Compound Interest

Average Error: 48.2 → 16.5

Time: 11.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -0.0749921289443250955:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \mathbf{elif}\;i \le 9.29801943706574769 \cdot 10^{-129}:\\ \;\;\;\;100 \cdot \left(\frac{\log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right) + i \cdot \left(i \cdot 0.5 + 1\right)}{-i} \cdot \left(-n\right)\right)\\ \mathbf{elif}\;i \le 3.8426241943682245 \cdot 10^{22}:\\ \;\;\;\;\frac{100 \cdot \left(\left(\left(\log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)\right) \cdot \left(\log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)\right) - \left(i \cdot \left(i \cdot 0.5 + 1\right)\right) \cdot \left(i \cdot \left(i \cdot 0.5 + 1\right)\right)\right) \cdot \left(-n\right)\right)}{\left(-i\right) \cdot \left(\log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right) - i \cdot \left(i \cdot 0.5 + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \end{array}\]

C

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

↓

double code(double i, double n) {
	double VAR;
	if ((i <= -0.0749921289443251)) {
		VAR = ((100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / i)) * n);
	} else {
		double VAR_1;
		if ((i <= 9.298019437065748e-129)) {
			VAR_1 = (100.0 * ((((log(1.0) * (n - (0.5 * pow(i, 2.0)))) + (i * ((i * 0.5) + 1.0))) / -i) * -n));
		} else {
			double VAR_2;
			if ((i <= 3.8426241943682245e+22)) {
				VAR_2 = ((100.0 * ((((log(1.0) * (n - (0.5 * pow(i, 2.0)))) * (log(1.0) * (n - (0.5 * pow(i, 2.0))))) - ((i * ((i * 0.5) + 1.0)) * (i * ((i * 0.5) + 1.0)))) * -n)) / (-i * ((log(1.0) * (n - (0.5 * pow(i, 2.0)))) - (i * ((i * 0.5) + 1.0)))));
			} else {
				VAR_2 = ((100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / i)) * n);
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus i

Bits error versus n

Target

Original	48.2
Target	47.6
Herbie	16.5

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

Derivation

1. Split input into 3 regimes

2. if i < -0.0749921289443251 or 3.8426241943682245e+22 < i

   1. Initial program 30.0

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

   3. Applied associate-/r/ 30.3

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

   4. Applied associate-*r* 30.3

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

   if -0.0749921289443251 < i < 9.298019437065748e-129

   1. Initial program 59.0

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

   2. Taylor expanded around 0 27.2

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

   3. Simplified 27.2

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

   5. Applied frac-2neg 27.2

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

   6. Applied associate-/r/ 7.8

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

   if 9.298019437065748e-129 < i < 3.8426241943682245e+22

   1. Initial program 53.0

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

   2. Taylor expanded around 0 26.3

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

   3. Simplified 26.3

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

   5. Applied frac-2neg 26.3

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

   6. Applied associate-/r/ 19.2

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

   8. Applied flip-+ 19.3

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

   9. Applied associate-/l/ 19.2

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

   10. Applied associate-*l/ 14.9

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

   11. Applied associate-*r/ 15.0

       \[\leadsto \color{blue}{\frac{100 \cdot \left(\left(\left(\log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)\right) \cdot \left(\log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)\right) - \left(i \cdot \left(i \cdot 0.5 + 1\right)\right) \cdot \left(i \cdot \left(i \cdot 0.5 + 1\right)\right)\right) \cdot \left(-n\right)\right)}{\left(-i\right) \cdot \left(\log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right) - i \cdot \left(i \cdot 0.5 + 1\right)\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 16.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.0749921289443250955:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \mathbf{elif}\;i \le 9.29801943706574769 \cdot 10^{-129}:\\ \;\;\;\;100 \cdot \left(\frac{\log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right) + i \cdot \left(i \cdot 0.5 + 1\right)}{-i} \cdot \left(-n\right)\right)\\ \mathbf{elif}\;i \le 3.8426241943682245 \cdot 10^{22}:\\ \;\;\;\;\frac{100 \cdot \left(\left(\left(\log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)\right) \cdot \left(\log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)\right) - \left(i \cdot \left(i \cdot 0.5 + 1\right)\right) \cdot \left(i \cdot \left(i \cdot 0.5 + 1\right)\right)\right) \cdot \left(-n\right)\right)}{\left(-i\right) \cdot \left(\log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right) - i \cdot \left(i \cdot 0.5 + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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))))