Average Error: 47.7 → 11.8
Time: 15.7s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq -2.1678153021847697 \cdot 10^{-06}:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 3.3504134620314933 \cdot 10^{-32}:\\ \;\;\;\;100 \cdot \left(\left(n + 0.5 \cdot \left(i \cdot n\right)\right) - i \cdot 0.5\right)\\ \mathbf{elif}\;i \leq 1.853747270743323 \cdot 10^{+248}:\\ \;\;\;\;100 \cdot \left(\left(\frac{\left(n \cdot n\right) \cdot \log i}{i} + 0.5 \cdot \left(\frac{{n}^{3} \cdot {\log i}^{2}}{i} + \frac{{n}^{3} \cdot {\log n}^{2}}{i}\right)\right) + \left(\frac{{n}^{3}}{i \cdot i} - \left(\frac{\left(n \cdot n\right) \cdot \log n}{i} + \frac{\log n \cdot \left(\log i \cdot {n}^{3}\right)}{i}\right)\right)\right)\\ \mathbf{elif}\;i \leq 2.852084085226162 \cdot 10^{+286}:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(\frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log n}^{2}\right) + \left(n \cdot \log i + \left(\frac{n \cdot n}{i} + 0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log i}^{2}\right)\right)\right)\right) - \log n \cdot \left(n + \left(n \cdot n\right) \cdot \log i\right)}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq -2.1678153021847697 \cdot 10^{-06}:\\
\;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\

\mathbf{elif}\;i \leq 3.3504134620314933 \cdot 10^{-32}:\\
\;\;\;\;100 \cdot \left(\left(n + 0.5 \cdot \left(i \cdot n\right)\right) - i \cdot 0.5\right)\\

\mathbf{elif}\;i \leq 1.853747270743323 \cdot 10^{+248}:\\
\;\;\;\;100 \cdot \left(\left(\frac{\left(n \cdot n\right) \cdot \log i}{i} + 0.5 \cdot \left(\frac{{n}^{3} \cdot {\log i}^{2}}{i} + \frac{{n}^{3} \cdot {\log n}^{2}}{i}\right)\right) + \left(\frac{{n}^{3}}{i \cdot i} - \left(\frac{\left(n \cdot n\right) \cdot \log n}{i} + \frac{\log n \cdot \left(\log i \cdot {n}^{3}\right)}{i}\right)\right)\right)\\

\mathbf{elif}\;i \leq 2.852084085226162 \cdot 10^{+286}:\\
\;\;\;\;100 \cdot \frac{n \cdot {\left(\frac{i}{n}\right)}^{n} - n}{i}\\

\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} + 0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log i}^{2}\right)\right)\right)\right) - \log n \cdot \left(n + \left(n \cdot n\right) \cdot \log i\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 -2.1678153021847697e-06)
   (* 100.0 (/ (- (exp i) 1.0) (/ i n)))
   (if (<= i 3.3504134620314933e-32)
     (* 100.0 (- (+ n (* 0.5 (* i n))) (* i 0.5)))
     (if (<= i 1.853747270743323e+248)
       (*
        100.0
        (+
         (+
          (/ (* (* n n) (log i)) i)
          (*
           0.5
           (+
            (/ (* (pow n 3.0) (pow (log i) 2.0)) i)
            (/ (* (pow n 3.0) (pow (log n) 2.0)) i))))
         (-
          (/ (pow n 3.0) (* i i))
          (+
           (/ (* (* n n) (log n)) i)
           (/ (* (log n) (* (log i) (pow n 3.0))) i)))))
       (if (<= i 2.852084085226162e+286)
         (* 100.0 (/ (- (* n (pow (/ i n) n)) n) i))
         (*
          100.0
          (/
           (-
            (+
             (* 0.5 (* (* n n) (pow (log n) 2.0)))
             (+
              (* n (log i))
              (+ (/ (* n n) i) (* 0.5 (* (* n n) (pow (log i) 2.0))))))
            (* (log n) (+ n (* (* n n) (log i)))))
           (/ 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 <= -2.1678153021847697e-06) {
		tmp = 100.0 * ((exp(i) - 1.0) / (i / n));
	} else if (i <= 3.3504134620314933e-32) {
		tmp = 100.0 * ((n + (0.5 * (i * n))) - (i * 0.5));
	} else if (i <= 1.853747270743323e+248) {
		tmp = 100.0 * (((((n * n) * log(i)) / i) + (0.5 * (((pow(n, 3.0) * pow(log(i), 2.0)) / i) + ((pow(n, 3.0) * pow(log(n), 2.0)) / i)))) + ((pow(n, 3.0) / (i * i)) - ((((n * n) * log(n)) / i) + ((log(n) * (log(i) * pow(n, 3.0))) / i))));
	} else if (i <= 2.852084085226162e+286) {
		tmp = 100.0 * (((n * pow((i / n), n)) - n) / i);
	} else {
		tmp = 100.0 * ((((0.5 * ((n * n) * pow(log(n), 2.0))) + ((n * log(i)) + (((n * n) / i) + (0.5 * ((n * n) * pow(log(i), 2.0)))))) - (log(n) * (n + ((n * n) * log(i))))) / (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

Original47.7
Target47.5
Herbie11.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 5 regimes

  2. if i < -2.1678153021847697e-6

    1. Initial program 27.9

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

    2. Taylor expanded around inf 12.2

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

    if -2.1678153021847697e-6 < i < 3.3504134620314933e-32

    1. Initial program 58.3

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

    2. Taylor expanded around 0 8.2

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

    3. Simplified8.2

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

    if 3.3504134620314933e-32 < i < 1.85374727074332292e248

    1. Initial program 37.2

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

    2. Taylor expanded around 0 22.7

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

    3. Simplified22.7

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

    if 1.85374727074332292e248 < i < 2.8520840852261622e286

    1. Initial program 32.1

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

    2. Taylor expanded around inf 30.7

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

    3. Simplified32.1

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

    if 2.8520840852261622e286 < i

    1. Initial program 33.5

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

    2. Taylor expanded around 0 28.3

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

    3. Simplified28.3

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log n}^{2}\right) + \left(n \cdot \log i + \left(\frac{n \cdot n}{i} + 0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log i}^{2}\right)\right)\right)\right) - \log n \cdot \left(n + \left(n \cdot n\right) \cdot \log i\right)}}{\frac{i}{n}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification11.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.1678153021847697 \cdot 10^{-06}:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 3.3504134620314933 \cdot 10^{-32}:\\ \;\;\;\;100 \cdot \left(\left(n + 0.5 \cdot \left(i \cdot n\right)\right) - i \cdot 0.5\right)\\ \mathbf{elif}\;i \leq 1.853747270743323 \cdot 10^{+248}:\\ \;\;\;\;100 \cdot \left(\left(\frac{\left(n \cdot n\right) \cdot \log i}{i} + 0.5 \cdot \left(\frac{{n}^{3} \cdot {\log i}^{2}}{i} + \frac{{n}^{3} \cdot {\log n}^{2}}{i}\right)\right) + \left(\frac{{n}^{3}}{i \cdot i} - \left(\frac{\left(n \cdot n\right) \cdot \log n}{i} + \frac{\log n \cdot \left(\log i \cdot {n}^{3}\right)}{i}\right)\right)\right)\\ \mathbf{elif}\;i \leq 2.852084085226162 \cdot 10^{+286}:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(\frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log n}^{2}\right) + \left(n \cdot \log i + \left(\frac{n \cdot n}{i} + 0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log i}^{2}\right)\right)\right)\right) - \log n \cdot \left(n + \left(n \cdot n\right) \cdot \log i\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021024 
(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))))