Average Error: 47.5 → 12.3
Time: 21.9s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq -0.00014452224618902238:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.006229383881859722:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + 0.16666666666666666 \cdot \left(i \cdot i\right)\right) + \left(n + 0.3333333333333333 \cdot \left(\frac{i}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{i}{\sqrt[3]{n}}\right)\right)\right) + -0.5 \cdot \left(i + i \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(i \cdot \frac{1}{n}\right)}^{n} - n}{i}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq -0.00014452224618902238:\\
\;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\

\mathbf{elif}\;i \leq 0.006229383881859722:\\
\;\;\;\;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(\frac{i}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{i}{\sqrt[3]{n}}\right)\right)\right) + -0.5 \cdot \left(i + i \cdot i\right)\right)\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \frac{n \cdot {\left(i \cdot \frac{1}{n}\right)}^{n} - n}{i}\\

\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 -0.00014452224618902238)
   (* 100.0 (/ (- (exp i) 1.0) (/ i n)))
   (if (<= i 0.006229383881859722)
     (*
      100.0
      (+
       (+
        (* n (+ (* i 0.5) (* 0.16666666666666666 (* i i))))
        (+
         n
         (*
          0.3333333333333333
          (* (/ i (* (cbrt n) (cbrt n))) (/ i (cbrt n))))))
       (* -0.5 (+ i (* i i)))))
     (* 100.0 (/ (- (* n (pow (* i (/ 1.0 n)) n)) n) 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 <= -0.00014452224618902238) {
		tmp = 100.0 * ((exp(i) - 1.0) / (i / n));
	} else if (i <= 0.006229383881859722) {
		tmp = 100.0 * (((n * ((i * 0.5) + (0.16666666666666666 * (i * i)))) + (n + (0.3333333333333333 * ((i / (cbrt(n) * cbrt(n))) * (i / cbrt(n)))))) + (-0.5 * (i + (i * i))));
	} else {
		tmp = 100.0 * (((n * pow((i * (1.0 / n)), n)) - n) / 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

Original47.5
Target47.8
Herbie12.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -1.44522246189022384e-4

    1. Initial program 28.3

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

    2. Taylor expanded around inf 11.0

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

    if -1.44522246189022384e-4 < i < 0.0062293838818597222

    1. Initial program 58.2

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

    2. Taylor expanded around 0 8.9

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

    3. Simplified8.9

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

    5. Applied add-cube-cbrt_binary64_35238.9

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

    6. Applied times-frac_binary64_34948.9

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

    if 0.0062293838818597222 < i

    1. Initial program 31.2

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

    2. Taylor expanded around inf 31.1

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

    3. Simplified31.3

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot {\left(\frac{1}{n} \cdot i\right)}^{n} - n}{i}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.00014452224618902238:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.006229383881859722:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + 0.16666666666666666 \cdot \left(i \cdot i\right)\right) + \left(n + 0.3333333333333333 \cdot \left(\frac{i}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{i}{\sqrt[3]{n}}\right)\right)\right) + -0.5 \cdot \left(i + i \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(i \cdot \frac{1}{n}\right)}^{n} - n}{i}\\ \end{array}\]

Reproduce

herbie shell --seed 2021093 
(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))))