n}}}\]
8. Applied times-frac28.8
  \[\leadsto \color{blue}{\frac{{\left(\sqrt{1 + \frac{i}{n}}\right)}^{n} + 1}{\sqrt[3]{\frac{i}{100 \cdot n}} \cdot \sqrt[3]{\frac{i}{100 \cdot n}}} \cdot \frac{{\left(\sqrt{1 + \frac{i}{n}}\right)}^{n} - 1}{\sqrt[3]{\frac{i}{100 \cdot n}}}}\]
if 1.7815974963192015e+308 < (* (/ (* (cbrt (+ i (* (* i i) (+ (* 1/6 i) 1/2)))) (+ (+ (* 1/6 (cbrt (pow i 4))) (* (cbrt (pow i 7)) 1/36)) (cbrt i))) i) (* (cbrt (+ i (* (+ (* i 1/6) 1/2) (* i i)))) (* 100 n)))
1. Initial program 31.2
  \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
2. Initial simplification31.3
  \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{100 \cdot n}}\]
3. Taylor expanded around inf 46.6
  \[\leadsto \frac{\color{blue}{e^{\left(\log \left(\frac{1}{n}\right) - \log \left(\frac{1}{i}\right)\right) \cdot n} - 1}}{\frac{i}{100 \cdot n}}\]
4. Simplified24.0
  \[\leadsto \frac{\color{blue}{{\left(\frac{i}{n}\right)}^{n} - 1}}{\frac{i}{100 \cdot n}}\]
Recombined 4 regimes into one program.
Final simplification14.3
\[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)} \cdot \left(n \cdot 100\right)\right) \cdot \frac{\left(\sqrt[3]{i} + \left(\sqrt[3]{{i}^{4}} \cdot \frac{1}{6} + \sqrt[3]{{i}^{7}} \cdot \frac{1}{36}\right)\right) \cdot \sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)}}{i} \le -1.770573189712183 \cdot 10^{+308}:\\ \;\;\;\;\left(\left(n \cdot 100\right) \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\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}\\ \mathbf{elif}\;\left(\sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)} \cdot \left(n \cdot 100\right)\right) \cdot \frac{\left(\sqrt[3]{i} + \left(\sqrt[3]{{i}^{4}} \cdot \frac{1}{6} + \sqrt[3]{{i}^{7}} \cdot \frac{1}{36}\right)\right) \cdot \sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)}}{i} \le -4.115528159098888 \cdot 10^{-225}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(\left(i \cdot 100\right) \cdot n\right)\\ \mathbf{elif}\;\left(\sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)} \cdot \left(n \cdot 100\right)\right) \cdot \frac{\left(\sqrt[3]{i} + \left(\sqrt[3]{{i}^{4}} \cdot \frac{1}{6} + \sqrt[3]{{i}^{7}} \cdot \frac{1}{36}\right)\right) \cdot \sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)}}{i} \le 3.704217697331561 \cdot 10^{-170}:\\ \;\;\;\;\frac{1 + {\left(\sqrt{1 + \frac{i}{n}}\right)}^{n}}{\sqrt[3]{\frac{i}{n \cdot 100}} \cdot \sqrt[3]{\frac{i}{n \cdot 100}}} \cdot \frac{{\left(\sqrt{1 + \frac{i}{n}}\right)}^{n} - 1}{\sqrt[3]{\frac{i}{n \cdot 100}}}\\ \mathbf{elif}\;\left(\sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)} \cdot \left(n \cdot 100\right)\right) \cdot \frac{\left(\sqrt[3]{i} + \left(\sqrt[3]{{i}^{4}} \cdot \frac{1}{6} + \sqrt[3]{{i}^{7}} \cdot \frac{1}{36}\right)\right) \cdot \sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)}}{i} \le 1.7815974963192015 \cdot 10^{+308}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(\left(i \cdot 100\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n \cdot 100}}\\ \end{array}\]

Error

Try it out

Target

Derivation

`if (* (/ (* (cbrt (+ i (* (* i i) (+ (* 1/6 i) 1/2)))) (+ (+ (* 1/6 (cbrt (pow i 4))) (* (cbrt (pow i 7)) 1/36)) (cbrt i))) i) (* (cbrt (+ i (* (+ (* i 1/6) 1/2) (* i i)))) (* 100 n))) < -1.770573189712183e+308`

`if -4.115528159098888e-225 < (* (/ (* (cbrt (+ i (* (* i i) (+ (* 1/6 i) 1/2)))) (+ (+ (* 1/6 (cbrt (pow i 4))) (* (cbrt (pow i 7)) 1/36)) (cbrt i))) i) (* (cbrt (+ i (* (+ (* i 1/6) 1/2) (* i i)))) (* 100 n))) < 3.704217697331561e-170`

`if 1.7815974963192015e+308 < (* (/ (* (cbrt (+ i (* (* i i) (+ (* 1/6 i) 1/2)))) (+ (+ (* 1/6 (cbrt (pow i 4))) (* (cbrt (pow i 7)) 1/36)) (cbrt i))) i) (* (cbrt (+ i (* (+ (* i 1/6) 1/2) (* i i)))) (* 100 n)))`

Runtime

In
Out