Compound Interest

Average Error: 47.3 → 16.6

Time: 13.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -1.32264855634611611:\\ \;\;\;\;\frac{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)\right) \cdot 100}{\frac{i}{n} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)}\\ \mathbf{elif}\;i \le 6.2316967628408393 \cdot 10^{-6}:\\ \;\;\;\;\left(100 \cdot \mathsf{fma}\left(0.5, i \cdot \log 1, -\mathsf{fma}\left(0.5, i, \frac{\log 1 \cdot n}{i} + 1\right)\right)\right) \cdot \left(-n\right)\\ \mathbf{elif}\;i \le 3.16763600985918069 \cdot 10^{238}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}}{i} \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} - \sqrt{1}\right) \cdot n\right)\right)\\ \mathbf{elif}\;i \le 1.8069577640268902 \cdot 10^{255}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}}{\frac{1}{n}}\\ \end{array}\]

C

double code(double i, double n) {
	return (100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n)));
}

↓

double code(double i, double n) {
	double temp;
	if ((i <= -1.322648556346116)) {
		temp = (((pow((1.0 + (i / n)), (2.0 * n)) + -(1.0 * 1.0)) * 100.0) / ((i / n) * (pow((1.0 + (i / n)), n) + 1.0)));
	} else {
		double temp_1;
		if ((i <= 6.231696762840839e-06)) {
			temp_1 = ((100.0 * fma(0.5, (i * log(1.0)), -fma(0.5, i, (((log(1.0) * n) / i) + 1.0)))) * -n);
		} else {
			double temp_2;
			if ((i <= 3.1676360098591807e+238)) {
				temp_2 = (100.0 * (((pow((1.0 + (i / n)), (n / 2.0)) + sqrt(1.0)) / i) * ((pow((1.0 + (i / n)), (n / 2.0)) - sqrt(1.0)) * n)));
			} else {
				double temp_3;
				if ((i <= 1.8069577640268902e+255)) {
					temp_3 = (100.0 * ((fma(1.0, i, fma(log(1.0), n, 1.0)) - 1.0) / (i / n)));
				} else {
					temp_3 = (((100.0 * (pow((1.0 + (i / n)), n) - 1.0)) / i) / (1.0 / n));
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus i

Bits error versus n

Target

Original	47.3
Target	47.5
Herbie	16.6

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

Derivation

1. Split input into 5 regimes

2. if i < -1.322648556346116

   1. Initial program 27.3

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

   3. Applied flip-- 27.3

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

   4. Applied associate-/l/ 27.3

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

   5. Applied associate-*r/ 27.2

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

   6. Simplified 27.2

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

   if -1.322648556346116 < i < 6.231696762840839e-06

   1. Initial program 57.8

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

   2. Taylor expanded around 0 26.0

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

   3. Simplified 26.0

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

   5. Applied frac-2neg 26.0

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

   6. Applied associate-/r/ 9.5

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

   7. Applied associate-*r* 9.5

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

   8. Taylor expanded around 0 9.5

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

   9. Simplified 9.5

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

   if 6.231696762840839e-06 < i < 3.1676360098591807e+238

   1. Initial program 30.8

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

   3. Applied div-inv 30.8

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

   4. Applied add-sqr-sqrt 30.8

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

   5. Applied sqr-pow 30.9

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

   6. Applied difference-of-squares 30.9

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

   7. Applied times-frac 30.9

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

   8. Simplified 30.8

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

   if 3.1676360098591807e+238 < i < 1.8069577640268902e+255

   1. Initial program 33.5

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

   2. Taylor expanded around 0 32.2

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

   3. Simplified 32.2

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

   if 1.8069577640268902e+255 < i

   1. Initial program 34.0

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

   3. Applied div-inv 34.0

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

   4. Applied associate-/r* 34.0

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

   5. Applied associate-*r/ 34.0

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

   6. Simplified 33.9

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

4. Final simplification 16.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.32264855634611611:\\ \;\;\;\;\frac{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)\right) \cdot 100}{\frac{i}{n} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)}\\ \mathbf{elif}\;i \le 6.2316967628408393 \cdot 10^{-6}:\\ \;\;\;\;\left(100 \cdot \mathsf{fma}\left(0.5, i \cdot \log 1, -\mathsf{fma}\left(0.5, i, \frac{\log 1 \cdot n}{i} + 1\right)\right)\right) \cdot \left(-n\right)\\ \mathbf{elif}\;i \le 3.16763600985918069 \cdot 10^{238}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}}{i} \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} - \sqrt{1}\right) \cdot n\right)\right)\\ \mathbf{elif}\;i \le 1.8069577640268902 \cdot 10^{255}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}}{\frac{1}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +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))))