Compound Interest

Average Error: 47.8 → 17.2

Time: 19.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -3.2254347733272635 \cdot 10^{-28}:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 4.0155833099615571 \cdot 10^{-213}:\\ \;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 23.786737204236545:\\ \;\;\;\;\frac{100}{i} \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 1.32981715900235243 \cdot 10^{143}:\\ \;\;\;\;100 \cdot \left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{\sqrt[3]{i} \cdot \sqrt[3]{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{\sqrt[3]{i}}{\sqrt[3]{n}}}\right)\\ \mathbf{elif}\;i \le 3.15042717040624542 \cdot 10^{243}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \end{array}\]

C

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

↓

double code(double i, double n) {
	double VAR;
	if ((i <= -3.2254347733272635e-28)) {
		VAR = ((double) (100.0 * ((double) (((double) (((double) pow(((double) (1.0 + ((double) (i / n)))), n)) - 1.0)) / ((double) (i / n))))));
	} else {
		double VAR_1;
		if ((i <= 4.015583309961557e-213)) {
			VAR_1 = ((double) (100.0 * ((double) (((double) (((double) (((double) (((double) (1.0 * i)) + ((double) (((double) (0.5 * ((double) pow(i, 2.0)))) + ((double) (((double) log(1.0)) * n)))))) - ((double) (0.5 * ((double) (((double) pow(i, 2.0)) * ((double) log(1.0)))))))) / i)) * n))));
		} else {
			double VAR_2;
			if ((i <= 23.786737204236545)) {
				VAR_2 = ((double) (((double) (100.0 / i)) * ((double) (((double) (((double) (((double) (1.0 * i)) + ((double) (((double) (0.5 * ((double) pow(i, 2.0)))) + ((double) (((double) log(1.0)) * n)))))) - ((double) (0.5 * ((double) (((double) pow(i, 2.0)) * ((double) log(1.0)))))))) / ((double) (1.0 / n))))));
			} else {
				double VAR_3;
				if ((i <= 1.3298171590023524e+143)) {
					VAR_3 = ((double) (100.0 * ((double) (((double) (((double) (((double) cbrt(((double) (((double) pow(((double) (1.0 + ((double) (i / n)))), n)) - 1.0)))) * ((double) cbrt(((double) (((double) pow(((double) (1.0 + ((double) (i / n)))), n)) - 1.0)))))) / ((double) (((double) (((double) cbrt(i)) * ((double) cbrt(i)))) / ((double) (((double) cbrt(n)) * ((double) cbrt(n)))))))) * ((double) (((double) cbrt(((double) (((double) pow(((double) (1.0 + ((double) (i / n)))), n)) - 1.0)))) / ((double) (((double) cbrt(i)) / ((double) cbrt(n))))))))));
				} else {
					double VAR_4;
					if ((i <= 3.1504271704062454e+243)) {
						VAR_4 = ((double) (100.0 * ((double) (((double) (((double) (((double) (1.0 * i)) + ((double) (((double) (((double) log(1.0)) * n)) + 1.0)))) - 1.0)) / ((double) (i / n))))));
					} else {
						VAR_4 = ((double) (((double) (100.0 * ((double) (((double) (((double) pow(((double) (1.0 + ((double) (i / n)))), n)) - 1.0)) / i)))) * n));
					}
					VAR_3 = VAR_4;
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus i

Bits error versus n

Target

Original	47.8
Target	47.7
Herbie	17.2

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

Derivation

1. Split input into 6 regimes

2. if i < -3.2254347733272635e-28

   1. Initial program 30.4

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

   if -3.2254347733272635e-28 < i < 4.0155833099615571e-213

   1. Initial program 59.6

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

   2. Taylor expanded around 0 27.7

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

   4. Applied associate-/r/ 6.8

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

   if 4.0155833099615571e-213 < i < 23.786737204236545

   1. Initial program 56.5

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

   2. Taylor expanded around 0 25.1

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

   4. Applied div-inv 25.1

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

   5. Applied *-un-lft-identity 25.1

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

   6. Applied times-frac 11.1

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

   7. Applied associate-*r* 11.2

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

   8. Simplified 11.2

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

   if 23.786737204236545 < i < 1.32981715900235243e143

   1. Initial program 29.5

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

   3. Applied add-cube-cbrt 29.7

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

   4. Applied add-cube-cbrt 29.7

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

   5. Applied times-frac 29.7

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

   6. Applied add-cube-cbrt 29.7

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

   7. Applied times-frac 29.7

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

   if 1.32981715900235243e143 < i < 3.15042717040624542e243

   1. Initial program 33.6

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

   2. Taylor expanded around 0 35.6

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

   if 3.15042717040624542e243 < i

   1. Initial program 33.3

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

   3. Applied associate-/r/ 33.3

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

   4. Applied associate-*r* 33.3

      \[\leadsto \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n}\]
3. Recombined 6 regimes into one program.

4. Final simplification 17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -3.2254347733272635 \cdot 10^{-28}:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 4.0155833099615571 \cdot 10^{-213}:\\ \;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 23.786737204236545:\\ \;\;\;\;\frac{100}{i} \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 1.32981715900235243 \cdot 10^{143}:\\ \;\;\;\;100 \cdot \left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{\sqrt[3]{i} \cdot \sqrt[3]{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{\sqrt[3]{i}}{\sqrt[3]{n}}}\right)\\ \mathbf{elif}\;i \le 3.15042717040624542 \cdot 10^{243}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2020169 
(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))))