Average Error: 47.5 → 11.6
Time: 31.4s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq -0.00018629600656972037:\\ \;\;\;\;100 \cdot \left(\frac{n \cdot \left(e^{i} + -1\right)}{i} - 0.5 \cdot \left(i \cdot e^{i}\right)\right)\\ \mathbf{elif}\;i \leq 5.60806670825752 \cdot 10^{-06}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + 0.16666666666666666 \cdot \left(i \cdot i\right)\right) + \left(n + 0.3333333333333333 \cdot \frac{i \cdot i}{n}\right)\right) + -0.5 \cdot \left(i + i \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log n}^{2}\right) + \left(n \cdot \log i + \left(\frac{n \cdot n}{i} + \left(\frac{\log i \cdot {n}^{3}}{i} + \left(0.5 \cdot \left({\log n}^{2} \cdot \left(\log i \cdot {n}^{3}\right) + \left(n \cdot n\right) \cdot {\log i}^{2}\right) + 0.16666666666666666 \cdot \left({n}^{3} \cdot {\log i}^{3}\right)\right)\right)\right)\right)\right) - \left(0.5 \cdot \frac{{n}^{3}}{i \cdot i} + \left(\log n \cdot \left(n + \left(n \cdot n\right) \cdot \log i\right) + \left(\frac{\log n \cdot {n}^{3}}{i} + \left(0.5 \cdot \left(\log n \cdot \left({n}^{3} \cdot {\log i}^{2}\right)\right) + 0.16666666666666666 \cdot \left({n}^{3} \cdot {\log n}^{3}\right)\right)\right)\right)\right)}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq -0.00018629600656972037:\\
\;\;\;\;100 \cdot \left(\frac{n \cdot \left(e^{i} + -1\right)}{i} - 0.5 \cdot \left(i \cdot e^{i}\right)\right)\\

\mathbf{elif}\;i \leq 5.60806670825752 \cdot 10^{-06}:\\
\;\;\;\;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 \frac{i \cdot i}{n}\right)\right) + -0.5 \cdot \left(i + i \cdot i\right)\right)\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \frac{\left(0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log n}^{2}\right) + \left(n \cdot \log i + \left(\frac{n \cdot n}{i} + \left(\frac{\log i \cdot {n}^{3}}{i} + \left(0.5 \cdot \left({\log n}^{2} \cdot \left(\log i \cdot {n}^{3}\right) + \left(n \cdot n\right) \cdot {\log i}^{2}\right) + 0.16666666666666666 \cdot \left({n}^{3} \cdot {\log i}^{3}\right)\right)\right)\right)\right)\right) - \left(0.5 \cdot \frac{{n}^{3}}{i \cdot i} + \left(\log n \cdot \left(n + \left(n \cdot n\right) \cdot \log i\right) + \left(\frac{\log n \cdot {n}^{3}}{i} + \left(0.5 \cdot \left(\log n \cdot \left({n}^{3} \cdot {\log i}^{2}\right)\right) + 0.16666666666666666 \cdot \left({n}^{3} \cdot {\log n}^{3}\right)\right)\right)\right)\right)}{\frac{i}{n}}\\

\end{array}
(FPCore (i n)
 :precision binary64
 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))
↓
(FPCore (i n)
 :precision binary64
 (if (<= i -0.00018629600656972037)
   (* 100.0 (- (/ (* n (+ (exp i) -1.0)) i) (* 0.5 (* i (exp i)))))
   (if (<= i 5.60806670825752e-06)
     (*
      100.0
      (+
       (+
        (* n (+ (* i 0.5) (* 0.16666666666666666 (* i i))))
        (+ n (* 0.3333333333333333 (/ (* i i) n))))
       (* -0.5 (+ i (* i i)))))
     (*
      100.0
      (/
       (-
        (+
         (* 0.5 (* (* n n) (pow (log n) 2.0)))
         (+
          (* n (log i))
          (+
           (/ (* n n) i)
           (+
            (/ (* (log i) (pow n 3.0)) i)
            (+
             (*
              0.5
              (+
               (* (pow (log n) 2.0) (* (log i) (pow n 3.0)))
               (* (* n n) (pow (log i) 2.0))))
             (* 0.16666666666666666 (* (pow n 3.0) (pow (log i) 3.0))))))))
        (+
         (* 0.5 (/ (pow n 3.0) (* i i)))
         (+
          (* (log n) (+ n (* (* n n) (log i))))
          (+
           (/ (* (log n) (pow n 3.0)) i)
           (+
            (* 0.5 (* (log n) (* (pow n 3.0) (pow (log i) 2.0))))
            (* 0.16666666666666666 (* (pow n 3.0) (pow (log n) 3.0))))))))
       (/ i n))))))
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 <= -0.00018629600656972037) {
		tmp = 100.0 * (((n * (exp(i) + -1.0)) / i) - (0.5 * (i * exp(i))));
	} else if (i <= 5.60806670825752e-06) {
		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 * ((((0.5 * ((n * n) * pow(log(n), 2.0))) + ((n * log(i)) + (((n * n) / i) + (((log(i) * pow(n, 3.0)) / i) + ((0.5 * ((pow(log(n), 2.0) * (log(i) * pow(n, 3.0))) + ((n * n) * pow(log(i), 2.0)))) + (0.16666666666666666 * (pow(n, 3.0) * pow(log(i), 3.0)))))))) - ((0.5 * (pow(n, 3.0) / (i * i))) + ((log(n) * (n + ((n * n) * log(i)))) + (((log(n) * pow(n, 3.0)) / i) + ((0.5 * (log(n) * (pow(n, 3.0) * pow(log(i), 2.0)))) + (0.16666666666666666 * (pow(n, 3.0) * pow(log(n), 3.0)))))))) / (i / n));
	}
	return tmp;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.5
Target47.3
Herbie11.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -1.8629600656972037e-4

    1. Initial program 27.7

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

    2. Taylor expanded around inf 13.1

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

    3. Simplified13.1

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

    if -1.8629600656972037e-4 < i < 5.60806670825751969e-6

    1. Initial program 58.4

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

    2. Taylor expanded around 0 8.7

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

    3. Simplified8.7

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

    if 5.60806670825751969e-6 < i

    1. Initial program 32.1

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

    2. Taylor expanded around 0 21.7

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

    3. Simplified21.7

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log n}^{2}\right) + \left(n \cdot \log i + \left(\frac{n \cdot n}{i} + \left(\frac{{n}^{3} \cdot \log i}{i} + \left(0.5 \cdot \left({\log n}^{2} \cdot \left({n}^{3} \cdot \log i\right) + \left(n \cdot n\right) \cdot {\log i}^{2}\right) + 0.16666666666666666 \cdot \left({n}^{3} \cdot {\log i}^{3}\right)\right)\right)\right)\right)\right) - \left(0.5 \cdot \frac{{n}^{3}}{i \cdot i} + \left(\log n \cdot \left(n + \left(n \cdot n\right) \cdot \log i\right) + \left(\frac{{n}^{3} \cdot \log n}{i} + \left(0.5 \cdot \left(\log n \cdot \left({n}^{3} \cdot {\log i}^{2}\right)\right) + 0.16666666666666666 \cdot \left({n}^{3} \cdot {\log n}^{3}\right)\right)\right)\right)\right)}}{\frac{i}{n}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.00018629600656972037:\\ \;\;\;\;100 \cdot \left(\frac{n \cdot \left(e^{i} + -1\right)}{i} - 0.5 \cdot \left(i \cdot e^{i}\right)\right)\\ \mathbf{elif}\;i \leq 5.60806670825752 \cdot 10^{-06}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + 0.16666666666666666 \cdot \left(i \cdot i\right)\right) + \left(n + 0.3333333333333333 \cdot \frac{i \cdot i}{n}\right)\right) + -0.5 \cdot \left(i + i \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log n}^{2}\right) + \left(n \cdot \log i + \left(\frac{n \cdot n}{i} + \left(\frac{\log i \cdot {n}^{3}}{i} + \left(0.5 \cdot \left({\log n}^{2} \cdot \left(\log i \cdot {n}^{3}\right) + \left(n \cdot n\right) \cdot {\log i}^{2}\right) + 0.16666666666666666 \cdot \left({n}^{3} \cdot {\log i}^{3}\right)\right)\right)\right)\right)\right) - \left(0.5 \cdot \frac{{n}^{3}}{i \cdot i} + \left(\log n \cdot \left(n + \left(n \cdot n\right) \cdot \log i\right) + \left(\frac{\log n \cdot {n}^{3}}{i} + \left(0.5 \cdot \left(\log n \cdot \left({n}^{3} \cdot {\log i}^{2}\right)\right) + 0.16666666666666666 \cdot \left({n}^{3} \cdot {\log n}^{3}\right)\right)\right)\right)\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))