Compound Interest

Average Error: 48.2 → 10.9

Time: 11.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -0.00014267079397400185:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.24071237325819259:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + 0.16666666666666666 \cdot \left(i \cdot i\right)\right) + \left(n + 0.3333333333333333 \cdot \left(i \cdot \frac{i}{n}\right)\right)\right) + -0.5 \cdot \left(i + i \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(\frac{{n}^{3}}{i \cdot i} + \left(0.5 \cdot \left({n}^{3} \cdot \frac{{\log i}^{2}}{i} + {n}^{3} \cdot \frac{{\log n}^{2}}{i}\right) + \frac{\log i \cdot \left(n \cdot n\right)}{i}\right)\right) - \left(\frac{\log n \cdot \left(n \cdot n\right)}{i} + \frac{\log n}{i} \cdot \left({n}^{3} \cdot \log i\right)\right)\right)\\ \end{array}\]

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 -0.00014267079397400185)
   (* 100.0 (/ (- (exp i) 1.0) (/ i n)))
   (if (<= i 0.24071237325819259)
     (*
      100.0
      (+
       (+
        (* n (+ (* i 0.5) (* 0.16666666666666666 (* i i))))
        (+ n (* 0.3333333333333333 (* i (/ i n)))))
       (* -0.5 (+ i (* i i)))))
     (*
      100.0
      (-
       (+
        (/ (pow n 3.0) (* i i))
        (+
         (*
          0.5
          (+
           (* (pow n 3.0) (/ (pow (log i) 2.0) i))
           (* (pow n 3.0) (/ (pow (log n) 2.0) i))))
         (/ (* (log i) (* n n)) i)))
       (+
        (/ (* (log n) (* n n)) i)
        (* (/ (log n) i) (* (pow n 3.0) (log 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 <= -0.00014267079397400185) {
		tmp = 100.0 * ((exp(i) - 1.0) / (i / n));
	} else if (i <= 0.24071237325819259) {
		tmp = 100.0 * (((n * ((i * 0.5) + (0.16666666666666666 * (i * i)))) + (n + (0.3333333333333333 * (i * (i / n))))) + (-0.5 * (i + (i * i))));
	} else {
		tmp = 100.0 * (((pow(n, 3.0) / (i * i)) + ((0.5 * ((pow(n, 3.0) * (pow(log(i), 2.0) / i)) + (pow(n, 3.0) * (pow(log(n), 2.0) / i)))) + ((log(i) * (n * n)) / i))) - (((log(n) * (n * n)) / i) + ((log(n) / i) * (pow(n, 3.0) * log(i)))));
	}
	return tmp;
}

Error

Bits error versus i

Bits error versus n

Target

Original	48.2
Target	47.4
Herbie	10.9

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

Derivation

1. Split input into 3 regimes

2. if i < -1.4267079397400185e-4

   1. Initial program 29.7

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

   2. Taylor expanded around inf 11.6

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

   if -1.4267079397400185e-4 < i < 0.240712373258192586

   1. Initial program 58.3

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

   2. Taylor expanded around 0 8.8

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

   3. Simplified 8.7

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

   if 0.240712373258192586 < i

   1. Initial program 32.7

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

   2. Taylor expanded around 0 19.9

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

   3. Simplified 19.9

      \[\leadsto 100 \cdot \color{blue}{\left(\left(\frac{{n}^{3}}{i \cdot i} + \left(0.5 \cdot \left(\frac{{\log i}^{2}}{i} \cdot {n}^{3} + \frac{{\log n}^{2}}{i} \cdot {n}^{3}\right) + \frac{\log i \cdot \left(n \cdot n\right)}{i}\right)\right) - \left(\frac{\log n \cdot \left(n \cdot n\right)}{i} + \frac{\log n}{i} \cdot \left(\log i \cdot {n}^{3}\right)\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.00014267079397400185:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.24071237325819259:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + 0.16666666666666666 \cdot \left(i \cdot i\right)\right) + \left(n + 0.3333333333333333 \cdot \left(i \cdot \frac{i}{n}\right)\right)\right) + -0.5 \cdot \left(i + i \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(\frac{{n}^{3}}{i \cdot i} + \left(0.5 \cdot \left({n}^{3} \cdot \frac{{\log i}^{2}}{i} + {n}^{3} \cdot \frac{{\log n}^{2}}{i}\right) + \frac{\log i \cdot \left(n \cdot n\right)}{i}\right)\right) - \left(\frac{\log n \cdot \left(n \cdot n\right)}{i} + \frac{\log n}{i} \cdot \left({n}^{3} \cdot \log i\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020344 
(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))))