Compound Interest

Average Error: 47.3 → 16.7

Time: 13.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -5.932755958531832 \cdot 10^{-10}:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.3385135141275714 \cdot 10^{-16}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i \cdot 1 + \left(n \cdot \log 1 + \left(i \cdot \left(i \cdot 0.5\right) - \left(i \cdot i\right) \cdot \left(\log 1 \cdot 0.5\right)\right)\right)}{i}\right)\\ \mathbf{elif}\;i \leq 5.460223498175969 \cdot 10^{+181}:\\ \;\;\;\;100 \cdot \left(\left(\left(\frac{\log 1}{i} \cdot \left(n \cdot n\right) + \left(\left(\left(\frac{\log 1}{i} \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)\right) \cdot 0.5 + \left(\left(\left({n}^{4} \cdot \frac{{\left(\log i\right)}^{3}}{i}\right) \cdot 0.16666666666666666 + \left(\left(n \cdot n\right) \cdot \frac{\log i}{i} + \left(\left(\frac{\log 1}{i} \cdot \left(\log i \cdot {n}^{3}\right) + 0.5 \cdot \left({\left(\log n\right)}^{2} \cdot \frac{{n}^{3}}{i}\right)\right) + \frac{\left({n}^{4} \cdot {\left(\log n\right)}^{2}\right) \cdot \log \left(\sqrt{i}\right)}{i}\right)\right)\right) + 0.5 \cdot \left({n}^{3} \cdot \frac{{\left(\log i\right)}^{2}}{i} + \frac{\log 1}{i} \cdot \left({n}^{4} \cdot {\left(\log i\right)}^{2}\right)\right)\right)\right) + 0.16666666666666666 \cdot \left({n}^{4} \cdot \frac{{\left(\log 1\right)}^{3}}{i}\right)\right)\right) + 0.5 \cdot \left(\frac{{\left(\log 1\right)}^{2}}{i} \cdot \left({n}^{4} \cdot \log i\right) + {n}^{3} \cdot \frac{{\left(\log 1\right)}^{2}}{i}\right)\right) + \left(\left(\frac{{n}^{4}}{i} \cdot {\left(\log n\right)}^{3}\right) \cdot -0.16666666666666666 - \left(\log n \cdot \frac{n \cdot n}{i} + \left(\frac{\log 1}{i} \cdot \left(\log n \cdot \left({n}^{4} \cdot \log i\right)\right) + \left(\frac{\log 1}{i} \cdot \left(\log n \cdot {n}^{3}\right) + \left(\left(\frac{\log i}{i} \cdot \left(\log n \cdot {n}^{3}\right) + 0.5 \cdot \left(\frac{{\left(\log 1\right)}^{2}}{i} \cdot \left({n}^{4} \cdot \log n\right)\right)\right) + 0.5 \cdot \left(\frac{{\left(\log i\right)}^{2}}{i} \cdot \left({n}^{4} \cdot \log n\right)\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\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 <= -5.932755958531832e-10)) {
		VAR = ((double) (100.0 * (((double) (((double) pow(((double) (1.0 + (i / n))), n)) - 1.0)) / (i / n))));
	} else {
		double VAR_1;
		if ((i <= 2.3385135141275714e-16)) {
			VAR_1 = ((double) (100.0 * ((double) (n * (((double) (((double) (i * 1.0)) + ((double) (((double) (n * ((double) log(1.0)))) + ((double) (((double) (i * ((double) (i * 0.5)))) - ((double) (((double) (i * i)) * ((double) (((double) log(1.0)) * 0.5)))))))))) / i)))));
		} else {
			double VAR_2;
			if ((i <= 5.460223498175969e+181)) {
				VAR_2 = ((double) (100.0 * ((double) (((double) (((double) (((double) ((((double) log(1.0)) / i) * ((double) (n * n)))) + ((double) (((double) (((double) (((double) ((((double) log(1.0)) / i) * ((double) (((double) pow(n, 4.0)) * ((double) pow(((double) log(n)), 2.0)))))) * 0.5)) + ((double) (((double) (((double) (((double) (((double) pow(n, 4.0)) * (((double) pow(((double) log(i)), 3.0)) / i))) * 0.16666666666666666)) + ((double) (((double) (((double) (n * n)) * (((double) log(i)) / i))) + ((double) (((double) (((double) ((((double) log(1.0)) / i) * ((double) (((double) log(i)) * ((double) pow(n, 3.0)))))) + ((double) (0.5 * ((double) (((double) pow(((double) log(n)), 2.0)) * (((double) pow(n, 3.0)) / i))))))) + (((double) (((double) (((double) pow(n, 4.0)) * ((double) pow(((double) log(n)), 2.0)))) * ((double) log(((double) sqrt(i)))))) / i))))))) + ((double) (0.5 * ((double) (((double) (((double) pow(n, 3.0)) * (((double) pow(((double) log(i)), 2.0)) / i))) + ((double) ((((double) log(1.0)) / i) * ((double) (((double) pow(n, 4.0)) * ((double) pow(((double) log(i)), 2.0)))))))))))))) + ((double) (0.16666666666666666 * ((double) (((double) pow(n, 4.0)) * (((double) pow(((double) log(1.0)), 3.0)) / i))))))))) + ((double) (0.5 * ((double) (((double) ((((double) pow(((double) log(1.0)), 2.0)) / i) * ((double) (((double) pow(n, 4.0)) * ((double) log(i)))))) + ((double) (((double) pow(n, 3.0)) * (((double) pow(((double) log(1.0)), 2.0)) / i))))))))) + ((double) (((double) (((double) ((((double) pow(n, 4.0)) / i) * ((double) pow(((double) log(n)), 3.0)))) * -0.16666666666666666)) - ((double) (((double) (((double) log(n)) * (((double) (n * n)) / i))) + ((double) (((double) ((((double) log(1.0)) / i) * ((double) (((double) log(n)) * ((double) (((double) pow(n, 4.0)) * ((double) log(i)))))))) + ((double) (((double) ((((double) log(1.0)) / i) * ((double) (((double) log(n)) * ((double) pow(n, 3.0)))))) + ((double) (((double) (((double) ((((double) log(i)) / i) * ((double) (((double) log(n)) * ((double) pow(n, 3.0)))))) + ((double) (0.5 * ((double) ((((double) pow(((double) log(1.0)), 2.0)) / i) * ((double) (((double) pow(n, 4.0)) * ((double) log(n)))))))))) + ((double) (0.5 * ((double) ((((double) pow(((double) log(i)), 2.0)) / i) * ((double) (((double) pow(n, 4.0)) * ((double) log(n))))))))))))))))))))));
			} else {
				VAR_2 = ((double) (n * ((double) (100.0 * (((double) (((double) pow(((double) (1.0 + (i / n))), n)) - 1.0)) / i)))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus i

Bits error versus n

Target

Original	47.3
Target	47.2
Herbie	16.7

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

Derivation

1. Split input into 4 regimes

2. if i < -5.9327559585318321e-10

   1. Initial program 29.1

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

   if -5.9327559585318321e-10 < i < 2.33851351412757138e-16

   1. Initial program 58.0

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

   2. Simplified 57.6

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

   3. Taylor expanded around 0 9.5

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

   4. Simplified 9.5

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

   if 2.33851351412757138e-16 < i < 5.4602234981759689e181

   1. Initial program 35.5

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

   2. Simplified 35.5

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

   3. Taylor expanded around 0 19.3

      \[\leadsto 100 \cdot \color{blue}{\left(\left(0.5 \cdot \frac{{\left(\log 1\right)}^{2} \cdot \left(\log i \cdot {n}^{4}\right)}{i} + \left(0.5 \cdot \frac{{\left(\log 1\right)}^{2} \cdot {n}^{3}}{i} + \left(0.16666666666666666 \cdot \frac{{\left(\log 1\right)}^{3} \cdot {n}^{4}}{i} + \left(\frac{\log 1 \cdot {n}^{2}}{i} + \left(0.5 \cdot \frac{\log 1 \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)}{i} + \left(0.5 \cdot \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i} + \left(0.5 \cdot \frac{\log 1 \cdot \left({\left(\log i\right)}^{2} \cdot {n}^{4}\right)}{i} + \left(0.16666666666666666 \cdot \frac{{\left(\log i\right)}^{3} \cdot {n}^{4}}{i} + \left(0.5 \cdot \frac{\log i \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)}{i} + \left(\frac{\log i \cdot {n}^{2}}{i} + \left(0.5 \cdot \frac{{n}^{3} \cdot {\left(\log n\right)}^{2}}{i} + \frac{\log 1 \cdot \left(\log i \cdot {n}^{3}\right)}{i}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(0.16666666666666666 \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(\frac{{n}^{2} \cdot \log n}{i} + \left(\frac{\log 1 \cdot \left(\log i \cdot \left({n}^{4} \cdot \log n\right)\right)}{i} + \left(0.5 \cdot \frac{{\left(\log i\right)}^{2} \cdot \left({n}^{4} \cdot \log n\right)}{i} + \left(\frac{\log 1 \cdot \left({n}^{3} \cdot \log n\right)}{i} + \left(\frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i} + 0.5 \cdot \frac{{\left(\log 1\right)}^{2} \cdot \left({n}^{4} \cdot \log n\right)}{i}\right)\right)\right)\right)\right)\right)\right)}\]

   4. Simplified 19.4

      \[\leadsto 100 \cdot \color{blue}{\left(\left(\left(\frac{\log 1}{i} \cdot \left(n \cdot n\right) + \left(\left(\left(\frac{\log 1}{i} \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)\right) \cdot 0.5 + \left(\left(\left(\frac{{\left(\log i\right)}^{3}}{i} \cdot {n}^{4}\right) \cdot 0.16666666666666666 + \left(\frac{\log i}{i} \cdot \left(n \cdot n\right) + \left(\left(\frac{\log 1}{i} \cdot \left(\log i \cdot {n}^{3}\right) + \left(\frac{{n}^{3}}{i} \cdot {\left(\log n\right)}^{2}\right) \cdot 0.5\right) + \frac{\log \left(\sqrt{i}\right) \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)}{i}\right)\right)\right) + 0.5 \cdot \left(\frac{{\left(\log i\right)}^{2}}{i} \cdot {n}^{3} + \frac{\log 1}{i} \cdot \left({n}^{4} \cdot {\left(\log i\right)}^{2}\right)\right)\right)\right) + \left(\frac{{\left(\log 1\right)}^{3}}{i} \cdot {n}^{4}\right) \cdot 0.16666666666666666\right)\right) + 0.5 \cdot \left(\frac{{\left(\log 1\right)}^{2}}{i} \cdot \left(\log i \cdot {n}^{4}\right) + \frac{{\left(\log 1\right)}^{2}}{i} \cdot {n}^{3}\right)\right) + \left(\left(\frac{{n}^{4}}{i} \cdot {\left(\log n\right)}^{3}\right) \cdot -0.16666666666666666 - \left(\frac{n \cdot n}{i} \cdot \log n + \left(\frac{\log 1}{i} \cdot \left(\log n \cdot \left(\log i \cdot {n}^{4}\right)\right) + \left(\frac{\log 1}{i} \cdot \left(\log n \cdot {n}^{3}\right) + \left(\left(\frac{\log i}{i} \cdot \left(\log n \cdot {n}^{3}\right) + \left(\frac{{\left(\log 1\right)}^{2}}{i} \cdot \left(\log n \cdot {n}^{4}\right)\right) \cdot 0.5\right) + \left(\frac{{\left(\log i\right)}^{2}}{i} \cdot \left(\log n \cdot {n}^{4}\right)\right) \cdot 0.5\right)\right)\right)\right)\right)\right)}\]

   if 5.4602234981759689e181 < i

   1. Initial program 32.3

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

   2. Simplified 32.3

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

   4. Applied pow1 32.3

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

   5. Applied pow1 32.3

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

   6. Applied pow-prod-down 32.3

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

   7. Applied pow1 32.3

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

   8. Applied pow-prod-down 32.3

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

   9. Simplified 32.3

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

4. Final simplification 16.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5.932755958531832 \cdot 10^{-10}:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.3385135141275714 \cdot 10^{-16}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i \cdot 1 + \left(n \cdot \log 1 + \left(i \cdot \left(i \cdot 0.5\right) - \left(i \cdot i\right) \cdot \left(\log 1 \cdot 0.5\right)\right)\right)}{i}\right)\\ \mathbf{elif}\;i \leq 5.460223498175969 \cdot 10^{+181}:\\ \;\;\;\;100 \cdot \left(\left(\left(\frac{\log 1}{i} \cdot \left(n \cdot n\right) + \left(\left(\left(\frac{\log 1}{i} \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)\right) \cdot 0.5 + \left(\left(\left({n}^{4} \cdot \frac{{\left(\log i\right)}^{3}}{i}\right) \cdot 0.16666666666666666 + \left(\left(n \cdot n\right) \cdot \frac{\log i}{i} + \left(\left(\frac{\log 1}{i} \cdot \left(\log i \cdot {n}^{3}\right) + 0.5 \cdot \left({\left(\log n\right)}^{2} \cdot \frac{{n}^{3}}{i}\right)\right) + \frac{\left({n}^{4} \cdot {\left(\log n\right)}^{2}\right) \cdot \log \left(\sqrt{i}\right)}{i}\right)\right)\right) + 0.5 \cdot \left({n}^{3} \cdot \frac{{\left(\log i\right)}^{2}}{i} + \frac{\log 1}{i} \cdot \left({n}^{4} \cdot {\left(\log i\right)}^{2}\right)\right)\right)\right) + 0.16666666666666666 \cdot \left({n}^{4} \cdot \frac{{\left(\log 1\right)}^{3}}{i}\right)\right)\right) + 0.5 \cdot \left(\frac{{\left(\log 1\right)}^{2}}{i} \cdot \left({n}^{4} \cdot \log i\right) + {n}^{3} \cdot \frac{{\left(\log 1\right)}^{2}}{i}\right)\right) + \left(\left(\frac{{n}^{4}}{i} \cdot {\left(\log n\right)}^{3}\right) \cdot -0.16666666666666666 - \left(\log n \cdot \frac{n \cdot n}{i} + \left(\frac{\log 1}{i} \cdot \left(\log n \cdot \left({n}^{4} \cdot \log i\right)\right) + \left(\frac{\log 1}{i} \cdot \left(\log n \cdot {n}^{3}\right) + \left(\left(\frac{\log i}{i} \cdot \left(\log n \cdot {n}^{3}\right) + 0.5 \cdot \left(\frac{{\left(\log 1\right)}^{2}}{i} \cdot \left({n}^{4} \cdot \log n\right)\right)\right) + 0.5 \cdot \left(\frac{{\left(\log i\right)}^{2}}{i} \cdot \left({n}^{4} \cdot \log n\right)\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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))))