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

↓

\[\begin{array}{l} \mathbf{if}\;e^{\log \left(\frac{i}{i} \cdot \left(\left(1 + i \cdot \frac{1}{2}\right) + \left(i \cdot i\right) \cdot \frac{1}{6}\right)\right)} \le 1639.7473225334033:\\ \;\;\;\;100 \cdot \left(e^{\left(\sqrt[3]{\log \left(\frac{\frac{1}{2} \cdot {i}^{2} + \left(\frac{1}{6} \cdot {i}^{3} + i\right)}{i}\right)} \cdot \sqrt[3]{\log \left(\frac{\frac{1}{2} \cdot {i}^{2} + \left(\frac{1}{6} \cdot {i}^{3} + i\right)}{i}\right)}\right) \cdot \sqrt[3]{\log \left(\frac{\frac{1}{2} \cdot {i}^{2} + \left(\frac{1}{6} \cdot {i}^{3} + i\right)}{i}\right)}} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\sqrt[3]{\frac{i}{n}}}\\ \end{array}\]

Original	47.6
Target	46.5
Herbie	17.4

Derivation

Split input into 2 regimes
if (exp (log (* (/ i i) (+ (+ 1 (* i 1/2)) (* (* i i) 1/6))))) < 1639.7473225334033
1. Initial program 57.3
  \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
2. Taylor expanded around 0 25.4
  \[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{2} \cdot {i}^{2} + \left(\frac{1}{6} \cdot {i}^{3} + i\right)}}{\frac{i}{n}}\]
3. Using strategy rm
4. Applied associate-/r/9.8
  \[\leadsto 100 \cdot \color{blue}{\left(\frac{\frac{1}{2} \cdot {i}^{2} + \left(\frac{1}{6} \cdot {i}^{3} + i\right)}{i} \cdot n\right)}\]
5. Using strategy rm
6. Applied add-exp-log9.8
  \[\leadsto 100 \cdot \left(\color{blue}{e^{\log \left(\frac{\frac{1}{2} \cdot {i}^{2} + \left(\frac{1}{6} \cdot {i}^{3} + i\right)}{i}\right)}} \cdot n\right)\]
7. Using strategy rm
8. Applied add-cube-cbrt9.8
  \[\leadsto 100 \cdot \left(e^{\color{blue}{\left(\sqrt[3]{\log \left(\frac{\frac{1}{2} \cdot {i}^{2} + \left(\frac{1}{6} \cdot {i}^{3} + i\right)}{i}\right)} \cdot \sqrt[3]{\log \left(\frac{\frac{1}{2} \cdot {i}^{2} + \left(\frac{1}{6} \cdot {i}^{3} + i\right)}{i}\right)}\right) \cdot \sqrt[3]{\log \left(\frac{\frac{1}{2} \cdot {i}^{2} + \left(\frac{1}{6} \cdot {i}^{3} + i\right)}{i}\right)}}} \cdot n\right)\]
if 1639.7473225334033 < (exp (log (* (/ i i) (+ (+ 1 (* i 1/2)) (* (* i i) 1/6)))))
1. Initial program 30.5
  \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
2. Using strategy rm
3. Applied add-cube-cbrt30.8
  \[\leadsto 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\color{blue}{\left(\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}\right) \cdot \sqrt[3]{\frac{i}{n}}}}\]
4. Applied add-cube-cbrt30.8
  \[\leadsto 100 \cdot \frac{\color{blue}{\left(\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right) \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}}{\left(\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}\right) \cdot \sqrt[3]{\frac{i}{n}}}\]
5. Applied times-frac30.8
  \[\leadsto 100 \cdot \color{blue}{\left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}} \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\sqrt[3]{\frac{i}{n}}}\right)}\]
6. Applied associate-*r*30.8
  \[\leadsto \color{blue}{\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\sqrt[3]{\frac{i}{n}}}}\]
Recombined 2 regimes into one program.

Error

Target

Derivation

`if (exp (log (* (/ i i) (+ (+ 1 (* i 1/2)) (* (* i i) 1/6))))) < 1639.7473225334033`

`if 1639.7473225334033 < (exp (log (* (/ i i) (+ (+ 1 (* i 1/2)) (* (* i i) 1/6)))))`

Runtime