Compound Interest

Average Error: 47.8 → 17.7

Time: 13.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -9.14504001401898701 \cdot 10^{-10}:\\ \;\;\;\;\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 8.6736436125377321 \cdot 10^{-26}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i \cdot 1 + \left(n \cdot \log 1 + \left(i \cdot \left(i \cdot 0.5\right)\right) \cdot \left(1 - \log 1\right)\right)}{i}\right)\\ \mathbf{elif}\;i \le 1.9138780675951828 \cdot 10^{154}:\\ \;\;\;\;100 \cdot \frac{\sqrt[3]{\left(n \cdot n\right) \cdot {i}^{-2}} \cdot \left(i \cdot \left(1 + i \cdot 0.5\right)\right) + \left(\log 1 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot {i}^{-2}}\right) \cdot \left(n - i \cdot \left(i \cdot 0.5\right)\right)}{\frac{\sqrt[3]{i}}{\sqrt[3]{n}}}\\ \mathbf{elif}\;i \le 2.0505733833233406 \cdot 10^{233}:\\ \;\;\;\;100 \cdot \frac{\left(1 + \left(i \cdot 1 + n \cdot \log 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} - n \cdot \frac{1}{i}\right)\\ \end{array}\]

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 <= -9.145040014018987e-10)) {
		VAR = ((double) (((double) (((double) pow(((double) (1.0 + ((double) (i / n)))), n)) - 1.0)) * ((double) (100.0 / ((double) (i / n))))));
	} else {
		double VAR_1;
		if ((i <= 8.673643612537732e-26)) {
			VAR_1 = ((double) (100.0 * ((double) (n * ((double) (((double) (((double) (i * 1.0)) + ((double) (((double) (n * ((double) log(1.0)))) + ((double) (((double) (i * ((double) (i * 0.5)))) * ((double) (1.0 - ((double) log(1.0)))))))))) / i))))));
		} else {
			double VAR_2;
			if ((i <= 1.9138780675951828e+154)) {
				VAR_2 = ((double) (100.0 * ((double) (((double) (((double) (((double) cbrt(((double) (((double) (n * n)) * ((double) pow(i, -2.0)))))) * ((double) (i * ((double) (1.0 + ((double) (i * 0.5)))))))) + ((double) (((double) (((double) log(1.0)) * ((double) cbrt(((double) (((double) (n * n)) * ((double) pow(i, -2.0)))))))) * ((double) (n - ((double) (i * ((double) (i * 0.5)))))))))) / ((double) (((double) cbrt(i)) / ((double) cbrt(n))))))));
			} else {
				double VAR_3;
				if ((i <= 2.0505733833233406e+233)) {
					VAR_3 = ((double) (100.0 * ((double) (((double) (((double) (1.0 + ((double) (((double) (i * 1.0)) + ((double) (n * ((double) log(1.0)))))))) - 1.0)) / ((double) (i / n))))));
				} else {
					VAR_3 = ((double) (100.0 * ((double) (((double) (n * ((double) (((double) pow(((double) (1.0 + ((double) (i / n)))), n)) / i)))) - ((double) (n * ((double) (1.0 / i))))))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus i

Bits error versus n

Target

Original	47.8
Target	47.1
Herbie	17.7

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

Derivation

1. Split input into 5 regimes

2. if i < -9.14504001401898701e-10

   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 pow1 30.4

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

   4. Applied pow1 30.4

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

   5. Applied pow-prod-down 30.4

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

   6. Simplified 30.3

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

   if -9.14504001401898701e-10 < i < 8.6736436125377321e-26

   1. Initial program 57.8

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

   2. Taylor expanded around 0 27.3

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

   3. Simplified 27.3

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

   5. Applied associate-/r/ 8.9

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

   6. Simplified 8.9

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

   if 8.6736436125377321e-26 < i < 1.9138780675951828e154

   1. Initial program 36.1

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

   2. Taylor expanded around 0 42.6

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

   3. Simplified 42.6

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

   5. Applied add-cube-cbrt 42.8

      \[\leadsto 100 \cdot \frac{1 \cdot i + \left(n \cdot \log 1 + \left(i \cdot \left(i \cdot 0.5\right) - \left(i \cdot i\right) \cdot \left(0.5 \cdot \log 1\right)\right)\right)}{\frac{i}{\color{blue}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}}}\]

   6. Applied add-cube-cbrt 42.8

      \[\leadsto 100 \cdot \frac{1 \cdot i + \left(n \cdot \log 1 + \left(i \cdot \left(i \cdot 0.5\right) - \left(i \cdot i\right) \cdot \left(0.5 \cdot \log 1\right)\right)\right)}{\frac{\color{blue}{\left(\sqrt[3]{i} \cdot \sqrt[3]{i}\right) \cdot \sqrt[3]{i}}}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}}\]

   7. Applied times-frac 42.8

      \[\leadsto 100 \cdot \frac{1 \cdot i + \left(n \cdot \log 1 + \left(i \cdot \left(i \cdot 0.5\right) - \left(i \cdot i\right) \cdot \left(0.5 \cdot \log 1\right)\right)\right)}{\color{blue}{\frac{\sqrt[3]{i} \cdot \sqrt[3]{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{\sqrt[3]{i}}{\sqrt[3]{n}}}}\]

   8. Applied associate-/r* 50.4

      \[\leadsto 100 \cdot \color{blue}{\frac{\frac{1 \cdot i + \left(n \cdot \log 1 + \left(i \cdot \left(i \cdot 0.5\right) - \left(i \cdot i\right) \cdot \left(0.5 \cdot \log 1\right)\right)\right)}{\frac{\sqrt[3]{i} \cdot \sqrt[3]{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}}}{\frac{\sqrt[3]{i}}{\sqrt[3]{n}}}}\]

   9. Simplified 50.4

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \frac{1 \cdot i + \left(n \cdot \log 1 + \left(i \cdot \left(i \cdot 0.5\right)\right) \cdot \left(1 - \log 1\right)\right)}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}}{\frac{\sqrt[3]{i}}{\sqrt[3]{n}}}\]

   10. Taylor expanded around 0 57.1

       \[\leadsto 100 \cdot \frac{\color{blue}{\left(0.5 \cdot \left({i}^{2} \cdot e^{\frac{1}{3} \cdot \left(\left(\log 1 + 2 \cdot \log n\right) - 2 \cdot \log i\right)}\right) + \left(e^{\frac{1}{3} \cdot \left(\left(\log 1 + 2 \cdot \log n\right) - 2 \cdot \log i\right)} \cdot \left(\log 1 \cdot n\right) + 1 \cdot \left(i \cdot e^{\frac{1}{3} \cdot \left(\left(\log 1 + 2 \cdot \log n\right) - 2 \cdot \log i\right)}\right)\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \left(e^{\frac{1}{3} \cdot \left(\left(\log 1 + 2 \cdot \log n\right) - 2 \cdot \log i\right)} \cdot \log 1\right)\right)}}{\frac{\sqrt[3]{i}}{\sqrt[3]{n}}}\]

   11. Simplified 32.7

       \[\leadsto 100 \cdot \frac{\color{blue}{\sqrt[3]{\left(n \cdot n\right) \cdot {i}^{-2}} \cdot \left(i \cdot \left(i \cdot 0.5 + 1\right)\right) + \left(\log 1 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot {i}^{-2}}\right) \cdot \left(n - i \cdot \left(i \cdot 0.5\right)\right)}}{\frac{\sqrt[3]{i}}{\sqrt[3]{n}}}\]

   if 1.9138780675951828e154 < i < 2.0505733833233406e233

   1. Initial program 30.1

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

   2. Taylor expanded around 0 38.5

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

   3. Simplified 38.5

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

   if 2.0505733833233406e233 < i

   1. Initial program 33.2

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

   3. Applied div-sub 33.2

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

   4. Simplified 33.9

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

   5. Simplified 33.7

      \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n - \color{blue}{\frac{1}{i} \cdot n}\right)\]
3. Recombined 5 regimes into one program.

4. Final simplification 17.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -9.14504001401898701 \cdot 10^{-10}:\\ \;\;\;\;\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 8.6736436125377321 \cdot 10^{-26}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i \cdot 1 + \left(n \cdot \log 1 + \left(i \cdot \left(i \cdot 0.5\right)\right) \cdot \left(1 - \log 1\right)\right)}{i}\right)\\ \mathbf{elif}\;i \le 1.9138780675951828 \cdot 10^{154}:\\ \;\;\;\;100 \cdot \frac{\sqrt[3]{\left(n \cdot n\right) \cdot {i}^{-2}} \cdot \left(i \cdot \left(1 + i \cdot 0.5\right)\right) + \left(\log 1 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot {i}^{-2}}\right) \cdot \left(n - i \cdot \left(i \cdot 0.5\right)\right)}{\frac{\sqrt[3]{i}}{\sqrt[3]{n}}}\\ \mathbf{elif}\;i \le 2.0505733833233406 \cdot 10^{233}:\\ \;\;\;\;100 \cdot \frac{\left(1 + \left(i \cdot 1 + n \cdot \log 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} - n \cdot \frac{1}{i}\right)\\ \end{array}\]

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))))