Compound Interest

Average Error: 47.6 → 17.6

Time: 15.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -0.0016013410645367089:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{-i}\right) \cdot \left(-n\right)\\ \mathbf{elif}\;i \le -1.03280553278684268 \cdot 10^{-218}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{i}\right) \cdot \frac{\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 1.11578736812586826 \cdot 10^{43}:\\ \;\;\;\;\frac{100 \cdot \left(1 \cdot i + 0.5 \cdot {i}^{2}\right) + \left(\left(\log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)\right) \cdot 1\right) \cdot 100}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)\right) \cdot 100}{\frac{i}{n} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)}\\ \end{array}\]

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 <= -0.001601341064536709)) {
		VAR = ((double) (((double) (100.0 * ((double) (((double) (((double) pow(((double) (1.0 + ((double) (i / n)))), n)) - 1.0)) / ((double) -(i)))))) * ((double) -(n))));
	} else {
		double VAR_1;
		if ((i <= -1.0328055327868427e-218)) {
			VAR_1 = ((double) (((double) (100.0 * ((double) (((double) (((double) cbrt(((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) cbrt(((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)))) * ((double) (((double) cbrt(((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_2;
			if ((i <= 1.1157873681258683e+43)) {
				VAR_2 = ((double) (((double) (((double) (((double) (100.0 * ((double) (((double) (1.0 * i)) + ((double) (0.5 * ((double) pow(i, 2.0)))))))) + ((double) (((double) (((double) (((double) log(1.0)) * ((double) (n - ((double) (0.5 * ((double) pow(i, 2.0)))))))) * 1.0)) * 100.0)))) / i)) * n));
			} else {
				VAR_2 = ((double) (((double) (((double) (((double) pow(((double) (1.0 + ((double) (i / n)))), ((double) (2.0 * n)))) + ((double) -(((double) (1.0 * 1.0)))))) * 100.0)) / ((double) (((double) (i / n)) * ((double) (((double) pow(((double) (1.0 + ((double) (i / n)))), n)) + 1.0))))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus i

Bits error versus n

Target

Original	47.6
Target	47.1
Herbie	17.6

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

Derivation

1. Split input into 4 regimes

2. if i < -0.001601341064536709

   1. Initial program 28.3

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

   3. Applied frac-2neg 28.3

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

   4. Applied associate-/r/ 29.1

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

   5. Applied associate-*r* 29.1

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

   if -0.001601341064536709 < i < -1.0328055327868427e-218

   1. Initial program 56.2

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

   2. Taylor expanded around 0 24.5

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

   4. Applied div-inv 24.6

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

   5. Applied add-cube-cbrt 25.3

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

   6. Applied times-frac 11.7

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

   7. Applied associate-*r* 11.8

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

   if -1.0328055327868427e-218 < i < 1.1157873681258683e+43

   1. Initial program 57.6

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

   2. Taylor expanded around 0 28.1

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

   4. Applied associate-/r/ 10.1

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

   5. Applied associate-*r* 10.1

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

   6. Simplified 10.2

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

   8. Applied associate-+r+ 10.2

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

   9. Applied associate--l+ 10.2

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

   10. Applied distribute-lft-in 10.2

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

   11. Simplified 10.2

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

   if 1.1157873681258683e+43 < i

   1. Initial program 32.9

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

   3. Applied flip-- 33.1

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

   4. Applied associate-/l/ 33.1

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

   5. Applied associate-*r/ 33.0

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

   6. Simplified 33.0

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

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.0016013410645367089:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{-i}\right) \cdot \left(-n\right)\\ \mathbf{elif}\;i \le -1.03280553278684268 \cdot 10^{-218}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{i}\right) \cdot \frac{\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 1.11578736812586826 \cdot 10^{43}:\\ \;\;\;\;\frac{100 \cdot \left(1 \cdot i + 0.5 \cdot {i}^{2}\right) + \left(\left(\log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)\right) \cdot 1\right) \cdot 100}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)\right) \cdot 100}{\frac{i}{n} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :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))))