Compound Interest

Average Error: 47.6 → 14.9

Time: 14.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -8.911795586685324 \cdot 10^{-08}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(1 + \frac{i}{n}\right)}^{n}}}{i}\right)\\ \mathbf{elif}\;i \leq -1.060168546551644 \cdot 10^{-187}:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot \left(i \cdot 1 + \left(n \cdot \log 1 + \left(i \cdot i\right) \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)\right)\right)}{i}\\ \mathbf{elif}\;i \leq 10.47714617646152:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\left(\sqrt[3]{\frac{i \cdot 1 + \left(n \cdot \log 1 + \left(i \cdot i\right) \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}} \cdot \sqrt[3]{\frac{i \cdot 1 + \left(n \cdot \log 1 + \left(i \cdot i\right) \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\frac{n \cdot \log 1 + i \cdot \left(1 + i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{n \cdot \log 1 + i \cdot \left(1 + i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}}} \cdot \sqrt[3]{\sqrt[3]{\frac{n \cdot \log 1 + i \cdot \left(1 + i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;49.99999999999999 \cdot \left(\frac{{\left(\log 1\right)}^{2}}{i} \cdot \left(\log i \cdot {n}^{4}\right)\right) + \left(50 \cdot \left(\frac{{\left(\log 1\right)}^{2}}{i} \cdot {n}^{3}\right) + \left(\left(16.666666666666664 \cdot \left({n}^{4} \cdot \frac{{\left(\log 1\right)}^{3}}{i}\right) + \left(100 \cdot \left(\frac{\log 1}{i} \cdot \left(n \cdot n\right)\right) + \left(\left(49.99999999999999 \cdot \left(\frac{\log 1}{i} \cdot \left({n}^{4} \cdot {\left(\log i\right)}^{2}\right)\right) + \left(16.666666666666664 \cdot \left({n}^{4} \cdot \frac{{\left(\log i\right)}^{3}}{i}\right) + \left(50 \cdot \left(\frac{\log i}{i} \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)\right) + \left(100 \cdot \left(\left(n \cdot n\right) \cdot \frac{\log i}{i}\right) + \left(50 \cdot \left({\left(\log n\right)}^{2} \cdot \frac{{n}^{3}}{i}\right) + 100 \cdot \left(\frac{\log 1}{i} \cdot \left(\log i \cdot {n}^{3}\right)\right)\right)\right)\right)\right)\right) + 50 \cdot \left(\frac{\log 1}{i} \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right) + {n}^{3} \cdot \frac{{\left(\log i\right)}^{2}}{i}\right)\right)\right)\right) - \left(16.666666666666664 \cdot \left(\frac{{n}^{4}}{i} \cdot {\left(\log n\right)}^{3}\right) + \left(\left(50 \cdot \left(\frac{{\left(\log i\right)}^{2}}{i} \cdot \left({n}^{4} \cdot \log n\right)\right) + \left(50 \cdot \left(\frac{{\left(\log 1\right)}^{2}}{i} \cdot \left({n}^{4} \cdot \log n\right)\right) + 100 \cdot \left(\frac{\log 1}{i} \cdot \left({n}^{3} \cdot \log n\right) + \frac{\log i}{i} \cdot \left({n}^{3} \cdot \log n\right)\right)\right)\right) + 100 \cdot \left(\log n \cdot \frac{n \cdot n}{i} + \frac{\log 1}{i} \cdot \left(\left(\log i \cdot {n}^{4}\right) \cdot \log n\right)\right)\right)\right)\right)\right)\\ \end{array}\]

C

double code(double i, double n) {
	return ((double) (100.0 * (((double) (((double) pow(((double) (1.0 + (i / n))), n)) - 1.0)) / (i / n))));
}

↓

double code(double i, double n) {
	double VAR;
	if ((i <= -8.911795586685324e-08)) {
		VAR = ((double) (100.0 * ((double) (n * ((((double) (((double) pow(((double) (1.0 + (i / n))), ((double) (n * 2.0)))) - ((double) (1.0 * 1.0)))) / ((double) (1.0 + ((double) pow(((double) (1.0 + (i / n))), n))))) / i)))));
	} else {
		double VAR_1;
		if ((i <= -1.060168546551644e-187)) {
			VAR_1 = (((double) (n * ((double) (100.0 * ((double) (((double) (i * 1.0)) + ((double) (((double) (n * ((double) log(1.0)))) + ((double) (((double) (i * i)) * ((double) (0.5 - ((double) (((double) log(1.0)) * 0.5)))))))))))))) / i);
		} else {
			double VAR_2;
			if ((i <= 10.47714617646152)) {
				VAR_2 = ((double) (100.0 * ((double) (n * ((double) (((double) (((double) cbrt((((double) (((double) (i * 1.0)) + ((double) (((double) (n * ((double) log(1.0)))) + ((double) (((double) (i * i)) * ((double) (0.5 - ((double) (((double) log(1.0)) * 0.5)))))))))) / i))) * ((double) cbrt((((double) (((double) (i * 1.0)) + ((double) (((double) (n * ((double) log(1.0)))) + ((double) (((double) (i * i)) * ((double) (0.5 - ((double) (((double) log(1.0)) * 0.5)))))))))) / i))))) * ((double) (((double) cbrt(((double) cbrt((((double) (((double) (n * ((double) log(1.0)))) + ((double) (i * ((double) (1.0 + ((double) (i * ((double) (0.5 - ((double) (((double) log(1.0)) * 0.5)))))))))))) / i))))) * ((double) (((double) cbrt(((double) cbrt((((double) (((double) (n * ((double) log(1.0)))) + ((double) (i * ((double) (1.0 + ((double) (i * ((double) (0.5 - ((double) (((double) log(1.0)) * 0.5)))))))))))) / i))))) * ((double) cbrt(((double) cbrt((((double) (((double) (n * ((double) log(1.0)))) + ((double) (i * ((double) (1.0 + ((double) (i * ((double) (0.5 - ((double) (((double) log(1.0)) * 0.5)))))))))))) / i)))))))))))))));
			} else {
				VAR_2 = ((double) (((double) (49.99999999999999 * ((double) ((((double) pow(((double) log(1.0)), 2.0)) / i) * ((double) (((double) log(i)) * ((double) pow(n, 4.0)))))))) + ((double) (((double) (50.0 * ((double) ((((double) pow(((double) log(1.0)), 2.0)) / i) * ((double) pow(n, 3.0)))))) + ((double) (((double) (((double) (16.666666666666664 * ((double) (((double) pow(n, 4.0)) * (((double) pow(((double) log(1.0)), 3.0)) / i))))) + ((double) (((double) (100.0 * ((double) ((((double) log(1.0)) / i) * ((double) (n * n)))))) + ((double) (((double) (((double) (49.99999999999999 * ((double) ((((double) log(1.0)) / i) * ((double) (((double) pow(n, 4.0)) * ((double) pow(((double) log(i)), 2.0)))))))) + ((double) (((double) (16.666666666666664 * ((double) (((double) pow(n, 4.0)) * (((double) pow(((double) log(i)), 3.0)) / i))))) + ((double) (((double) (50.0 * ((double) ((((double) log(i)) / i) * ((double) (((double) pow(n, 4.0)) * ((double) pow(((double) log(n)), 2.0)))))))) + ((double) (((double) (100.0 * ((double) (((double) (n * n)) * (((double) log(i)) / i))))) + ((double) (((double) (50.0 * ((double) (((double) pow(((double) log(n)), 2.0)) * (((double) pow(n, 3.0)) / i))))) + ((double) (100.0 * ((double) ((((double) log(1.0)) / i) * ((double) (((double) log(i)) * ((double) pow(n, 3.0)))))))))))))))))) + ((double) (50.0 * ((double) (((double) ((((double) log(1.0)) / i) * ((double) (((double) pow(n, 4.0)) * ((double) pow(((double) log(n)), 2.0)))))) + ((double) (((double) pow(n, 3.0)) * (((double) pow(((double) log(i)), 2.0)) / i))))))))))))) - ((double) (((double) (16.666666666666664 * ((double) ((((double) pow(n, 4.0)) / i) * ((double) pow(((double) log(n)), 3.0)))))) + ((double) (((double) (((double) (50.0 * ((double) ((((double) pow(((double) log(i)), 2.0)) / i) * ((double) (((double) pow(n, 4.0)) * ((double) log(n)))))))) + ((double) (((double) (50.0 * ((double) ((((double) pow(((double) log(1.0)), 2.0)) / i) * ((double) (((double) pow(n, 4.0)) * ((double) log(n)))))))) + ((double) (100.0 * ((double) (((double) ((((double) log(1.0)) / i) * ((double) (((double) pow(n, 3.0)) * ((double) log(n)))))) + ((double) ((((double) log(i)) / i) * ((double) (((double) pow(n, 3.0)) * ((double) log(n)))))))))))))) + ((double) (100.0 * ((double) (((double) (((double) log(n)) * (((double) (n * n)) / i))) + ((double) ((((double) log(1.0)) / i) * ((double) (((double) (((double) log(i)) * ((double) pow(n, 4.0)))) * ((double) log(n))))))))))))))))))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus i

Bits error versus n

Target

Original	47.6
Target	47.6
Herbie	14.9

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

Derivation

1. Split input into 4 regimes

2. if i < -8.911795586685324e-8

   1. Initial program Error: 28.4 bits

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

   2. Simplified Error: 28.8 bits

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

   4. Applied flip-- Error: 28.8 bits

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

   5. Simplified Error: 28.8 bits

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

   6. Simplified Error: 28.8 bits

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

   if -8.911795586685324e-8 < i < -1.060168546551644e-187

   1. Initial program Error: 55.1 bits

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

   2. Simplified Error: 54.9 bits

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

   3. Taylor expanded around 0 Error: 14.0 bits

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

   4. Simplified Error: 14.0 bits

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

   6. Applied associate-*r/ Error: 10.3 bits

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

   7. Applied associate-*r/ Error: 10.5 bits

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

   8. Simplified Error: 10.5 bits

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

   if -1.060168546551644e-187 < i < 10.47714617646152

   1. Initial program Error: 59.2 bits

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

   2. Simplified Error: 58.8 bits

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

   3. Taylor expanded around 0 Error: 7.6 bits

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

   4. Simplified Error: 7.6 bits

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

   6. Applied add-cube-cbrt Error: 7.7 bits

      \[\leadsto 100 \cdot \left(n \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1 \cdot i + \left(n \cdot \log 1 + \left(i \cdot \left(i \cdot 0.5\right) - \left(i \cdot i\right) \cdot \left(0.5 \cdot \log 1\right)\right)\right)}{i}} \cdot \sqrt[3]{\frac{1 \cdot i + \left(n \cdot \log 1 + \left(i \cdot \left(i \cdot 0.5\right) - \left(i \cdot i\right) \cdot \left(0.5 \cdot \log 1\right)\right)\right)}{i}}\right) \cdot \sqrt[3]{\frac{1 \cdot i + \left(n \cdot \log 1 + \left(i \cdot \left(i \cdot 0.5\right) - \left(i \cdot i\right) \cdot \left(0.5 \cdot \log 1\right)\right)\right)}{i}}\right)}\right)\]

   7. Simplified Error: 7.7 bits

      \[\leadsto 100 \cdot \left(n \cdot \left(\color{blue}{\left(\sqrt[3]{\frac{i \cdot 1 + \left(\log 1 \cdot n + \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)}{i}} \cdot \sqrt[3]{\frac{i \cdot 1 + \left(\log 1 \cdot n + \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)}{i}}\right)} \cdot \sqrt[3]{\frac{1 \cdot i + \left(n \cdot \log 1 + \left(i \cdot \left(i \cdot 0.5\right) - \left(i \cdot i\right) \cdot \left(0.5 \cdot \log 1\right)\right)\right)}{i}}\right)\right)\]

   8. Simplified Error: 7.7 bits

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

   10. Applied add-cube-cbrt Error: 7.7 bits

       \[\leadsto 100 \cdot \left(n \cdot \left(\left(\sqrt[3]{\frac{i \cdot 1 + \left(\log 1 \cdot n + \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)}{i}} \cdot \sqrt[3]{\frac{i \cdot 1 + \left(\log 1 \cdot n + \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)}{i}}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\frac{i \cdot 1 + \left(\log 1 \cdot n + \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)}{i}}} \cdot \sqrt[3]{\sqrt[3]{\frac{i \cdot 1 + \left(\log 1 \cdot n + \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)}{i}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{i \cdot 1 + \left(\log 1 \cdot n + \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)}{i}}}\right)}\right)\right)\]

   11. Simplified Error: 7.7 bits

       \[\leadsto 100 \cdot \left(n \cdot \left(\left(\sqrt[3]{\frac{i \cdot 1 + \left(\log 1 \cdot n + \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)}{i}} \cdot \sqrt[3]{\frac{i \cdot 1 + \left(\log 1 \cdot n + \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)}{i}}\right) \cdot \left(\color{blue}{\left(\sqrt[3]{\sqrt[3]{\frac{\log 1 \cdot n + i \cdot \left(1 + i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\log 1 \cdot n + i \cdot \left(1 + i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}}}\right)} \cdot \sqrt[3]{\sqrt[3]{\frac{i \cdot 1 + \left(\log 1 \cdot n + \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)}{i}}}\right)\right)\right)\]

   12. Simplified Error: 7.7 bits

       \[\leadsto 100 \cdot \left(n \cdot \left(\left(\sqrt[3]{\frac{i \cdot 1 + \left(\log 1 \cdot n + \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)}{i}} \cdot \sqrt[3]{\frac{i \cdot 1 + \left(\log 1 \cdot n + \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)}{i}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{\log 1 \cdot n + i \cdot \left(1 + i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\log 1 \cdot n + i \cdot \left(1 + i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}}}\right) \cdot \color{blue}{\sqrt[3]{\sqrt[3]{\frac{\log 1 \cdot n + i \cdot \left(1 + i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}}}}\right)\right)\right)\]

   if 10.47714617646152 < i

   1. Initial program Error: 29.4 bits

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

   2. Simplified Error: 29.4 bits

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

   3. Taylor expanded around 0 Error: 22.3 bits

      \[\leadsto \color{blue}{\left(49.99999999999999 \cdot \frac{{\left(\log 1\right)}^{2} \cdot \left(\log i \cdot {n}^{4}\right)}{i} + \left(50 \cdot \frac{{\left(\log 1\right)}^{2} \cdot {n}^{3}}{i} + \left(16.666666666666664 \cdot \frac{{\left(\log 1\right)}^{3} \cdot {n}^{4}}{i} + \left(100 \cdot \frac{\log 1 \cdot {n}^{2}}{i} + \left(50 \cdot \frac{\log 1 \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)}{i} + \left(50 \cdot \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i} + \left(49.99999999999999 \cdot \frac{\log 1 \cdot \left({\left(\log i\right)}^{2} \cdot {n}^{4}\right)}{i} + \left(16.666666666666664 \cdot \frac{{\left(\log i\right)}^{3} \cdot {n}^{4}}{i} + \left(50 \cdot \frac{\log i \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)}{i} + \left(100 \cdot \frac{\log i \cdot {n}^{2}}{i} + \left(50 \cdot \frac{{n}^{3} \cdot {\left(\log n\right)}^{2}}{i} + 100 \cdot \frac{\log 1 \cdot \left(\log i \cdot {n}^{3}\right)}{i}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(16.666666666666664 \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(100 \cdot \frac{{n}^{2} \cdot \log n}{i} + \left(100 \cdot \frac{\log 1 \cdot \left(\log i \cdot \left({n}^{4} \cdot \log n\right)\right)}{i} + \left(50 \cdot \frac{{\left(\log i\right)}^{2} \cdot \left({n}^{4} \cdot \log n\right)}{i} + \left(100 \cdot \frac{\log 1 \cdot \left({n}^{3} \cdot \log n\right)}{i} + \left(100 \cdot \frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i} + 50 \cdot \frac{{\left(\log 1\right)}^{2} \cdot \left({n}^{4} \cdot \log n\right)}{i}\right)\right)\right)\right)\right)\right)}\]

   4. Simplified Error: 22.4 bits

      \[\leadsto \color{blue}{49.99999999999999 \cdot \left(\frac{{\left(\log 1\right)}^{2}}{i} \cdot \left(\log i \cdot {n}^{4}\right)\right) + \left(50 \cdot \left(\frac{{\left(\log 1\right)}^{2}}{i} \cdot {n}^{3}\right) + \left(\left(16.666666666666664 \cdot \left(\frac{{\left(\log 1\right)}^{3}}{i} \cdot {n}^{4}\right) + \left(100 \cdot \left(\frac{\log 1}{i} \cdot \left(n \cdot n\right)\right) + \left(\left(49.99999999999999 \cdot \left(\frac{\log 1}{i} \cdot \left({n}^{4} \cdot {\left(\log i\right)}^{2}\right)\right) + \left(16.666666666666664 \cdot \left(\frac{{\left(\log i\right)}^{3}}{i} \cdot {n}^{4}\right) + \left(50 \cdot \left(\frac{\log i}{i} \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)\right) + \left(100 \cdot \left(\frac{\log i}{i} \cdot \left(n \cdot n\right)\right) + \left(50 \cdot \left(\frac{{n}^{3}}{i} \cdot {\left(\log n\right)}^{2}\right) + 100 \cdot \left(\frac{\log 1}{i} \cdot \left(\log i \cdot {n}^{3}\right)\right)\right)\right)\right)\right)\right) + 50 \cdot \left(\frac{\log 1}{i} \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right) + \frac{{\left(\log i\right)}^{2}}{i} \cdot {n}^{3}\right)\right)\right)\right) - \left(16.666666666666664 \cdot \left(\frac{{n}^{4}}{i} \cdot {\left(\log n\right)}^{3}\right) + \left(\left(50 \cdot \left(\frac{{\left(\log i\right)}^{2}}{i} \cdot \left(\log n \cdot {n}^{4}\right)\right) + \left(50 \cdot \left(\frac{{\left(\log 1\right)}^{2}}{i} \cdot \left(\log n \cdot {n}^{4}\right)\right) + 100 \cdot \left(\frac{\log 1}{i} \cdot \left(\log n \cdot {n}^{3}\right) + \frac{\log i}{i} \cdot \left(\log n \cdot {n}^{3}\right)\right)\right)\right) + 100 \cdot \left(\frac{n \cdot n}{i} \cdot \log n + \frac{\log 1}{i} \cdot \left(\log n \cdot \left(\log i \cdot {n}^{4}\right)\right)\right)\right)\right)\right)\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification Error: 14.9 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -8.911795586685324 \cdot 10^{-08}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(1 + \frac{i}{n}\right)}^{n}}}{i}\right)\\ \mathbf{elif}\;i \leq -1.060168546551644 \cdot 10^{-187}:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot \left(i \cdot 1 + \left(n \cdot \log 1 + \left(i \cdot i\right) \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)\right)\right)}{i}\\ \mathbf{elif}\;i \leq 10.47714617646152:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\left(\sqrt[3]{\frac{i \cdot 1 + \left(n \cdot \log 1 + \left(i \cdot i\right) \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}} \cdot \sqrt[3]{\frac{i \cdot 1 + \left(n \cdot \log 1 + \left(i \cdot i\right) \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\frac{n \cdot \log 1 + i \cdot \left(1 + i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{n \cdot \log 1 + i \cdot \left(1 + i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}}} \cdot \sqrt[3]{\sqrt[3]{\frac{n \cdot \log 1 + i \cdot \left(1 + i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;49.99999999999999 \cdot \left(\frac{{\left(\log 1\right)}^{2}}{i} \cdot \left(\log i \cdot {n}^{4}\right)\right) + \left(50 \cdot \left(\frac{{\left(\log 1\right)}^{2}}{i} \cdot {n}^{3}\right) + \left(\left(16.666666666666664 \cdot \left({n}^{4} \cdot \frac{{\left(\log 1\right)}^{3}}{i}\right) + \left(100 \cdot \left(\frac{\log 1}{i} \cdot \left(n \cdot n\right)\right) + \left(\left(49.99999999999999 \cdot \left(\frac{\log 1}{i} \cdot \left({n}^{4} \cdot {\left(\log i\right)}^{2}\right)\right) + \left(16.666666666666664 \cdot \left({n}^{4} \cdot \frac{{\left(\log i\right)}^{3}}{i}\right) + \left(50 \cdot \left(\frac{\log i}{i} \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)\right) + \left(100 \cdot \left(\left(n \cdot n\right) \cdot \frac{\log i}{i}\right) + \left(50 \cdot \left({\left(\log n\right)}^{2} \cdot \frac{{n}^{3}}{i}\right) + 100 \cdot \left(\frac{\log 1}{i} \cdot \left(\log i \cdot {n}^{3}\right)\right)\right)\right)\right)\right)\right) + 50 \cdot \left(\frac{\log 1}{i} \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right) + {n}^{3} \cdot \frac{{\left(\log i\right)}^{2}}{i}\right)\right)\right)\right) - \left(16.666666666666664 \cdot \left(\frac{{n}^{4}}{i} \cdot {\left(\log n\right)}^{3}\right) + \left(\left(50 \cdot \left(\frac{{\left(\log i\right)}^{2}}{i} \cdot \left({n}^{4} \cdot \log n\right)\right) + \left(50 \cdot \left(\frac{{\left(\log 1\right)}^{2}}{i} \cdot \left({n}^{4} \cdot \log n\right)\right) + 100 \cdot \left(\frac{\log 1}{i} \cdot \left({n}^{3} \cdot \log n\right) + \frac{\log i}{i} \cdot \left({n}^{3} \cdot \log n\right)\right)\right)\right) + 100 \cdot \left(\log n \cdot \frac{n \cdot n}{i} + \frac{\log 1}{i} \cdot \left(\left(\log i \cdot {n}^{4}\right) \cdot \log n\right)\right)\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(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))))