Compound Interest

Average Error: 47.3 → 15.7

Time: 12.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -4.331633902717184 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.051881246670273766:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{n \cdot \log 1 + \left(i \cdot \left(\sqrt[3]{1} + \sqrt[3]{1} \cdot \left(i \cdot \left(0.3333333333333333 - i \cdot 0.027777777777777776\right)\right)\right)\right) \cdot \sqrt[3]{1 + i \cdot 0.5}}{i} - n \cdot \left(i \cdot \left(\log 1 \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;i \leq 7.413989741085994 \cdot 10^{+291}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 + \left(n \cdot \log 1 + i \cdot 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

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 (/ i n) n) 1.0) (/ i n)))
   (if (<= i 0.051881246670273766)
     (*
      100.0
      (-
       (*
        n
        (/
         (+
          (* n (log 1.0))
          (*
           (*
            i
            (+
             (cbrt 1.0)
             (*
              (cbrt 1.0)
              (* i (- 0.3333333333333333 (* i 0.027777777777777776))))))
           (cbrt (+ 1.0 (* i 0.5)))))
         i))
       (* n (* i (* (log 1.0) 0.5)))))
     (if (<= i 7.413989741085994e+291)
       (* 100.0 (* n (/ (- (pow (/ i n) n) 1.0) i)))
       (* 100.0 (/ (- (+ 1.0 (+ (* n (log 1.0)) (* i 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((i / n), n)) - 1.0)) / (i / n))));
	} else {
		double tmp_1;
		if ((i <= 0.051881246670273766)) {
			tmp_1 = ((double) (100.0 * ((double) (((double) (n * (((double) (((double) (n * ((double) log(1.0)))) + ((double) (((double) (i * ((double) (((double) cbrt(1.0)) + ((double) (((double) cbrt(1.0)) * ((double) (i * ((double) (0.3333333333333333 - ((double) (i * 0.027777777777777776)))))))))))) * ((double) cbrt(((double) (1.0 + ((double) (i * 0.5)))))))))) / i))) - ((double) (n * ((double) (i * ((double) (((double) log(1.0)) * 0.5))))))))));
		} else {
			double tmp_2;
			if ((i <= 7.413989741085994e+291)) {
				tmp_2 = ((double) (100.0 * ((double) (n * (((double) (((double) pow((i / n), n)) - 1.0)) / i)))));
			} else {
				tmp_2 = ((double) (100.0 * (((double) (((double) (1.0 + ((double) (((double) (n * ((double) log(1.0)))) + ((double) (i * 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	15.7

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

Derivation

1. Split input into 4 regimes

2. if i < -4.3316339027171839e28

   1. Initial program Error: 25.6 bits

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

   2. Taylor expanded around inf Error: 64.0 bits

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

   3. Simplified Error: 17.0 bits

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

   if -4.3316339027171839e28 < i < 0.051881246670273766

   1. Initial program Error: 57.8 bits

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

   2. Taylor expanded around 0 Error: 27.8 bits

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

   3. Simplified Error: 27.8 bits

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

   5. Applied associate-+r- Error: 27.8 bits

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

   6. Applied associate-+r- Error: 27.8 bits

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

   7. Applied div-sub Error: 27.8 bits

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

   8. Simplified Error: 27.0 bits

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

   9. Simplified Error: 11.8 bits

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

   11. Applied add-cube-cbrt Error: 11.8 bits

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

   12. Applied associate-*r* Error: 11.8 bits

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

   13. Taylor expanded around 0 Error: 11.7 bits

       \[\leadsto 100 \cdot \left(n \cdot \frac{n \cdot \log 1 + \left(i \cdot \color{blue}{\left(\left({1}^{0.3333333333333333} + 0.3333333333333333 \cdot \left(i \cdot {1}^{0.3333333333333333}\right)\right) - 0.027777777777777776 \cdot \left({i}^{2} \cdot {1}^{0.3333333333333333}\right)\right)}\right) \cdot \sqrt[3]{1 + i \cdot 0.5}}{i} - n \cdot \frac{i \cdot \left(\log 1 \cdot 0.5\right)}{1}\right)\]

   14. Simplified Error: 11.7 bits

       \[\leadsto 100 \cdot \left(n \cdot \frac{n \cdot \log 1 + \left(i \cdot \color{blue}{\left(\sqrt[3]{1} + \sqrt[3]{1} \cdot \left(i \cdot \left(0.3333333333333333 - i \cdot 0.027777777777777776\right)\right)\right)}\right) \cdot \sqrt[3]{1 + i \cdot 0.5}}{i} - n \cdot \frac{i \cdot \left(\log 1 \cdot 0.5\right)}{1}\right)\]

   if 0.051881246670273766 < i < 7.4139897410859938e291

   1. Initial program Error: 31.5 bits

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

   2. Taylor expanded around inf Error: 29.6 bits

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

   3. Simplified Error: 31.7 bits

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

   if 7.4139897410859938e291 < i

   1. Initial program Error: 28.0 bits

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

   2. Taylor expanded around 0 Error: 36.3 bits

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

   3. Simplified Error: 36.3 bits

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

4. Final simplification Error: 15.7 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -4.331633902717184 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.051881246670273766:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{n \cdot \log 1 + \left(i \cdot \left(\sqrt[3]{1} + \sqrt[3]{1} \cdot \left(i \cdot \left(0.3333333333333333 - i \cdot 0.027777777777777776\right)\right)\right)\right) \cdot \sqrt[3]{1 + i \cdot 0.5}}{i} - n \cdot \left(i \cdot \left(\log 1 \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;i \leq 7.413989741085994 \cdot 10^{+291}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 + \left(n \cdot \log 1 + i \cdot 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

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))))