Compound Interest

Average Error: 47.4 → 16.1

Time: 14.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -0.218788485238502056:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 0.00561970824232101585:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{\mathsf{fma}\left(1, {\left(1 + \frac{i}{n}\right)}^{n} + 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) \cdot i}{n}}\\ \end{array}\]

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 <= -0.21878848523850206)) {
		VAR = ((double) (100.0 * ((double) (((double) (((double) (((double) pow(((double) (1.0 + ((double) (i / n)))), ((double) (2.0 * n)))) + ((double) -(((double) (1.0 * 1.0)))))) / ((double) (((double) pow(((double) (1.0 + ((double) (i / n)))), n)) + 1.0)))) / ((double) (i / n))))));
	} else {
		double VAR_1;
		if ((i <= 0.005619708242321016)) {
			VAR_1 = ((double) (100.0 * ((double) (((double) (((double) fma(i, 1.0, ((double) (((double) fma(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 {
			VAR_1 = ((double) (100.0 * ((double) (((double) (((double) pow(((double) pow(((double) (1.0 + ((double) (i / n)))), n)), 3.0)) - ((double) pow(1.0, 3.0)))) / ((double) (((double) (((double) fma(1.0, ((double) (((double) pow(((double) (1.0 + ((double) (i / n)))), n)) + 1.0)), ((double) pow(((double) (1.0 + ((double) (i / n)))), ((double) (2.0 * n)))))) * i)) / n))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus i

Bits error versus n

Target

Original	47.4
Target	47.4
Herbie	16.1

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

Derivation

1. Split input into 3 regimes

2. if i < -0.21878848523850206

   1. Initial program 27.7

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

   3. Applied flip-- 27.7

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

   4. Simplified 27.7

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

   if -0.21878848523850206 < i < 0.005619708242321016

   1. Initial program 58.3

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

   2. Taylor expanded around 0 26.2

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

   3. Simplified 26.2

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

   5. Applied associate-/r/ 8.8

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

   if 0.005619708242321016 < i

   1. Initial program 30.2

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

   3. Applied flip3-- 30.3

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

   4. Applied associate-/l/ 30.3

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

   5. Simplified 30.3

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

4. Final simplification 16.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.218788485238502056:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 0.00561970824232101585:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{\mathsf{fma}\left(1, {\left(1 + \frac{i}{n}\right)}^{n} + 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) \cdot i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 +o rules:numerics
(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))))