Compound Interest

Average Error: 47.8 → 14.7

Time: 52.5min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -0.61196979809143093:\\ \;\;\;\;\left(\frac{100}{i} \cdot n\right) \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)\\ \mathbf{elif}\;i \le 2.63341232799653732 \cdot 10^{-7}:\\ \;\;\;\;\left(100 \cdot \frac{i \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \log 1\right) + 1\right) + \log 1 \cdot n}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]

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 <= -0.6119697980914309)) {
		VAR = ((double) (((double) ((100.0 / i) * n)) * ((double) (((double) pow((i / n), n)) - 1.0))));
	} else {
		double VAR_1;
		if ((i <= 2.6334123279965373e-07)) {
			VAR_1 = ((double) (((double) (100.0 * (((double) (((double) (i * ((double) (((double) (i * ((double) (0.5 - ((double) (0.5 * ((double) log(1.0)))))))) + 1.0)))) + ((double) (((double) log(1.0)) * n)))) / i))) * n));
		} else {
			VAR_1 = ((double) (100.0 * ((((double) (((double) pow(((double) (1.0 + (i / n))), ((double) (2.0 * n)))) - ((double) (1.0 * 1.0)))) / ((double) (((double) pow(((double) (1.0 + (i / n))), n)) + 1.0))) / (i / n))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus i

Bits error versus n

Target

Original	47.8
Target	47.4
Herbie	14.7

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

Derivation

1. Split input into 3 regimes

2. if i < -0.61196979809143093

   1. Initial program 27.4

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

   2. Taylor expanded around -inf 30.5

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

   3. Simplified 17.9

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

   if -0.61196979809143093 < i < 2.63341232799653732e-7

   1. Initial program 58.0

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

   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

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

   5. Applied associate-/r/ 9.4

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

   6. Applied associate-*r* 9.4

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

   if 2.63341232799653732e-7 < i

   1. Initial program 34.4

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

   3. Applied flip-- 34.4

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

   4. Simplified 34.3

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

4. Final simplification 14.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.61196979809143093:\\ \;\;\;\;\left(\frac{100}{i} \cdot n\right) \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)\\ \mathbf{elif}\;i \le 2.63341232799653732 \cdot 10^{-7}:\\ \;\;\;\;\left(100 \cdot \frac{i \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \log 1\right) + 1\right) + \log 1 \cdot n}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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))))