Compound Interest

Average Error: 48.1 → 14.3

Time: 10.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -43209245.05120845:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.014357544782884098:\\ \;\;\;\;100 \cdot \left(0.5 \cdot \left(i \cdot n\right) + \left(n + 0.16666666666666666 \cdot \left(i \cdot \left(i \cdot n\right)\right)\right)\right)\\ \mathbf{elif}\;i \leq 2.8968362884498612 \cdot 10^{+166}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 6.755767338475522 \cdot 10^{+227}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} + -1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))

↓

(FPCore (i n)
 :precision binary64
 (if (<= i -43209245.05120845)
   (* 100.0 (/ (- (pow (/ i n) n) 1.0) (/ i n)))
   (if (<= i 0.014357544782884098)
     (* 100.0 (+ (* 0.5 (* i n)) (+ n (* 0.16666666666666666 (* i (* i n))))))
     (if (<= i 2.8968362884498612e+166)
       0.0
       (if (<= i 6.755767338475522e+227)
         (*
          100.0
          (/
           (/
            (+ (pow (+ (/ i n) 1.0) (* n 2.0)) -1.0)
            (+ 1.0 (pow (+ (/ i n) 1.0) n)))
           (/ i n)))
         0.0)))))

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 tmp;
	if (i <= -43209245.05120845) {
		tmp = 100.0 * ((pow((i / n), n) - 1.0) / (i / n));
	} else if (i <= 0.014357544782884098) {
		tmp = 100.0 * ((0.5 * (i * n)) + (n + (0.16666666666666666 * (i * (i * n)))));
	} else if (i <= 2.8968362884498612e+166) {
		tmp = 0.0;
	} else if (i <= 6.755767338475522e+227) {
		tmp = 100.0 * (((pow(((i / n) + 1.0), (n * 2.0)) + -1.0) / (1.0 + pow(((i / n) + 1.0), n))) / (i / n));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Error

Bits error versus i

Bits error versus n

Target

Original	48.1
Target	47.6
Herbie	14.3

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

Derivation

1. Split input into 4 regimes

2. if i < -43209245.051208451

   1. Initial program 27.7

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

   2. Taylor expanded around inf 64.0

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

   3. Simplified 18.5

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

   if -43209245.051208451 < i < 0.014357544782884098

   1. Initial program 58.0

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

   2. Taylor expanded around 0 26.6

      \[\leadsto 100 \cdot \frac{\color{blue}{0.16666666666666666 \cdot {i}^{3} + \left(0.5 \cdot {i}^{2} + i\right)}}{\frac{i}{n}}\]

   3. Simplified 26.6

      \[\leadsto 100 \cdot \frac{\color{blue}{{i}^{3} \cdot 0.16666666666666666 + \left(i + \left(i \cdot i\right) \cdot 0.5\right)}}{\frac{i}{n}}\]

   4. Taylor expanded around 0 9.4

      \[\leadsto 100 \cdot \color{blue}{\left(0.5 \cdot \left(i \cdot n\right) + \left(n + 0.16666666666666666 \cdot \left({i}^{2} \cdot n\right)\right)\right)}\]

   5. Simplified 9.4

      \[\leadsto 100 \cdot \color{blue}{\left(0.5 \cdot \left(i \cdot n\right) + \left(n + 0.16666666666666666 \cdot \left(i \cdot \left(i \cdot n\right)\right)\right)\right)}\]

   if 0.014357544782884098 < i < 2.89683628844986125e166 or 6.75576733847552218e227 < i

   1. Initial program 33.2

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

   2. Taylor expanded around 0 30.4

      \[\leadsto \color{blue}{0}\]

   if 2.89683628844986125e166 < i < 6.75576733847552218e227

   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 flip--_binary64 34.0

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

   4. Simplified 34.0

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

   5. Simplified 34.0

      \[\leadsto 100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(n \cdot 2\right)} + -1}{\color{blue}{1 + {\left(1 + \frac{i}{n}\right)}^{n}}}}{\frac{i}{n}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 14.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -43209245.05120845:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.014357544782884098:\\ \;\;\;\;100 \cdot \left(0.5 \cdot \left(i \cdot n\right) + \left(n + 0.16666666666666666 \cdot \left(i \cdot \left(i \cdot n\right)\right)\right)\right)\\ \mathbf{elif}\;i \leq 2.8968362884498612 \cdot 10^{+166}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 6.755767338475522 \cdot 10^{+227}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} + -1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020220 
(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))))