Compound Interest

Average Error: 47.7 → 16.9

Time: 14.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -0.27000945627389827:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, {\left(\sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, -1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 0.34323219986681708:\\ \;\;\;\;\left(100 \cdot \left(\left(\sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}}\right) \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}}}\right)\right) \cdot n\\ \mathbf{elif}\;i \le 3.6088222865594129 \cdot 10^{217}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.30620374507854063 \cdot 10^{237}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{n}{i}\right)\\ \end{array}\]

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 VAR;
	if ((i <= -0.2700094562738983)) {
		VAR = (100.0 * (fma(pow((cbrt((1.0 + (i / n))) * cbrt((1.0 + (i / n)))), n), pow(cbrt((1.0 + (i / n))), n), -1.0) / (i / n)));
	} else {
		double VAR_1;
		if ((i <= 0.3432321998668171)) {
			VAR_1 = ((100.0 * ((cbrt((fma(i, 1.0, (fma(0.5, pow(i, 2.0), (log(1.0) * n)) - (0.5 * (pow(i, 2.0) * log(1.0))))) / i)) * cbrt((fma(i, 1.0, (fma(0.5, pow(i, 2.0), (log(1.0) * n)) - (0.5 * (pow(i, 2.0) * log(1.0))))) / i))) * cbrt(((cbrt((fma(i, 1.0, (fma(0.5, pow(i, 2.0), (log(1.0) * n)) - (0.5 * (pow(i, 2.0) * log(1.0))))) / i)) * cbrt((fma(i, 1.0, (fma(0.5, pow(i, 2.0), (log(1.0) * n)) - (0.5 * (pow(i, 2.0) * log(1.0))))) / i))) * cbrt((fma(i, 1.0, (fma(0.5, pow(i, 2.0), (log(1.0) * n)) - (0.5 * (pow(i, 2.0) * log(1.0))))) / i)))))) * n);
		} else {
			double VAR_2;
			if ((i <= 3.608822286559413e+217)) {
				VAR_2 = (100.0 * (((pow((1.0 + (i / n)), (2.0 * n)) + -(1.0 * 1.0)) / (pow((1.0 + (i / n)), n) + 1.0)) / (i / n)));
			} else {
				double VAR_3;
				if ((i <= 1.3062037450785406e+237)) {
					VAR_3 = (100.0 * ((fma(1.0, i, fma(log(1.0), n, 1.0)) - 1.0) / (i / n)));
				} else {
					VAR_3 = (100.0 * ((pow((1.0 + (i / n)), n) - 1.0) * (n / 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	47.7
Target	46.9
Herbie	16.9

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

Derivation

1. Split input into 5 regimes

2. if i < -0.2700094562738983

   1. Initial program 28.8

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

   3. Applied add-cube-cbrt 28.9

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

   4. Applied unpow-prod-down 28.9

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

   5. Applied fma-neg 28.9

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, {\left(\sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, -1\right)}}{\frac{i}{n}}\]

   if -0.2700094562738983 < i < 0.3432321998668171

   1. Initial program 57.8

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

   2. Taylor expanded around 0 27.1

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

   3. Simplified 27.1

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

   5. Applied associate-/r/ 9.4

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

   6. Applied associate-*r* 9.4

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

   8. Applied add-cube-cbrt 9.4

      \[\leadsto \left(100 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}}\right) \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}}\right)}\right) \cdot n\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 9.4

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

   if 0.3432321998668171 < i < 3.608822286559413e+217

   1. Initial program 31.4

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

   3. Applied flip-- 31.5

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

   4. Simplified 31.4

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

   if 3.608822286559413e+217 < i < 1.3062037450785406e+237

   1. Initial program 32.0

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

   2. Taylor expanded around 0 36.2

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

   3. Simplified 36.2

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

   if 1.3062037450785406e+237 < i

   1. Initial program 33.1

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

   3. Applied div-inv 33.1

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

   4. Simplified 33.0

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

4. Final simplification 16.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.27000945627389827:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, {\left(\sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, -1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 0.34323219986681708:\\ \;\;\;\;\left(100 \cdot \left(\left(\sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}}\right) \cdot \sqrt[3]{\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}}}\right)\right) \cdot n\\ \mathbf{elif}\;i \le 3.6088222865594129 \cdot 10^{217}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.30620374507854063 \cdot 10^{237}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{n}{i}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 +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))))