Compound Interest

Average Error: 47.6 → 15.3

Time: 13.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -3.6971897647249273 \cdot 10^{-17}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{i}\right)\\ \mathbf{elif}\;i \le 0.33679197925217047:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i \cdot 1 + \left(n \cdot \log 1 + \log \left({\left({\left(e^{i}\right)}^{i}\right)}^{\left(0.5 - \log 1 \cdot 0.5\right)}\right)\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \end{array}\]

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 <= -3.697189764724927e-17)) {
		VAR = ((double) (100.0 * ((double) (n * ((double) (((double) (((double) pow(((double) (i / n)), n)) - 1.0)) / i))))));
	} else {
		double VAR_1;
		if ((i <= 0.33679197925217047)) {
			VAR_1 = ((double) (100.0 * ((double) (n * ((double) (((double) (((double) (i * 1.0)) + ((double) (((double) (n * ((double) log(1.0)))) + ((double) log(((double) pow(((double) pow(((double) exp(i)), i)), ((double) (0.5 - ((double) (((double) log(1.0)) * 0.5)))))))))))) / i))))));
		} else {
			VAR_1 = ((double) (100.0 * ((double) (((double) (((double) (((double) pow(((double) (((double) (i / n)) + 1.0)), ((double) (n * 2.0)))) - ((double) (1.0 * 1.0)))) / ((double) (1.0 + ((double) pow(((double) (((double) (i / n)) + 1.0)), n)))))) / ((double) (i / n))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus i

Bits error versus n

Target

Original	47.6
Target	47.3
Herbie	15.3

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

Derivation

1. Split input into 3 regimes

2. if i < -3.6971897647249273e-17

   1. Initial program 30.6

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

   2. Taylor expanded around inf 64.0

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

   3. Simplified 22.0

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

   if -3.6971897647249273e-17 < i < 0.33679197925217047

   1. Initial program 58.1

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

   2. Taylor expanded around 0 26.8

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

   3. Simplified 26.8

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

   5. Applied associate-/r/ 9.2

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

   6. Simplified 9.2

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

   8. Applied add-log-exp 9.3

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

   9. Simplified 9.3

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

   if 0.33679197925217047 < i

   1. Initial program 30.4

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

   3. Applied flip-- 30.4

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

   4. Simplified 30.4

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

   5. Simplified 30.4

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

4. Final simplification 15.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -3.6971897647249273 \cdot 10^{-17}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{i}\right)\\ \mathbf{elif}\;i \le 0.33679197925217047:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i \cdot 1 + \left(n \cdot \log 1 + \log \left({\left({\left(e^{i}\right)}^{i}\right)}^{\left(0.5 - \log 1 \cdot 0.5\right)}\right)\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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))))