Compound Interest

Average Error: 47.3 → 14.5

Time: 20.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -2.0660858770764103 \cdot 10^{-06}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left(-1 + {\left(\frac{i}{n}\right)}^{n}\right)\right)\\ \mathbf{elif}\;i \leq 3.042363773132192:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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 -2.0660858770764103e-06)
   (* 100.0 (* (/ n i) (+ -1.0 (pow (/ i n) n))))
   (if (<= i 3.042363773132192)
     (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
     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 <= -2.0660858770764103e-06) {
		tmp = 100.0 * ((n / i) * (-1.0 + pow((i / n), n)));
	} else if (i <= 3.042363773132192) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Error

Bits error versus i

Bits error versus n

Target

Original	47.3
Target	47.0
Herbie	14.5

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

Derivation

1. Split input into 3 regimes

2. if i < -2.06608587707641033e-6

   1. Initial program 27.2

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

   2. Taylor expanded around inf 64.0

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

   3. Simplified 19.4

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

   if -2.06608587707641033e-6 < i < 3.042363773132192

   1. Initial program 57.9

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

   2. Taylor expanded around 0 26.6

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

   3. Simplified 26.6

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

   4. Taylor expanded around 0 9.6

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + \left(100 \cdot n + 16.666666666666668 \cdot \left({i}^{2} \cdot n\right)\right)}\]

   5. Simplified 9.6

      \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot \left(50 + 16.666666666666668 \cdot i\right)\right)}\]

   if 3.042363773132192 < i

   1. Initial program 31.9

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

   2. Taylor expanded around 0 30.1

      \[\leadsto \color{blue}{0}\]
3. Recombined 3 regimes into one program.

4. Final simplification 14.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.0660858770764103 \cdot 10^{-06}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left(-1 + {\left(\frac{i}{n}\right)}^{n}\right)\right)\\ \mathbf{elif}\;i \leq 3.042363773132192:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020277 
(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))))