Average Error: 47.9 → 15.0
Time: 13.2s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq -20604699981894388:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.0526062084788294 \cdot 10^{-06}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq -20604699981894388:\\
\;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\

\mathbf{elif}\;i \leq 1.0526062084788294 \cdot 10^{-06}:\\
\;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} - 1\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 -20604699981894388.0)
   (* 100.0 (/ (- (pow (/ i n) n) 1.0) (/ i n)))
   (if (<= i 1.0526062084788294e-06)
     (* 100.0 (+ n (* n (* i (+ 0.5 (* i 0.16666666666666666))))))
     (/ (* 100.0 (- (pow (+ (/ i n) 1.0) n) 1.0)) (/ 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 <= -20604699981894388.0) {
		tmp = 100.0 * ((pow((i / n), n) - 1.0) / (i / n));
	} else if (i <= 1.0526062084788294e-06) {
		tmp = 100.0 * (n + (n * (i * (0.5 + (i * 0.16666666666666666)))));
	} else {
		tmp = (100.0 * (pow(((i / n) + 1.0), n) - 1.0)) / (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

Original47.9
Target47.5
Herbie15.0
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -20604699981894388

    1. Initial program 27.4

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

    2. Taylor expanded around inf 64.0

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

    3. Simplified17.9

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

    if -20604699981894388 < i < 1.0526062084788294e-6

    1. Initial program 58.1

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

    2. Taylor expanded around 0 26.4

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

    3. Simplified26.4

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

    4. Taylor expanded around 0 10.3

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

    5. Simplified10.3

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

    if 1.0526062084788294e-6 < i

    1. Initial program 32.0

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
    2. Using strategy rm

    3. Applied associate-*r/_binary64_274832.0

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

    4. Simplified32.0

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

  4. Final simplification15.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -20604699981894388:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.0526062084788294 \cdot 10^{-06}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020343 
(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))))