Average Error: 47.7 → 11.6
Time: 13.1s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq -0.00016572415090070473:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.000957179281730772:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + \log \left(e^{0.16666666666666666 \cdot \left(i \cdot i\right)}\right)\right) + \left(n + 0.3333333333333333 \cdot \left(i \cdot \frac{i}{n}\right)\right)\right) + -0.5 \cdot \left(i + i \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(\frac{{n}^{3}}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{3} \cdot {\log i}^{2}}{i} + \left(0.5 \cdot \frac{{n}^{3} \cdot {\log n}^{2}}{i} + \frac{\log i \cdot {n}^{2}}{i}\right)\right)\right) - \left(\frac{\log n \cdot {n}^{2}}{i} + \frac{\log n \cdot \left({n}^{3} \cdot \log i\right)}{i}\right)\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq -0.00016572415090070473:\\
\;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\

\mathbf{elif}\;i \leq 0.000957179281730772:\\
\;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + \log \left(e^{0.16666666666666666 \cdot \left(i \cdot i\right)}\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}^{2}} + \left(0.5 \cdot \frac{{n}^{3} \cdot {\log i}^{2}}{i} + \left(0.5 \cdot \frac{{n}^{3} \cdot {\log n}^{2}}{i} + \frac{\log i \cdot {n}^{2}}{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)\\

\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 -0.00016572415090070473)
   (* 100.0 (/ (- (exp i) 1.0) (/ i n)))
   (if (<= i 0.000957179281730772)
     (*
      100.0
      (+
       (+
        (* n (+ (* i 0.5) (log (exp (* 0.16666666666666666 (* i i))))))
        (+ n (* 0.3333333333333333 (* i (/ i n)))))
       (* -0.5 (+ i (* i i)))))
     (*
      100.0
      (-
       (+
        (/ (pow n 3.0) (pow i 2.0))
        (+
         (* 0.5 (/ (* (pow n 3.0) (pow (log i) 2.0)) i))
         (+
          (* 0.5 (/ (* (pow n 3.0) (pow (log n) 2.0)) i))
          (/ (* (log i) (pow n 2.0)) i))))
       (+
        (/ (* (log n) (pow n 2.0)) i)
        (/ (* (log n) (* (pow n 3.0) (log i))) i)))))))
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 <= -0.00016572415090070473) {
		tmp = 100.0 * ((exp(i) - 1.0) / (i / n));
	} else if (i <= 0.000957179281730772) {
		tmp = 100.0 * (((n * ((i * 0.5) + log(exp(0.16666666666666666 * (i * i))))) + (n + (0.3333333333333333 * (i * (i / n))))) + (-0.5 * (i + (i * i))));
	} else {
		tmp = 100.0 * (((pow(n, 3.0) / pow(i, 2.0)) + ((0.5 * ((pow(n, 3.0) * pow(log(i), 2.0)) / i)) + ((0.5 * ((pow(n, 3.0) * pow(log(n), 2.0)) / i)) + ((log(i) * pow(n, 2.0)) / i)))) - (((log(n) * pow(n, 2.0)) / i) + ((log(n) * (pow(n, 3.0) * log(i))) / i)));
	}
	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

Original47.7
Target47.4
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 < -1.65724150900704734e-4

    1. Initial program 28.2

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

    2. Taylor expanded around inf 12.1

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

    if -1.65724150900704734e-4 < i < 9.5717928173077202e-4

    1. Initial program 58.1

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

    2. Taylor expanded around 0 9.3

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

    3. Simplified9.3

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

    5. Applied *-un-lft-identity_binary649.3

      \[\leadsto 100 \cdot \left(\left(n \cdot \left(0.5 \cdot i + 0.16666666666666666 \cdot \left(i \cdot i\right)\right) + \left(n + 0.3333333333333333 \cdot \frac{i \cdot i}{\color{blue}{1 \cdot n}}\right)\right) + -0.5 \cdot \left(i + i \cdot i\right)\right)\]

    6. Applied times-frac_binary649.2

      \[\leadsto 100 \cdot \left(\left(n \cdot \left(0.5 \cdot i + 0.16666666666666666 \cdot \left(i \cdot i\right)\right) + \left(n + 0.3333333333333333 \cdot \color{blue}{\left(\frac{i}{1} \cdot \frac{i}{n}\right)}\right)\right) + -0.5 \cdot \left(i + i \cdot i\right)\right)\]

    7. Simplified9.2

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

    9. Applied add-log-exp_binary649.2

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

    if 9.5717928173077202e-4 < i

    1. Initial program 32.8

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

    2. Taylor expanded around 0 21.6

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

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.00016572415090070473:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.000957179281730772:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + \log \left(e^{0.16666666666666666 \cdot \left(i \cdot i\right)}\right)\right) + \left(n + 0.3333333333333333 \cdot \left(i \cdot \frac{i}{n}\right)\right)\right) + -0.5 \cdot \left(i + i \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(\frac{{n}^{3}}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{3} \cdot {\log i}^{2}}{i} + \left(0.5 \cdot \frac{{n}^{3} \cdot {\log n}^{2}}{i} + \frac{\log i \cdot {n}^{2}}{i}\right)\right)\right) - \left(\frac{\log n \cdot {n}^{2}}{i} + \frac{\log n \cdot \left({n}^{3} \cdot \log i\right)}{i}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021173 
(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))))