Compound Interest

Average Error: 42.7 → 31.4

Time: 18.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -2.56140166535216897 \cdot 10^{135}:\\ \;\;\;\;\frac{100 \cdot \frac{\log \left(e^{\mathsf{fma}\left(-1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.3992561866449662 \cdot 10^{-10}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.51591290926460688 \cdot 10^{-160}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 6.02622510223326963 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{100 \cdot \mathsf{fma}\left(-1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}{i \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)}}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 8532543483832934860000:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 7.39555913873958208 \cdot 10^{219}:\\ \;\;\;\;\frac{\frac{100 \cdot \mathsf{fma}\left(-1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}{i \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

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 <= -2.561401665352169e+135)) {
		temp = ((100.0 * (log(exp(fma(-1.0, 1.0, pow((1.0 + (i / n)), (2.0 * n))))) / (pow((1.0 + (i / n)), n) + 1.0))) / (i / n));
	} else {
		double temp_1;
		if ((i <= -1.3992561866449662e-10)) {
			temp_1 = ((100.0 * (pow((i / n), n) - 1.0)) / (i / n));
		} else {
			double temp_2;
			if ((i <= 2.515912909264607e-160)) {
				temp_2 = (100.0 * (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 temp_3;
				if ((i <= 6.02622510223327e-125)) {
					temp_3 = (((100.0 * fma(-1.0, 1.0, pow((1.0 + (i / n)), (2.0 * n)))) / (i * (pow((1.0 + (i / n)), n) + 1.0))) / (1.0 / n));
				} else {
					double temp_4;
					if ((i <= 8.532543483832935e+21)) {
						temp_4 = (100.0 * (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 temp_5;
						if ((i <= 7.395559138739582e+219)) {
							temp_5 = (((100.0 * fma(-1.0, 1.0, pow((1.0 + (i / n)), (2.0 * n)))) / (i * (pow((1.0 + (i / n)), n) + 1.0))) / (1.0 / n));
						} else {
							temp_5 = (100.0 * ((fma(1.0, i, fma(log(1.0), n, 1.0)) - 1.0) / (i / n)));
						}
						temp_4 = temp_5;
					}
					temp_3 = temp_4;
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.7
Target	42.6
Herbie	31.4

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

Derivation

1. Split input into 5 regimes

2. if i < -2.561401665352169e+135

   1. Initial program 15.4

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

   3. Applied associate-*r/ 15.4

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

   5. Applied flip-- 15.4

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

   6. Simplified 15.4

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

   8. Applied add-log-exp 15.4

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

   if -2.561401665352169e+135 < i < -1.3992561866449662e-10

   1. Initial program 41.3

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

   3. Applied associate-*r/ 41.3

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

   4. Taylor expanded around inf 64.0

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

   5. Simplified 27.6

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

   if -1.3992561866449662e-10 < i < 2.515912909264607e-160 or 6.02622510223327e-125 < i < 8.532543483832935e+21

   1. Initial program 49.9

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

   2. Taylor expanded around 0 33.7

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

   3. Simplified 33.7

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

   if 2.515912909264607e-160 < i < 6.02622510223327e-125 or 8.532543483832935e+21 < i < 7.395559138739582e+219

   1. Initial program 38.2

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

   3. Applied associate-*r/ 38.1

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

   5. Applied flip-- 38.1

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

   6. Simplified 38.1

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

   8. Applied add-log-exp 38.3

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

   10. Applied div-inv 38.3

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

   11. Applied associate-/r* 38.3

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

   12. Simplified 38.0

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

   if 7.395559138739582e+219 < i

   1. Initial program 30.6

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

   2. Taylor expanded around 0 34.7

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

   3. Simplified 34.7

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

4. Final simplification 31.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.56140166535216897 \cdot 10^{135}:\\ \;\;\;\;\frac{100 \cdot \frac{\log \left(e^{\mathsf{fma}\left(-1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.3992561866449662 \cdot 10^{-10}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.51591290926460688 \cdot 10^{-160}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 6.02622510223326963 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{100 \cdot \mathsf{fma}\left(-1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}{i \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)}}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 8532543483832934860000:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 7.39555913873958208 \cdot 10^{219}:\\ \;\;\;\;\frac{\frac{100 \cdot \mathsf{fma}\left(-1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}{i \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))))