Compound Interest

Average Error: 48.1 → 14.6

Time: 12.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -0.1758689209773507:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 206.5505471855869:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\frac{\sqrt[3]{i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - 0.5 \cdot \left(i \cdot i\right)\right)} \cdot \sqrt[3]{i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - 0.5 \cdot \left(i \cdot i\right)\right)}}{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \frac{\sqrt[3]{i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - 0.5 \cdot \left(i \cdot i\right)\right)}}{\sqrt[3]{i}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \end{array}\]

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 -0.1758689209773507)
   (* 100.0 (/ (- (pow (/ i n) n) 1.0) (/ i n)))
   (if (<= i 206.5505471855869)
     (*
      100.0
      (*
       n
       (*
        (/
         (*
          (cbrt
           (+ (* i (+ 1.0 (* i 0.5))) (* (log 1.0) (- n (* 0.5 (* i i))))))
          (cbrt
           (+ (* i (+ 1.0 (* i 0.5))) (* (log 1.0) (- n (* 0.5 (* i i)))))))
         (* (cbrt i) (cbrt i)))
        (/
         (cbrt (+ (* i (+ 1.0 (* i 0.5))) (* (log 1.0) (- n (* 0.5 (* i i))))))
         (cbrt i)))))
     (*
      100.0
      (/
       (/
        (- (pow (+ (/ i n) 1.0) (* n 2.0)) (* 1.0 1.0))
        (+ 1.0 (pow (+ (/ i n) 1.0) n)))
       (/ 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 <= -0.1758689209773507)) {
		tmp = ((double) (100.0 * (((double) (((double) pow((i / n), n)) - 1.0)) / (i / n))));
	} else {
		double tmp_1;
		if ((i <= 206.5505471855869)) {
			tmp_1 = ((double) (100.0 * ((double) (n * ((double) ((((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)))))))))))) * ((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)))))))))))))) / ((double) (((double) cbrt(i)) * ((double) cbrt(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)))))))))))) / ((double) cbrt(i)))))))));
		} else {
			tmp_1 = ((double) (100.0 * ((((double) (((double) pow(((double) ((i / n) + 1.0)), ((double) (n * 2.0)))) - ((double) (1.0 * 1.0)))) / ((double) (1.0 + ((double) pow(((double) ((i / n) + 1.0)), n))))) / (i / n))));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus i

Bits error versus n

Target

Original	48.1
Target	47.8
Herbie	14.6

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

Derivation

1. Split input into 3 regimes

2. if i < -0.175868920977350712

   1. Initial program 28.5

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

   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 19.3

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

   if -0.175868920977350712 < i < 206.550547185586908

   1. Initial program 58.0

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

   2. Taylor expanded around 0 26.4

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

   3. Simplified 26.4

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

   5. Applied associate-/r/_binary64 9.5

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

   7. Applied add-cube-cbrt_binary64 10.7

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

   8. Applied add-cube-cbrt_binary64 9.5

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

   9. Applied times-frac_binary64 9.5

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

   10. Simplified 9.5

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

   11. Simplified 9.5

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

   if 206.550547185586908 < i

   1. Initial program 32.2

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

   3. Applied flip--_binary64 32.3

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

   4. Simplified 32.2

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

   5. Simplified 32.2

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

4. Final simplification 14.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.1758689209773507:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 206.5505471855869:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\frac{\sqrt[3]{i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - 0.5 \cdot \left(i \cdot i\right)\right)} \cdot \sqrt[3]{i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - 0.5 \cdot \left(i \cdot i\right)\right)}}{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \frac{\sqrt[3]{i \cdot \left(1 + i \cdot 0.5\right) + \log 1 \cdot \left(n - 0.5 \cdot \left(i \cdot i\right)\right)}}{\sqrt[3]{i}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(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))))