Compound Interest

Average Error: 47.3 → 16.4

Time: 11.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -4.331633902717184 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{\frac{-1}{n}}{\frac{-1}{i}}\right)}^{n}}{\frac{i}{n}} - \frac{n}{i}\right)\\ \mathbf{elif}\;i \leq 1.1307223325030768 \cdot 10^{+27}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\sqrt[3]{\frac{i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - 0.5 \cdot \left(i \cdot i\right)\right)}{i}} \cdot \left(\sqrt[3]{\frac{i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - 0.5 \cdot \left(i \cdot i\right)\right)}{i}} \cdot \sqrt[3]{\frac{i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - 0.5 \cdot \left(i \cdot i\right)\right)}{i}}\right)\right)\right)\\ \mathbf{elif}\;i \leq 1.091638204103338 \cdot 10^{+297}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(i \cdot 1 + \left(1 + n \cdot \log 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))

↓

(FPCore (i n)
 :precision binary64
 (if (<= i -4.331633902717184e+28)
   (* 100.0 (- (/ (pow (/ (/ -1.0 n) (/ -1.0 i)) n) (/ i n)) (/ n i)))
   (if (<= i 1.1307223325030768e+27)
     (*
      100.0
      (*
       n
       (*
        (cbrt
         (/ (+ (* i (+ 1.0 (* i 0.5))) (* (log 1.0) (- n (* 0.5 (* i i))))) i))
        (*
         (cbrt
          (/
           (+ (* i (+ 1.0 (* i 0.5))) (* (log 1.0) (- n (* 0.5 (* i i)))))
           i))
         (cbrt
          (/
           (+ (* i (+ 1.0 (* i 0.5))) (* (log 1.0) (- n (* 0.5 (* i i)))))
           i))))))
     (if (<= i 1.091638204103338e+297)
       (* n (* 100.0 (/ (- (pow (+ (/ i n) 1.0) n) 1.0) i)))
       (* 100.0 (/ (- (+ (* i 1.0) (+ 1.0 (* n (log 1.0)))) 1.0) (/ i n)))))))

C

double code(double i, double n) {
	return ((double) (100.0 * (((double) (((double) pow(((double) (1.0 + (i / n))), n)) - 1.0)) / (i / n))));
}

↓

double code(double i, double n) {
	double tmp;
	if ((i <= -4.331633902717184e+28)) {
		tmp = ((double) (100.0 * ((double) ((((double) pow(((-1.0 / n) / (-1.0 / i)), n)) / (i / n)) - (n / i)))));
	} else {
		double tmp_1;
		if ((i <= 1.1307223325030768e+27)) {
			tmp_1 = ((double) (100.0 * ((double) (n * ((double) (((double) cbrt((((double) (((double) (i * ((double) (1.0 + ((double) (i * 0.5)))))) + ((double) (((double) log(1.0)) * ((double) (n - ((double) (0.5 * ((double) (i * i)))))))))) / i))) * ((double) (((double) cbrt((((double) (((double) (i * ((double) (1.0 + ((double) (i * 0.5)))))) + ((double) (((double) log(1.0)) * ((double) (n - ((double) (0.5 * ((double) (i * i)))))))))) / i))) * ((double) cbrt((((double) (((double) (i * ((double) (1.0 + ((double) (i * 0.5)))))) + ((double) (((double) log(1.0)) * ((double) (n - ((double) (0.5 * ((double) (i * i)))))))))) / i)))))))))));
		} else {
			double tmp_2;
			if ((i <= 1.091638204103338e+297)) {
				tmp_2 = ((double) (n * ((double) (100.0 * (((double) (((double) pow(((double) ((i / n) + 1.0)), n)) - 1.0)) / i)))));
			} else {
				tmp_2 = ((double) (100.0 * (((double) (((double) (((double) (i * 1.0)) + ((double) (1.0 + ((double) (n * ((double) log(1.0)))))))) - 1.0)) / (i / n))));
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus i

Bits error versus n

Target

Original	47.3
Target	46.9
Herbie	16.4

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

Derivation

1. Split input into 4 regimes

2. if i < -4.3316339027171839e28

   1. Initial program 25.6

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

   2. Taylor expanded around -inf 29.1

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

   3. Simplified 19.0

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

   if -4.3316339027171839e28 < i < 1.13072233250307679e27

   1. Initial program 57.4

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

   2. Taylor expanded around 0 28.3

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

   3. Simplified 28.3

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

   5. Applied associate-/r/_binary64 12.6

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

   6. Simplified 12.6

      \[\leadsto 100 \cdot \left(\color{blue}{\frac{i \cdot \left(1 + i \cdot 0.5\right) + \left(n \cdot \log 1 - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}} \cdot n\right)\]
   7. Using strategy rm

   8. Applied add-cube-cbrt_binary64 12.7

      \[\leadsto 100 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\frac{i \cdot \left(1 + i \cdot 0.5\right) + \left(n \cdot \log 1 - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}} \cdot \sqrt[3]{\frac{i \cdot \left(1 + i \cdot 0.5\right) + \left(n \cdot \log 1 - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}}\right) \cdot \sqrt[3]{\frac{i \cdot \left(1 + i \cdot 0.5\right) + \left(n \cdot \log 1 - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}}\right)} \cdot n\right)\]

   9. Simplified 12.7

      \[\leadsto 100 \cdot \left(\left(\color{blue}{\left(\sqrt[3]{\frac{i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - \left(i \cdot i\right) \cdot 0.5\right)}{i}} \cdot \sqrt[3]{\frac{i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - \left(i \cdot i\right) \cdot 0.5\right)}{i}}\right)} \cdot \sqrt[3]{\frac{i \cdot \left(1 + i \cdot 0.5\right) + \left(n \cdot \log 1 - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}}\right) \cdot n\right)\]

   10. Simplified 12.7

       \[\leadsto 100 \cdot \left(\left(\left(\sqrt[3]{\frac{i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - \left(i \cdot i\right) \cdot 0.5\right)}{i}} \cdot \sqrt[3]{\frac{i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - \left(i \cdot i\right) \cdot 0.5\right)}{i}}\right) \cdot \color{blue}{\sqrt[3]{\frac{i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - \left(i \cdot i\right) \cdot 0.5\right)}{i}}}\right) \cdot n\right)\]

   if 1.13072233250307679e27 < i < 1.091638204103338e297

   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/_binary64 31.1

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

   4. Applied associate-*r*_binary64 31.1

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

   5. Simplified 31.1

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

   if 1.091638204103338e297 < i

   1. Initial program 28.8

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

   2. Taylor expanded around 0 35.7

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

   3. Simplified 35.7

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

4. Final simplification 16.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -4.331633902717184 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{\frac{-1}{n}}{\frac{-1}{i}}\right)}^{n}}{\frac{i}{n}} - \frac{n}{i}\right)\\ \mathbf{elif}\;i \leq 1.1307223325030768 \cdot 10^{+27}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\sqrt[3]{\frac{i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - 0.5 \cdot \left(i \cdot i\right)\right)}{i}} \cdot \left(\sqrt[3]{\frac{i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - 0.5 \cdot \left(i \cdot i\right)\right)}{i}} \cdot \sqrt[3]{\frac{i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - 0.5 \cdot \left(i \cdot i\right)\right)}{i}}\right)\right)\right)\\ \mathbf{elif}\;i \leq 1.091638204103338 \cdot 10^{+297}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(i \cdot 1 + \left(1 + n \cdot \log 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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))))