Compound Interest

Average Error: 43.0 → 21.4

Time: 16.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -0.0281548125542239701 \lor \neg \left(i \le 9.343203234098088 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(\mathsf{fma}\left(i, 0.5, \mathsf{fma}\left(\frac{1}{2}, \log 1 \cdot n, 1\right) - \sqrt{1}\right) \cdot n\right) \cdot \left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}\right)}{i}\\ \end{array}\]

C

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 <= -0.02815481255422397) || !(i <= 9.343203234098088e-18))) {
		temp = ((100.0 * (pow((1.0 + (i / n)), n) - 1.0)) / (i / n));
	} else {
		temp = (100.0 * (((fma(i, 0.5, (fma(0.5, (log(1.0) * n), 1.0) - sqrt(1.0))) * n) * (pow((1.0 + (i / n)), (n / 2.0)) + sqrt(1.0))) / i));
	}
	return temp;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.0
Target	42.4
Herbie	21.4

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

Derivation

1. Split input into 2 regimes

2. if i < -0.02815481255422397 or 9.343203234098088e-18 < i

   1. Initial program 31.1

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

   3. Applied associate-*r/ 31.1

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

   if -0.02815481255422397 < i < 9.343203234098088e-18

   1. Initial program 50.4

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

   3. Applied div-inv 50.4

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

   4. Applied add-sqr-sqrt 50.4

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

   5. Applied sqr-pow 50.4

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

   6. Applied difference-of-squares 50.4

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

   7. Applied times-frac 50.1

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

   8. Applied associate-*r* 50.1

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

   9. Taylor expanded around 0 50.0

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

   10. Simplified 16.0

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

   12. Applied associate-*l* 15.6

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

   13. Simplified 15.6

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

   15. Applied associate-*r/ 15.4

       \[\leadsto 100 \cdot \color{blue}{\frac{\left(\mathsf{fma}\left(i, 0.5, \mathsf{fma}\left(\frac{1}{2}, \log 1 \cdot n, 1\right) - \sqrt{1}\right) \cdot n\right) \cdot \left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}\right)}{i}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 21.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.0281548125542239701 \lor \neg \left(i \le 9.343203234098088 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(\mathsf{fma}\left(i, 0.5, \mathsf{fma}\left(\frac{1}{2}, \log 1 \cdot n, 1\right) - \sqrt{1}\right) \cdot n\right) \cdot \left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}\right)}{i}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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))))