Compound Interest

Average Error: 47.7 → 11.3

Time: 15.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -0.001121243698947904:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.135690279498289:\\ \;\;\;\;100 \cdot \left(n \cdot \left(i \cdot 0.5 + \left(0.16666666666666666 \cdot {i}^{2} + \left(1 + 0.041666666666666664 \cdot {i}^{3}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{{n}^{3}}{{i}^{2}} + \left(50 \cdot \frac{{n}^{3} \cdot {\log n}^{2}}{i} + \left(100 \cdot \frac{\log i \cdot {n}^{2}}{i} + 50 \cdot \frac{{n}^{3} \cdot {\log i}^{2}}{i}\right)\right)\right) - \left(100 \cdot \frac{\log n \cdot {n}^{2}}{i} + 100 \cdot \frac{\log n \cdot \left({n}^{3} \cdot \log i\right)}{i}\right)\\ \end{array}\]

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 -0.001121243698947904)
   (* 100.0 (/ (- (exp i) 1.0) (/ i n)))
   (if (<= i 0.135690279498289)
     (*
      100.0
      (*
       n
       (+
        (* i 0.5)
        (+
         (* 0.16666666666666666 (pow i 2.0))
         (+ 1.0 (* 0.041666666666666664 (pow i 3.0)))))))
     (-
      (+
       (* 100.0 (/ (pow n 3.0) (pow i 2.0)))
       (+
        (* 50.0 (/ (* (pow n 3.0) (pow (log n) 2.0)) i))
        (+
         (* 100.0 (/ (* (log i) (pow n 2.0)) i))
         (* 50.0 (/ (* (pow n 3.0) (pow (log i) 2.0)) i)))))
      (+
       (* 100.0 (/ (* (log n) (pow n 2.0)) i))
       (* 100.0 (/ (* (log n) (* (pow n 3.0) (log i))) i)))))))

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 <= -0.001121243698947904) {
		tmp = 100.0 * ((exp(i) - 1.0) / (i / n));
	} else if (i <= 0.135690279498289) {
		tmp = 100.0 * (n * ((i * 0.5) + ((0.16666666666666666 * pow(i, 2.0)) + (1.0 + (0.041666666666666664 * pow(i, 3.0))))));
	} else {
		tmp = ((100.0 * (pow(n, 3.0) / pow(i, 2.0))) + ((50.0 * ((pow(n, 3.0) * pow(log(n), 2.0)) / i)) + ((100.0 * ((log(i) * pow(n, 2.0)) / i)) + (50.0 * ((pow(n, 3.0) * pow(log(i), 2.0)) / i))))) - ((100.0 * ((log(n) * pow(n, 2.0)) / i)) + (100.0 * ((log(n) * (pow(n, 3.0) * log(i))) / i)));
	}
	return tmp;
}

Error

Bits error versus i

Bits error versus n

Target

Original	47.7
Target	47.5
Herbie	11.3

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

Derivation

1. Split input into 3 regimes

2. if i < -0.001121243698947904

   1. Initial program 27.7

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

   2. Taylor expanded around inf 11.5

      \[\leadsto 100 \cdot \frac{\color{blue}{e^{i}} - 1}{\frac{i}{n}}\]

   if -0.001121243698947904 < i < 0.13569027949828899

   1. Initial program 58.1

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

   2. Taylor expanded around 0 13.6

      \[\leadsto 100 \cdot \color{blue}{\left(\left(0.5 \cdot \left(i \cdot n\right) + \left(0.3333333333333333 \cdot \frac{{i}^{2}}{n} + \left(0.16666666666666666 \cdot \left({i}^{2} \cdot n\right) + \left(0.4583333333333333 \cdot \frac{{i}^{3}}{n} + \left(n + 0.041666666666666664 \cdot \left({i}^{3} \cdot n\right)\right)\right)\right)\right)\right) - \left(0.25 \cdot {i}^{3} + \left(0.25 \cdot \frac{{i}^{3}}{{n}^{2}} + \left(0.5 \cdot i + 0.5 \cdot {i}^{2}\right)\right)\right)\right)}\]

   3. Simplified 13.6

      \[\leadsto 100 \cdot \color{blue}{\left(\left(0.5 \cdot \left(n \cdot i\right) + \left(0.3333333333333333 \cdot \frac{i \cdot i}{n} + \left(0.16666666666666666 \cdot \left(n \cdot \left(i \cdot i\right)\right) + \left(0.4583333333333333 \cdot \frac{{i}^{3}}{n} + \left(n + 0.041666666666666664 \cdot \left(n \cdot {i}^{3}\right)\right)\right)\right)\right)\right) - \left(0.25 \cdot \left({i}^{3} + \frac{{i}^{3}}{n \cdot n}\right) + 0.5 \cdot \left(i + i \cdot i\right)\right)\right)}\]

   4. Taylor expanded around inf 9.3

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \left(0.5 \cdot i + \left(0.16666666666666666 \cdot {i}^{2} + \left(0.041666666666666664 \cdot {i}^{3} + 1\right)\right)\right)\right)}\]

   if 0.13569027949828899 < i

   1. Initial program 32.9

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

   2. Taylor expanded around 0 20.8

      \[\leadsto \color{blue}{\left(100 \cdot \frac{{n}^{3}}{{i}^{2}} + \left(50 \cdot \frac{{\log n}^{2} \cdot {n}^{3}}{i} + \left(100 \cdot \frac{\log i \cdot {n}^{2}}{i} + 50 \cdot \frac{{\log i}^{2} \cdot {n}^{3}}{i}\right)\right)\right) - \left(100 \cdot \frac{\log n \cdot {n}^{2}}{i} + 100 \cdot \frac{\log n \cdot \left({n}^{3} \cdot \log i\right)}{i}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.001121243698947904:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.135690279498289:\\ \;\;\;\;100 \cdot \left(n \cdot \left(i \cdot 0.5 + \left(0.16666666666666666 \cdot {i}^{2} + \left(1 + 0.041666666666666664 \cdot {i}^{3}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{{n}^{3}}{{i}^{2}} + \left(50 \cdot \frac{{n}^{3} \cdot {\log n}^{2}}{i} + \left(100 \cdot \frac{\log i \cdot {n}^{2}}{i} + 50 \cdot \frac{{n}^{3} \cdot {\log i}^{2}}{i}\right)\right)\right) - \left(100 \cdot \frac{\log n \cdot {n}^{2}}{i} + 100 \cdot \frac{\log n \cdot \left({n}^{3} \cdot \log i\right)}{i}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021174 
(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))))