Compound Interest

Average Error: 47.6 → 14.4

Time: 12.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -4.486060090746444 \cdot 10^{+21}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -3.558185175373979 \cdot 10^{-154}:\\ \;\;\;\;\sqrt[3]{\left(n \cdot \left(i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - i \cdot \left(i \cdot 0.5\right)\right)\right)\right) \cdot \frac{100}{i}} \cdot \left(\sqrt[3]{\left(n \cdot \left(i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - i \cdot \left(i \cdot 0.5\right)\right)\right)\right) \cdot \frac{100}{i}} \cdot \sqrt[3]{\left(n \cdot \left(i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - i \cdot \left(i \cdot 0.5\right)\right)\right)\right) \cdot \frac{100}{i}}\right)\\ \mathbf{elif}\;i \leq 250.08617759702352:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i \cdot \left(1 + i \cdot 0.5\right) + \left(n \cdot \log 1 - \log 1 \cdot \left(0.5 \cdot \left(i \cdot i\right)\right)\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}} \cdot \left(100 \cdot \sqrt{\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}}\right)\\ \end{array}\]

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 -4.486060090746444e+21)
   (* 100.0 (/ (- (pow (/ i n) n) 1.0) (/ i n)))
   (if (<= i -3.558185175373979e-154)
     (*
      (cbrt
       (*
        (* n (+ (* i (+ 1.0 (* i 0.5))) (* (log 1.0) (- n (* i (* i 0.5))))))
        (/ 100.0 i)))
      (*
       (cbrt
        (*
         (* n (+ (* i (+ 1.0 (* i 0.5))) (* (log 1.0) (- n (* i (* i 0.5))))))
         (/ 100.0 i)))
       (cbrt
        (*
         (* n (+ (* i (+ 1.0 (* i 0.5))) (* (log 1.0) (- n (* i (* i 0.5))))))
         (/ 100.0 i)))))
     (if (<= i 250.08617759702352)
       (*
        100.0
        (*
         n
         (/
          (+
           (* i (+ 1.0 (* i 0.5)))
           (- (* n (log 1.0)) (* (log 1.0) (* 0.5 (* i i)))))
          i)))
       (*
        (sqrt (/ (- (pow (+ (/ i n) 1.0) n) 1.0) (/ i n)))
        (* 100.0 (sqrt (/ (- (pow (+ (/ i n) 1.0) n) 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 <= -4.486060090746444e+21)) {
		tmp = ((double) (100.0 * (((double) (((double) pow((i / n), n)) - 1.0)) / (i / n))));
	} else {
		double tmp_1;
		if ((i <= -3.558185175373979e-154)) {
			tmp_1 = ((double) (((double) cbrt(((double) (((double) (n * ((double) (((double) (i * ((double) (1.0 + ((double) (i * 0.5)))))) + ((double) (((double) log(1.0)) * ((double) (n - ((double) (i * ((double) (i * 0.5)))))))))))) * (100.0 / i))))) * ((double) (((double) cbrt(((double) (((double) (n * ((double) (((double) (i * ((double) (1.0 + ((double) (i * 0.5)))))) + ((double) (((double) log(1.0)) * ((double) (n - ((double) (i * ((double) (i * 0.5)))))))))))) * (100.0 / i))))) * ((double) cbrt(((double) (((double) (n * ((double) (((double) (i * ((double) (1.0 + ((double) (i * 0.5)))))) + ((double) (((double) log(1.0)) * ((double) (n - ((double) (i * ((double) (i * 0.5)))))))))))) * (100.0 / i)))))))));
		} else {
			double tmp_2;
			if ((i <= 250.08617759702352)) {
				tmp_2 = ((double) (100.0 * ((double) (n * (((double) (((double) (i * ((double) (1.0 + ((double) (i * 0.5)))))) + ((double) (((double) (n * ((double) log(1.0)))) - ((double) (((double) log(1.0)) * ((double) (0.5 * ((double) (i * i)))))))))) / i)))));
			} else {
				tmp_2 = ((double) (((double) sqrt((((double) (((double) pow(((double) ((i / n) + 1.0)), n)) - 1.0)) / (i / n)))) * ((double) (100.0 * ((double) sqrt((((double) (((double) pow(((double) ((i / n) + 1.0)), n)) - 1.0)) / (i / n))))))));
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus i

Bits error versus n

Target

Original	47.6
Target	47.4
Herbie	14.4

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

Derivation

1. Split input into 4 regimes

2. if i < -4.4860600907464442e21

   1. Initial program 26.3

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

   2. Taylor expanded around inf 64.0

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

   3. Simplified 16.1

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

   if -4.4860600907464442e21 < i < -3.55818517537397883e-154

   1. Initial program 53.5

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

   2. Taylor expanded around 0 28.1

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

   3. Simplified 28.1

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

   5. Applied div-inv_binary64 28.1

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

   6. Applied *-un-lft-identity_binary64 28.1

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

   7. Applied times-frac_binary64 17.4

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

   8. Simplified 17.4

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

   10. Applied add-cube-cbrt_binary64 18.0

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

   11. Simplified 18.0

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

   12. Simplified 18.0

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

   if -3.55818517537397883e-154 < i < 250.086177597023521

   1. Initial program 59.0

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

   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 8.3

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

   6. Simplified 8.3

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

   if 250.086177597023521 < 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 add-sqr-sqrt_binary64 31.2

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

   4. Applied associate-*r*_binary64 31.2

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

   5. Simplified 31.2

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

4. Final simplification 14.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -4.486060090746444 \cdot 10^{+21}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -3.558185175373979 \cdot 10^{-154}:\\ \;\;\;\;\sqrt[3]{\left(n \cdot \left(i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - i \cdot \left(i \cdot 0.5\right)\right)\right)\right) \cdot \frac{100}{i}} \cdot \left(\sqrt[3]{\left(n \cdot \left(i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - i \cdot \left(i \cdot 0.5\right)\right)\right)\right) \cdot \frac{100}{i}} \cdot \sqrt[3]{\left(n \cdot \left(i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - i \cdot \left(i \cdot 0.5\right)\right)\right)\right) \cdot \frac{100}{i}}\right)\\ \mathbf{elif}\;i \leq 250.08617759702352:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i \cdot \left(1 + i \cdot 0.5\right) + \left(n \cdot \log 1 - \log 1 \cdot \left(0.5 \cdot \left(i \cdot i\right)\right)\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}} \cdot \left(100 \cdot \sqrt{\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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))))