Average Error: 47.3 → 16.6
Time: 13.7s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.32264855634611611:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 6.2316967628408393 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 3.16763600985918069 \cdot 10^{238}:\\ \;\;\;\;100 \cdot \left(\frac{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + \sqrt{1}}{i} \cdot \left(\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} - \sqrt{1}\right) \cdot n\right)\right)\\ \mathbf{elif}\;i \le 1.8069577640268902 \cdot 10^{255}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + \sqrt{1}}{i} \cdot \left(\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} - \sqrt{1}\right) \cdot n\right)\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -1.32264855634611611:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 6.2316967628408393 \cdot 10^{-6}:\\
\;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\right)\\

\mathbf{elif}\;i \le 3.16763600985918069 \cdot 10^{238}:\\
\;\;\;\;100 \cdot \left(\frac{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + \sqrt{1}}{i} \cdot \left(\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} - \sqrt{1}\right) \cdot n\right)\right)\\

\mathbf{elif}\;i \le 1.8069577640268902 \cdot 10^{255}:\\
\;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\

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

\end{array}
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 temp;
	if ((i <= -1.322648556346116)) {
		temp = (100.0 * (((pow((1.0 + (i / n)), (2.0 * n)) + -(1.0 * 1.0)) / (pow((1.0 + (i / n)), n) + 1.0)) / (i / n)));
	} else {
		double temp_1;
		if ((i <= 6.231696762840839e-06)) {
			temp_1 = (100.0 * ((fma(i, 1.0, (fma(0.5, pow(i, 2.0), (log(1.0) * n)) - (0.5 * (pow(i, 2.0) * log(1.0))))) / i) * n));
		} else {
			double temp_2;
			if ((i <= 3.1676360098591807e+238)) {
				temp_2 = (100.0 * (((sqrt(pow((1.0 + (i / n)), n)) + sqrt(1.0)) / i) * ((sqrt(pow((1.0 + (i / n)), n)) - sqrt(1.0)) * n)));
			} else {
				double temp_3;
				if ((i <= 1.8069577640268902e+255)) {
					temp_3 = (100.0 * ((fma(1.0, i, fma(log(1.0), n, 1.0)) - 1.0) / (i / n)));
				} else {
					temp_3 = (100.0 * (((sqrt(pow((1.0 + (i / n)), n)) + sqrt(1.0)) / i) * ((sqrt(pow((1.0 + (i / n)), n)) - sqrt(1.0)) * n)));
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.3
Target47.5
Herbie16.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -1.322648556346116

    1. Initial program 27.3

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

    3. Applied flip--27.3

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

    4. Simplified27.2

      \[\leadsto 100 \cdot \frac{\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\]

    if -1.322648556346116 < i < 6.231696762840839e-06

    1. Initial program 57.8

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

    2. Taylor expanded around 0 26.0

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{i}{n}}\]

    3. Simplified26.0

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}}{\frac{i}{n}}\]
    4. Using strategy rm

    5. Applied associate-/r/9.5

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\right)}\]

    if 6.231696762840839e-06 < i < 3.1676360098591807e+238 or 1.8069577640268902e+255 < i

    1. Initial program 31.4

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

    3. Applied div-inv31.5

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

    4. Applied add-sqr-sqrt31.5

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

    5. Applied add-sqr-sqrt31.5

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

    6. Applied difference-of-squares31.5

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

    7. Applied times-frac31.5

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

    8. Simplified31.5

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

    if 3.1676360098591807e+238 < i < 1.8069577640268902e+255

    1. Initial program 33.5

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

    2. Taylor expanded around 0 32.2

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

    3. Simplified32.2

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right)} - 1}{\frac{i}{n}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.32264855634611611:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 6.2316967628408393 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 3.16763600985918069 \cdot 10^{238}:\\ \;\;\;\;100 \cdot \left(\frac{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + \sqrt{1}}{i} \cdot \left(\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} - \sqrt{1}\right) \cdot n\right)\right)\\ \mathbf{elif}\;i \le 1.8069577640268902 \cdot 10^{255}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + \sqrt{1}}{i} \cdot \left(\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} - \sqrt{1}\right) \cdot n\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :herbie-target
  (* 100 (/ (- (exp (* n (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) 1) (/ i n)))

  (* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))))