Average Error: 47.5 → 11.7
Time: 17.8s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq -1.1481220983977971 \cdot 10^{-08}:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.48544185323136774:\\ \;\;\;\;\left(n \cdot \left(i \cdot 50 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + \left(100 \cdot n + \left(\frac{i}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{i}{\sqrt[3]{n}}\right) \cdot 33.333333333333336\right)\right) + -50 \cdot \left(i + i \cdot i\right)\\ \mathbf{elif}\;i \leq 6.229817388872846 \cdot 10^{+176}:\\ \;\;\;\;100 \cdot \left(\left(\frac{{n}^{3}}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{3} \cdot {\log n}^{2}}{i} + \left(0.16666666666666666 \cdot \frac{{n}^{4} \cdot {\log i}^{3}}{i} + \left(0.5 \cdot \frac{{\log n}^{2} \cdot \left({n}^{4} \cdot \log i\right)}{i} + \left(\frac{{n}^{4} \cdot \log i}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{3} \cdot {\log i}^{2}}{i} + \frac{\log i \cdot {n}^{2}}{i}\right)\right)\right)\right)\right)\right) - \left(\frac{\log n \cdot \left({n}^{3} \cdot \log i\right)}{i} + \left(0.16666666666666666 \cdot \frac{{n}^{4} \cdot {\log n}^{3}}{i} + \left(\frac{\log n \cdot {n}^{4}}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{4}}{{i}^{3}} + \left(0.5 \cdot \frac{\log n \cdot \left({n}^{4} \cdot {\log i}^{2}\right)}{i} + \frac{\log n \cdot {n}^{2}}{i}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{1}{\frac{i}{n}}\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq -1.1481220983977971 \cdot 10^{-08}:\\
\;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\

\mathbf{elif}\;i \leq 0.48544185323136774:\\
\;\;\;\;\left(n \cdot \left(i \cdot 50 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + \left(100 \cdot n + \left(\frac{i}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{i}{\sqrt[3]{n}}\right) \cdot 33.333333333333336\right)\right) + -50 \cdot \left(i + i \cdot i\right)\\

\mathbf{elif}\;i \leq 6.229817388872846 \cdot 10^{+176}:\\
\;\;\;\;100 \cdot \left(\left(\frac{{n}^{3}}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{3} \cdot {\log n}^{2}}{i} + \left(0.16666666666666666 \cdot \frac{{n}^{4} \cdot {\log i}^{3}}{i} + \left(0.5 \cdot \frac{{\log n}^{2} \cdot \left({n}^{4} \cdot \log i\right)}{i} + \left(\frac{{n}^{4} \cdot \log i}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{3} \cdot {\log i}^{2}}{i} + \frac{\log i \cdot {n}^{2}}{i}\right)\right)\right)\right)\right)\right) - \left(\frac{\log n \cdot \left({n}^{3} \cdot \log i\right)}{i} + \left(0.16666666666666666 \cdot \frac{{n}^{4} \cdot {\log n}^{3}}{i} + \left(\frac{\log n \cdot {n}^{4}}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{4}}{{i}^{3}} + \left(0.5 \cdot \frac{\log n \cdot \left({n}^{4} \cdot {\log i}^{2}\right)}{i} + \frac{\log n \cdot {n}^{2}}{i}\right)\right)\right)\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{1}{\frac{i}{n}}\right)\\

\end{array}
(FPCore (i n)
 :precision binary64
 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))
↓
(FPCore (i n)
 :precision binary64
 (if (<= i -1.1481220983977971e-08)
   (* 100.0 (/ (- (exp i) 1.0) (/ i n)))
   (if (<= i 0.48544185323136774)
     (+
      (+
       (* n (+ (* i 50.0) (* 16.666666666666668 (* i i))))
       (+
        (* 100.0 n)
        (* (* (/ i (* (cbrt n) (cbrt n))) (/ i (cbrt n))) 33.333333333333336)))
      (* -50.0 (+ i (* i i))))
     (if (<= i 6.229817388872846e+176)
       (*
        100.0
        (-
         (+
          (/ (pow n 3.0) (pow i 2.0))
          (+
           (* 0.5 (/ (* (pow n 3.0) (pow (log n) 2.0)) i))
           (+
            (* 0.16666666666666666 (/ (* (pow n 4.0) (pow (log i) 3.0)) i))
            (+
             (* 0.5 (/ (* (pow (log n) 2.0) (* (pow n 4.0) (log i))) i))
             (+
              (/ (* (pow n 4.0) (log i)) (pow i 2.0))
              (+
               (* 0.5 (/ (* (pow n 3.0) (pow (log i) 2.0)) i))
               (/ (* (log i) (pow n 2.0)) i)))))))
         (+
          (/ (* (log n) (* (pow n 3.0) (log i))) i)
          (+
           (* 0.16666666666666666 (/ (* (pow n 4.0) (pow (log n) 3.0)) i))
           (+
            (/ (* (log n) (pow n 4.0)) (pow i 2.0))
            (+
             (* 0.5 (/ (pow n 4.0) (pow i 3.0)))
             (+
              (* 0.5 (/ (* (log n) (* (pow n 4.0) (pow (log i) 2.0))) i))
              (/ (* (log n) (pow n 2.0)) i))))))))
       (* 100.0 (* (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ 1.0 (/ i n))))))))
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 <= -1.1481220983977971e-08) {
		tmp = 100.0 * ((exp(i) - 1.0) / (i / n));
	} else if (i <= 0.48544185323136774) {
		tmp = ((n * ((i * 50.0) + (16.666666666666668 * (i * i)))) + ((100.0 * n) + (((i / (cbrt(n) * cbrt(n))) * (i / cbrt(n))) * 33.333333333333336))) + (-50.0 * (i + (i * i)));
	} else if (i <= 6.229817388872846e+176) {
		tmp = 100.0 * (((pow(n, 3.0) / pow(i, 2.0)) + ((0.5 * ((pow(n, 3.0) * pow(log(n), 2.0)) / i)) + ((0.16666666666666666 * ((pow(n, 4.0) * pow(log(i), 3.0)) / i)) + ((0.5 * ((pow(log(n), 2.0) * (pow(n, 4.0) * log(i))) / i)) + (((pow(n, 4.0) * log(i)) / pow(i, 2.0)) + ((0.5 * ((pow(n, 3.0) * pow(log(i), 2.0)) / i)) + ((log(i) * pow(n, 2.0)) / i))))))) - (((log(n) * (pow(n, 3.0) * log(i))) / i) + ((0.16666666666666666 * ((pow(n, 4.0) * pow(log(n), 3.0)) / i)) + (((log(n) * pow(n, 4.0)) / pow(i, 2.0)) + ((0.5 * (pow(n, 4.0) / pow(i, 3.0))) + ((0.5 * ((log(n) * (pow(n, 4.0) * pow(log(i), 2.0))) / i)) + ((log(n) * pow(n, 2.0)) / i)))))));
	} else {
		tmp = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) * (1.0 / (i / n)));
	}
	return tmp;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.5
Target47.4
Herbie11.7
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -1.1481220983977971e-8

    1. Initial program 28.2

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

    2. Taylor expanded around inf 12.6

      \[\leadsto 100 \cdot \frac{\color{blue}{e^{i}} - 1}{\frac{i}{n}}\]

    if -1.1481220983977971e-8 < i < 0.485441853231367737

    1. Initial program 58.0

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

    2. Taylor expanded around 0 9.1

      \[\leadsto \color{blue}{\left(50 \cdot \left(i \cdot n\right) + \left(16.666666666666668 \cdot \left({i}^{2} \cdot n\right) + \left(33.333333333333336 \cdot \frac{{i}^{2}}{n} + 100 \cdot n\right)\right)\right) - \left(50 \cdot {i}^{2} + 50 \cdot i\right)}\]

    3. Simplified9.1

      \[\leadsto \color{blue}{\left(n \cdot \left(i \cdot 50 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + \left(n \cdot 100 + \frac{i \cdot i}{n} \cdot 33.333333333333336\right)\right) + -50 \cdot \left(i + i \cdot i\right)}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt_binary64_31829.1

      \[\leadsto \left(n \cdot \left(i \cdot 50 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + \left(n \cdot 100 + \frac{i \cdot i}{\color{blue}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}} \cdot 33.333333333333336\right)\right) + -50 \cdot \left(i + i \cdot i\right)\]

    6. Applied times-frac_binary64_31539.1

      \[\leadsto \left(n \cdot \left(i \cdot 50 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + \left(n \cdot 100 + \color{blue}{\left(\frac{i}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{i}{\sqrt[3]{n}}\right)} \cdot 33.333333333333336\right)\right) + -50 \cdot \left(i + i \cdot i\right)\]

    if 0.485441853231367737 < i < 6.22981738887284585e176

    1. Initial program 33.1

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

    2. Taylor expanded around 0 13.3

      \[\leadsto 100 \cdot \color{blue}{\left(\left(\frac{{n}^{3}}{{i}^{2}} + \left(0.5 \cdot \frac{{\log n}^{2} \cdot {n}^{3}}{i} + \left(0.16666666666666666 \cdot \frac{{n}^{4} \cdot {\log i}^{3}}{i} + \left(0.5 \cdot \frac{{\log n}^{2} \cdot \left(\log i \cdot {n}^{4}\right)}{i} + \left(\frac{\log i \cdot {n}^{4}}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{3} \cdot {\log i}^{2}}{i} + \frac{{n}^{2} \cdot \log i}{i}\right)\right)\right)\right)\right)\right) - \left(\frac{\log n \cdot \left(\log i \cdot {n}^{3}\right)}{i} + \left(0.16666666666666666 \cdot \frac{{\log n}^{3} \cdot {n}^{4}}{i} + \left(\frac{\log n \cdot {n}^{4}}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{4}}{{i}^{3}} + \left(0.5 \cdot \frac{\log n \cdot \left({\log i}^{2} \cdot {n}^{4}\right)}{i} + \frac{\log n \cdot {n}^{2}}{i}\right)\right)\right)\right)\right)\right)}\]

    if 6.22981738887284585e176 < i

    1. Initial program 31.2

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

    3. Applied div-inv_binary64_314431.2

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

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.1481220983977971 \cdot 10^{-08}:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.48544185323136774:\\ \;\;\;\;\left(n \cdot \left(i \cdot 50 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + \left(100 \cdot n + \left(\frac{i}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{i}{\sqrt[3]{n}}\right) \cdot 33.333333333333336\right)\right) + -50 \cdot \left(i + i \cdot i\right)\\ \mathbf{elif}\;i \leq 6.229817388872846 \cdot 10^{+176}:\\ \;\;\;\;100 \cdot \left(\left(\frac{{n}^{3}}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{3} \cdot {\log n}^{2}}{i} + \left(0.16666666666666666 \cdot \frac{{n}^{4} \cdot {\log i}^{3}}{i} + \left(0.5 \cdot \frac{{\log n}^{2} \cdot \left({n}^{4} \cdot \log i\right)}{i} + \left(\frac{{n}^{4} \cdot \log i}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{3} \cdot {\log i}^{2}}{i} + \frac{\log i \cdot {n}^{2}}{i}\right)\right)\right)\right)\right)\right) - \left(\frac{\log n \cdot \left({n}^{3} \cdot \log i\right)}{i} + \left(0.16666666666666666 \cdot \frac{{n}^{4} \cdot {\log n}^{3}}{i} + \left(\frac{\log n \cdot {n}^{4}}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{4}}{{i}^{3}} + \left(0.5 \cdot \frac{\log n \cdot \left({n}^{4} \cdot {\log i}^{2}\right)}{i} + \frac{\log n \cdot {n}^{2}}{i}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{1}{\frac{i}{n}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021007 
(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))))