Compound Interest

Average Error: 47.5 → 11.5

Time: 24.5s

Precision: binary64

Cost: 173442

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -0.00022620816475513194:\\ \;\;\;\;100 \cdot \left(\frac{{\left(e^{i \cdot 0.5}\right)}^{2}}{\frac{i}{n}} - \frac{n}{i}\right) + \left(i \cdot {\left(e^{i \cdot 0.5}\right)}^{2}\right) \cdot -50\\ \mathbf{elif}\;i \leq 5.60806670825752 \cdot 10^{-06}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + \left(i \cdot i\right) \cdot 0.16666666666666666\right) + \left(n + 0.3333333333333333 \cdot \frac{i}{\frac{n}{i}}\right)\right) + \left(i + i \cdot i\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log n}^{2}\right) + \left(n \cdot \log i + \left(\frac{n}{\frac{i}{n}} + \left({n}^{3} \cdot \frac{\log i}{i} + \left(0.5 \cdot \left({\log n}^{2} \cdot \left(\log i \cdot {n}^{3}\right) + \left(n \cdot n\right) \cdot {\log i}^{2}\right) + 0.16666666666666666 \cdot \left({n}^{3} \cdot {\log i}^{3}\right)\right)\right)\right)\right)\right) - \left(0.5 \cdot \frac{{n}^{3}}{i \cdot i} + \left(\log n \cdot \left(n + \left(n \cdot n\right) \cdot \log i\right) + \left({n}^{3} \cdot \frac{\log n}{i} + \left(0.5 \cdot \left(\log n \cdot \left({n}^{3} \cdot {\log i}^{2}\right)\right) + 0.16666666666666666 \cdot \left({n}^{3} \cdot {\log n}^{3}\right)\right)\right)\right)\right)}{\frac{i}{n}}\\ \end{array}\]

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 -0.00022620816475513194)
   (+
    (* 100.0 (- (/ (pow (exp (* i 0.5)) 2.0) (/ i n)) (/ n i)))
    (* (* i (pow (exp (* i 0.5)) 2.0)) -50.0))
   (if (<= i 5.60806670825752e-06)
     (*
      100.0
      (+
       (+
        (* n (+ (* i 0.5) (* (* i i) 0.16666666666666666)))
        (+ n (* 0.3333333333333333 (/ i (/ n i)))))
       (* (+ i (* i i)) -0.5)))
     (*
      100.0
      (/
       (-
        (+
         (* 0.5 (* (* n n) (pow (log n) 2.0)))
         (+
          (* n (log i))
          (+
           (/ n (/ i n))
           (+
            (* (pow n 3.0) (/ (log i) i))
            (+
             (*
              0.5
              (+
               (* (pow (log n) 2.0) (* (log i) (pow n 3.0)))
               (* (* n n) (pow (log i) 2.0))))
             (* 0.16666666666666666 (* (pow n 3.0) (pow (log i) 3.0))))))))
        (+
         (* 0.5 (/ (pow n 3.0) (* i i)))
         (+
          (* (log n) (+ n (* (* n n) (log i))))
          (+
           (* (pow n 3.0) (/ (log n) i))
           (+
            (* 0.5 (* (log n) (* (pow n 3.0) (pow (log i) 2.0))))
            (* 0.16666666666666666 (* (pow n 3.0) (pow (log n) 3.0))))))))
       (/ i n))))))

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 tmp;
	if (i <= -0.00022620816475513194) {
		tmp = (100.0 * ((pow(exp(i * 0.5), 2.0) / (i / n)) - (n / i))) + ((i * pow(exp(i * 0.5), 2.0)) * -50.0);
	} else if (i <= 5.60806670825752e-06) {
		tmp = 100.0 * (((n * ((i * 0.5) + ((i * i) * 0.16666666666666666))) + (n + (0.3333333333333333 * (i / (n / i))))) + ((i + (i * i)) * -0.5));
	} else {
		tmp = 100.0 * ((((0.5 * ((n * n) * pow(log(n), 2.0))) + ((n * log(i)) + ((n / (i / n)) + ((pow(n, 3.0) * (log(i) / i)) + ((0.5 * ((pow(log(n), 2.0) * (log(i) * pow(n, 3.0))) + ((n * n) * pow(log(i), 2.0)))) + (0.16666666666666666 * (pow(n, 3.0) * pow(log(i), 3.0)))))))) - ((0.5 * (pow(n, 3.0) / (i * i))) + ((log(n) * (n + ((n * n) * log(i)))) + ((pow(n, 3.0) * (log(n) / i)) + ((0.5 * (log(n) * (pow(n, 3.0) * pow(log(i), 2.0)))) + (0.16666666666666666 * (pow(n, 3.0) * pow(log(n), 3.0)))))))) / (i / n));
	}
	return tmp;
}

Error

Bits error versus i

Bits error versus n

Target

Original	47.5
Target	47.3
Herbie	11.5

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

Alternatives

Alternative 1
Error	11.5
Cost	48194

\[\begin{array}{l} \mathbf{if}\;i \leq -0.00018130698679654392:\\ \;\;\;\;100 \cdot \left(\frac{{\left(e^{i \cdot 0.5}\right)}^{2}}{\frac{i}{n}} - \frac{n}{i}\right) + \left(i \cdot {\left(e^{i \cdot 0.5}\right)}^{2}\right) \cdot -50\\ \mathbf{elif}\;i \leq 0.00011620513023260118:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + \left(i \cdot i\right) \cdot 0.16666666666666666\right) + \left(n + 0.3333333333333333 \cdot \frac{i}{\frac{n}{i}}\right)\right) + \left(i + i \cdot i\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log n}^{2}\right) + \left(n \cdot \log i + \left(\frac{n}{\frac{i}{n}} + 0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log i}^{2}\right)\right)\right)\right) - \log n \cdot \left(n + \left(n \cdot n\right) \cdot \log i\right)}{\frac{i}{n}}\\ \end{array}\]

Alternative 2
Error	12.4
Cost	27393

\[\begin{array}{l} \mathbf{if}\;i \leq -0.0001763179670233675:\\ \;\;\;\;100 \cdot \left(\frac{{\left(e^{i \cdot 0.5}\right)}^{2}}{\frac{i}{n}} - \frac{n}{i}\right) + \left(i \cdot {\left(e^{i \cdot 0.5}\right)}^{2}\right) \cdot -50\\ \mathbf{elif}\;i \leq 2.4990061673644405 \cdot 10^{-05}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + \left(i \cdot i\right) \cdot 0.16666666666666666\right) + \left(n + 0.3333333333333333 \cdot \frac{i}{\frac{n}{i}}\right)\right) + \left(i + i \cdot i\right) \cdot -0.5\right)\\ \mathbf{elif}\;i \leq 2.7913736004264033 \cdot 10^{+82}:\\ \;\;\;\;100 \cdot \frac{n \cdot \log \left(\frac{i}{n}\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot e^{n \cdot \left(\log i - \log n\right)} - n}{i}\\ \end{array}\]

Alternative 3
Error	12.3
Cost	20995

\[\begin{array}{l} \mathbf{if}\;i \leq -0.00019627404611607328:\\ \;\;\;\;100 \cdot \left(\frac{-1 + e^{i}}{\frac{i}{n}} - 0.5 \cdot \left(i \cdot e^{i}\right)\right)\\ \mathbf{elif}\;i \leq 2.5992345702905444 \cdot 10^{-05}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + \left(i \cdot i\right) \cdot 0.16666666666666666\right) + \left(n + 0.3333333333333333 \cdot \frac{i}{\frac{n}{i}}\right)\right) + \left(i + i \cdot i\right) \cdot -0.5\right)\\ \mathbf{elif}\;i \leq 2.5380492857080553 \cdot 10^{+82}:\\ \;\;\;\;100 \cdot \frac{n \cdot \log \left(\frac{i}{n}\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot e^{n \cdot \left(\log i - \log n\right)} - n}{i}\\ \end{array}\]

Alternative 4
Error	12.5
Cost	15363

\[\begin{array}{l} \mathbf{if}\;i \leq -0.00014139482861113235:\\ \;\;\;\;100 \cdot \left(\frac{-1 + e^{i}}{\frac{i}{n}} - 0.5 \cdot \left(i \cdot e^{i}\right)\right)\\ \mathbf{elif}\;i \leq 0.0001142005621740791:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + \left(i \cdot i\right) \cdot 0.16666666666666666\right) + \left(n + 0.3333333333333333 \cdot \frac{i}{\frac{n}{i}}\right)\right) + \left(i + i \cdot i\right) \cdot -0.5\right)\\ \mathbf{elif}\;i \leq 8.143489155155343 \cdot 10^{+82}:\\ \;\;\;\;100 \cdot \frac{n \cdot \log \left(\frac{i}{n}\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{-1 + {\left(\frac{i}{n} + 1\right)}^{\left(2 \cdot n\right)}}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \end{array}\]

Alternative 5
Error	12.5
Cost	14081

\[\begin{array}{l} \mathbf{if}\;i \leq -0.00014139482861113235:\\ \;\;\;\;100 \cdot \left(\frac{-1 + e^{i}}{\frac{i}{n}} - 0.5 \cdot \left(i \cdot e^{i}\right)\right)\\ \mathbf{elif}\;i \leq 3.349452163194337 \cdot 10^{-05}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + \left(i \cdot i\right) \cdot 0.16666666666666666\right) + \left(n + 0.3333333333333333 \cdot \frac{i}{\frac{n}{i}}\right)\right) + \left(i + i \cdot i\right) \cdot -0.5\right)\\ \mathbf{elif}\;i \leq 6.454660390366355 \cdot 10^{+82}:\\ \;\;\;\;100 \cdot \frac{n \cdot \log \left(\frac{i}{n}\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(-1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}{\frac{i}{n}}\\ \end{array}\]

Alternative 6
Error	12.5
Cost	8259

\[\begin{array}{l} \mathbf{if}\;i \leq -0.0007548462965901222:\\ \;\;\;\;100 \cdot \frac{-1 + e^{i}}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 8.613660935476997 \cdot 10^{-05}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + \left(i \cdot i\right) \cdot 0.16666666666666666\right) + \left(n + 0.3333333333333333 \cdot \frac{i}{\frac{n}{i}}\right)\right) + \left(i + i \cdot i\right) \cdot -0.5\right)\\ \mathbf{elif}\;i \leq 5.756101709849182 \cdot 10^{+82}:\\ \;\;\;\;100 \cdot \frac{n \cdot \log \left(\frac{i}{n}\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(-1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)}{\frac{i}{n}}\\ \end{array}\]

Alternative 7
Error	12.5
Cost	8259

\[\begin{array}{l} \mathbf{if}\;i \leq -0.0007548462965901222:\\ \;\;\;\;100 \cdot \frac{-1 + e^{i}}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.00011820969829112327:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + \left(i \cdot i\right) \cdot 0.16666666666666666\right) + \left(n + 0.3333333333333333 \cdot \frac{i}{\frac{n}{i}}\right)\right) + \left(i + i \cdot i\right) \cdot -0.5\right)\\ \mathbf{elif}\;i \leq 5.5872188333702836 \cdot 10^{+82}:\\ \;\;\;\;100 \cdot \frac{n \cdot \log \left(\frac{i}{n}\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{-1 + {\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}\\ \end{array}\]

Alternative 8
Error	12.4
Cost	8067

\[\begin{array}{l} \mathbf{if}\;i \leq -0.0007548462965901222:\\ \;\;\;\;100 \cdot \frac{-1 + e^{i}}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.00011620513023260118:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + \left(i \cdot i\right) \cdot 0.16666666666666666\right) + \left(n + 0.3333333333333333 \cdot \frac{i}{\frac{n}{i}}\right)\right) + \left(i + i \cdot i\right) \cdot -0.5\right)\\ \mathbf{elif}\;i \leq 3.729507192102396 \cdot 10^{+82}:\\ \;\;\;\;100 \cdot \frac{n \cdot \log \left(\frac{i}{n}\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 9
Error	12.7
Cost	7297

\[\begin{array}{l} \mathbf{if}\;i \leq -0.0007548462965901222:\\ \;\;\;\;100 \cdot \frac{-1 + e^{i}}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.02899917788426517:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + \left(i \cdot i\right) \cdot 0.16666666666666666\right) + \left(n + 0.3333333333333333 \cdot \frac{i}{\frac{n}{i}}\right)\right) + \left(i + i \cdot i\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 10
Error	19.9
Cost	2626

\[\begin{array}{l} \mathbf{if}\;i \leq -128661073283528.75:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.03077535197274432:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + \left(i \cdot i\right) \cdot 0.16666666666666666\right) + \left(n + 0.3333333333333333 \cdot \frac{i}{\frac{n}{i}}\right)\right) + \left(i + i \cdot i\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 11
Error	20.0
Cost	1474

\[\begin{array}{l} \mathbf{if}\;i \leq -0.00018130698679654392:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.5593825984714659:\\ \;\;\;\;100 \cdot \left(\left(n + 0.5 \cdot \left(i \cdot n\right)\right) - i \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 12
Error	20.0
Cost	1346

\[\begin{array}{l} \mathbf{if}\;i \leq -0.00024117522407466127:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.24775172253953445:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right) - i \cdot 50\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 13
Error	20.2
Cost	834

\[\begin{array}{l} \mathbf{if}\;i \leq -124381688850220.47:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.1153288977306012:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 14
Error	50.8
Cost	64

\[0\]

Alternative 15
Error	61.7
Cost	64

\[1\]

Derivation

1. Split input into 3 regimes

2. if i < -2.2620816475513194e-4

   1. Initial program 27.7

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

   3. Applied div-inv_binary64_1439 27.7

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

   4. Applied sqr-pow_binary64_1414 27.8

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

   5. Applied difference-of-sqr-1_binary64_1412 27.8

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

   6. Applied times-frac_binary64_1448 28.3

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

   7. Simplified 28.3

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

   8. Simplified 28.2

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

   9. Taylor expanded around inf 13.1

      \[\leadsto \color{blue}{100 \cdot \frac{{\left(e^{0.5 \cdot i}\right)}^{2} \cdot n}{i} - \left(100 \cdot \frac{n}{i} + 50 \cdot \left(i \cdot {\left(e^{0.5 \cdot i}\right)}^{2}\right)\right)}\]

   10. Simplified 13.1

       \[\leadsto \color{blue}{100 \cdot \left(\frac{{\left(e^{i \cdot 0.5}\right)}^{2}}{\frac{i}{n}} - \frac{n}{i}\right) + -50 \cdot \left(i \cdot {\left(e^{i \cdot 0.5}\right)}^{2}\right)}\]

   11. Simplified 13.1

       \[\leadsto \color{blue}{100 \cdot \left(\frac{{\left(e^{i \cdot 0.5}\right)}^{2}}{\frac{i}{n}} - \frac{n}{i}\right) + \left(i \cdot {\left(e^{i \cdot 0.5}\right)}^{2}\right) \cdot -50}\]

   if -2.2620816475513194e-4 < i < 5.60806670825751969e-6

   1. Initial program 58.4

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

   2. Taylor expanded around 0 8.7

      \[\leadsto 100 \cdot \color{blue}{\left(\left(0.5 \cdot \left(i \cdot n\right) + \left(0.16666666666666666 \cdot \left({i}^{2} \cdot n\right) + \left(0.3333333333333333 \cdot \frac{{i}^{2}}{n} + n\right)\right)\right) - \left(0.5 \cdot {i}^{2} + 0.5 \cdot i\right)\right)}\]

   3. Simplified 8.6

      \[\leadsto 100 \cdot \color{blue}{\left(\left(n \cdot \left(i \cdot 0.5 + 0.16666666666666666 \cdot \left(i \cdot i\right)\right) + \left(n + 0.3333333333333333 \cdot \frac{i}{\frac{n}{i}}\right)\right) + -0.5 \cdot \left(i + i \cdot i\right)\right)}\]

   4. Simplified 8.6

      \[\leadsto \color{blue}{100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + \left(i \cdot i\right) \cdot 0.16666666666666666\right) + \left(n + 0.3333333333333333 \cdot \frac{i}{\frac{n}{i}}\right)\right) + \left(i + i \cdot i\right) \cdot -0.5\right)}\]

   if 5.60806670825751969e-6 < i

   1. Initial program 32.1

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

   2. Taylor expanded around 0 21.7

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(0.5 \cdot \left({\log n}^{2} \cdot {n}^{2}\right) + \left(\log i \cdot n + \left(\frac{{n}^{2}}{i} + \left(\frac{\log i \cdot {n}^{3}}{i} + \left(0.5 \cdot \left({\log n}^{2} \cdot \left({n}^{3} \cdot \log i\right)\right) + \left(0.5 \cdot \left({\log i}^{2} \cdot {n}^{2}\right) + 0.16666666666666666 \cdot \left({n}^{3} \cdot {\log i}^{3}\right)\right)\right)\right)\right)\right)\right) - \left(0.5 \cdot \frac{{n}^{3}}{{i}^{2}} + \left(\log n \cdot n + \left(\log n \cdot \left(\log i \cdot {n}^{2}\right) + \left(\frac{\log n \cdot {n}^{3}}{i} + \left(0.16666666666666666 \cdot \left({\log n}^{3} \cdot {n}^{3}\right) + 0.5 \cdot \left(\log n \cdot \left({n}^{3} \cdot {\log i}^{2}\right)\right)\right)\right)\right)\right)\right)}}{\frac{i}{n}}\]

   3. Simplified 21.7

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log n}^{2}\right) + \left(n \cdot \log i + \left(\frac{n}{\frac{i}{n}} + \left(\frac{\log i}{i} \cdot {n}^{3} + \left(0.5 \cdot \left({\log n}^{2} \cdot \left(\log i \cdot {n}^{3}\right) + \left(n \cdot n\right) \cdot {\log i}^{2}\right) + 0.16666666666666666 \cdot \left({n}^{3} \cdot {\log i}^{3}\right)\right)\right)\right)\right)\right) - \left(0.5 \cdot \frac{{n}^{3}}{i \cdot i} + \left(\log n \cdot \left(n + \log i \cdot \left(n \cdot n\right)\right) + \left(\frac{\log n}{i} \cdot {n}^{3} + \left(0.5 \cdot \left(\log n \cdot \left({n}^{3} \cdot {\log i}^{2}\right)\right) + 0.16666666666666666 \cdot \left({n}^{3} \cdot {\log n}^{3}\right)\right)\right)\right)\right)}}{\frac{i}{n}}\]

   4. Simplified 21.7

      \[\leadsto \color{blue}{100 \cdot \frac{\left(0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log n}^{2}\right) + \left(n \cdot \log i + \left(\frac{n}{\frac{i}{n}} + \left({n}^{3} \cdot \frac{\log i}{i} + \left(0.5 \cdot \left({\log n}^{2} \cdot \left({n}^{3} \cdot \log i\right) + \left(n \cdot n\right) \cdot {\log i}^{2}\right) + 0.16666666666666666 \cdot \left({n}^{3} \cdot {\log i}^{3}\right)\right)\right)\right)\right)\right) - \left(0.5 \cdot \frac{{n}^{3}}{i \cdot i} + \left(\log n \cdot \left(n + \left(n \cdot n\right) \cdot \log i\right) + \left({n}^{3} \cdot \frac{\log n}{i} + \left(0.5 \cdot \left(\log n \cdot \left({n}^{3} \cdot {\log i}^{2}\right)\right) + 0.16666666666666666 \cdot \left({n}^{3} \cdot {\log n}^{3}\right)\right)\right)\right)\right)}{\frac{i}{n}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 11.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.00022620816475513194:\\ \;\;\;\;100 \cdot \left(\frac{{\left(e^{i \cdot 0.5}\right)}^{2}}{\frac{i}{n}} - \frac{n}{i}\right) + \left(i \cdot {\left(e^{i \cdot 0.5}\right)}^{2}\right) \cdot -50\\ \mathbf{elif}\;i \leq 5.60806670825752 \cdot 10^{-06}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(i \cdot 0.5 + \left(i \cdot i\right) \cdot 0.16666666666666666\right) + \left(n + 0.3333333333333333 \cdot \frac{i}{\frac{n}{i}}\right)\right) + \left(i + i \cdot i\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(0.5 \cdot \left(\left(n \cdot n\right) \cdot {\log n}^{2}\right) + \left(n \cdot \log i + \left(\frac{n}{\frac{i}{n}} + \left({n}^{3} \cdot \frac{\log i}{i} + \left(0.5 \cdot \left({\log n}^{2} \cdot \left(\log i \cdot {n}^{3}\right) + \left(n \cdot n\right) \cdot {\log i}^{2}\right) + 0.16666666666666666 \cdot \left({n}^{3} \cdot {\log i}^{3}\right)\right)\right)\right)\right)\right) - \left(0.5 \cdot \frac{{n}^{3}}{i \cdot i} + \left(\log n \cdot \left(n + \left(n \cdot n\right) \cdot \log i\right) + \left({n}^{3} \cdot \frac{\log n}{i} + \left(0.5 \cdot \left(\log n \cdot \left({n}^{3} \cdot {\log i}^{2}\right)\right) + 0.16666666666666666 \cdot \left({n}^{3} \cdot {\log n}^{3}\right)\right)\right)\right)\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))