Compound Interest

Average Error: 47.8 → 15.3

Time: 12.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -7.85699781958539598 \cdot 10^{-32}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 5.6810687224704382 \cdot 10^{-5}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i \cdot 1 + \left(n \cdot \log 1 + \left(i \cdot i\right) \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}\right)\\ \mathbf{elif}\;i \le 1.84223808258969376 \cdot 10^{236}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} - n \cdot \frac{1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 + \left(i \cdot 1 + n \cdot \log 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

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 <= -7.856997819585396e-32)) {
		VAR = ((double) (100.0 * (((double) (((double) pow((i / n), n)) - 1.0)) / (i / n))));
	} else {
		double VAR_1;
		if ((i <= 5.681068722470438e-05)) {
			VAR_1 = ((double) (100.0 * ((double) (n * (((double) (((double) (i * 1.0)) + ((double) (((double) (n * ((double) log(1.0)))) + ((double) (((double) (i * i)) * ((double) (0.5 - ((double) (((double) log(1.0)) * 0.5)))))))))) / i)))));
		} else {
			double VAR_2;
			if ((i <= 1.8422380825896938e+236)) {
				VAR_2 = ((double) (100.0 * ((double) (((double) (n * (((double) pow(((double) ((i / n) + 1.0)), n)) / i))) - ((double) (n * (1.0 / i)))))));
			} else {
				VAR_2 = ((double) (100.0 * (((double) (((double) (1.0 + ((double) (((double) (i * 1.0)) + ((double) (n * ((double) log(1.0)))))))) - 1.0)) / (i / n))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus i

Bits error versus n

Target

Original	47.8
Target	47.8
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 4 regimes

2. if i < -7.85699781958539598e-32

   1. Initial program 30.9

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

   2. Taylor expanded around inf 64.0

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

   3. Simplified 22.9

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

   if -7.85699781958539598e-32 < i < 5.6810687224704382e-5

   1. Initial program 58.8

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

   2. Taylor expanded around 0 26.8

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

   3. Simplified 26.8

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

   5. Applied associate-/r/ 8.2

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

   6. Simplified 8.2

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

   if 5.6810687224704382e-5 < i < 1.84223808258969376e236

   1. Initial program 31.0

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

   3. Applied div-sub 31.0

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

   4. Simplified 34.8

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

   5. Simplified 31.0

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

   if 1.84223808258969376e236 < i

   1. Initial program 31.7

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

   2. Taylor expanded around 0 33.2

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

   3. Simplified 33.2

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

4. Final simplification 15.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -7.85699781958539598 \cdot 10^{-32}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 5.6810687224704382 \cdot 10^{-5}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i \cdot 1 + \left(n \cdot \log 1 + \left(i \cdot i\right) \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}\right)\\ \mathbf{elif}\;i \le 1.84223808258969376 \cdot 10^{236}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} - n \cdot \frac{1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 + \left(i \cdot 1 + n \cdot \log 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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))))