Average Error: 47.6 → 11.5
Time: 12.6s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq -0.0001425252087604258:\\ \;\;\;\;100 \cdot \frac{e^{i} + -1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.2208532522868418:\\ \;\;\;\;100 \cdot \left(\left(0.16666666666666666 \cdot \left(n \cdot \left(i \cdot i\right)\right) + \left(n + \left(0.5 \cdot \left(i \cdot n\right) + 0.3333333333333333 \cdot \frac{i \cdot i}{n}\right)\right)\right) - 0.5 \cdot \left(i + i \cdot i\right)\right)\\ \mathbf{elif}\;i \leq 1.4546833800995257 \cdot 10^{+152} \lor \neg \left(i \leq 6.596829234964693 \cdot 10^{+202}\right):\\ \;\;\;\;\left(\frac{{n}^{3} \cdot {\log i}^{2}}{i} \cdot 50 + \left(100 \cdot \left(\frac{\log i \cdot \left(n \cdot n\right)}{i} + \frac{{n}^{3}}{i \cdot i}\right) + 50 \cdot \frac{{n}^{3} \cdot {\log n}^{2}}{i}\right)\right) - 100 \cdot \left(\frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i} + \frac{\left(n \cdot n\right) \cdot \log n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;10 \cdot \left(10 \cdot \frac{-1 + {\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}\right)\\ \end{array} \]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq -0.0001425252087604258:\\
\;\;\;\;100 \cdot \frac{e^{i} + -1}{\frac{i}{n}}\\

\mathbf{elif}\;i \leq 0.2208532522868418:\\
\;\;\;\;100 \cdot \left(\left(0.16666666666666666 \cdot \left(n \cdot \left(i \cdot i\right)\right) + \left(n + \left(0.5 \cdot \left(i \cdot n\right) + 0.3333333333333333 \cdot \frac{i \cdot i}{n}\right)\right)\right) - 0.5 \cdot \left(i + i \cdot i\right)\right)\\

\mathbf{elif}\;i \leq 1.4546833800995257 \cdot 10^{+152} \lor \neg \left(i \leq 6.596829234964693 \cdot 10^{+202}\right):\\
\;\;\;\;\left(\frac{{n}^{3} \cdot {\log i}^{2}}{i} \cdot 50 + \left(100 \cdot \left(\frac{\log i \cdot \left(n \cdot n\right)}{i} + \frac{{n}^{3}}{i \cdot i}\right) + 50 \cdot \frac{{n}^{3} \cdot {\log n}^{2}}{i}\right)\right) - 100 \cdot \left(\frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i} + \frac{\left(n \cdot n\right) \cdot \log n}{i}\right)\\

\mathbf{else}:\\
\;\;\;\;10 \cdot \left(10 \cdot \frac{-1 + {\left(\frac{i}{n} + 1\right)}^{n}}{\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 -0.0001425252087604258)
   (* 100.0 (/ (+ (exp i) -1.0) (/ i n)))
   (if (<= i 0.2208532522868418)
     (*
      100.0
      (-
       (+
        (* 0.16666666666666666 (* n (* i i)))
        (+ n (+ (* 0.5 (* i n)) (* 0.3333333333333333 (/ (* i i) n)))))
       (* 0.5 (+ i (* i i)))))
     (if (or (<= i 1.4546833800995257e+152)
             (not (<= i 6.596829234964693e+202)))
       (-
        (+
         (* (/ (* (pow n 3.0) (pow (log i) 2.0)) i) 50.0)
         (+
          (* 100.0 (+ (/ (* (log i) (* n n)) i) (/ (pow n 3.0) (* i i))))
          (* 50.0 (/ (* (pow n 3.0) (pow (log n) 2.0)) i))))
        (*
         100.0
         (+
          (/ (* (log i) (* (pow n 3.0) (log n))) i)
          (/ (* (* n n) (log n)) i))))
       (* 10.0 (* 10.0 (/ (+ -1.0 (pow (+ (/ i n) 1.0) n)) (/ 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 <= -0.0001425252087604258) {
		tmp = 100.0 * ((exp(i) + -1.0) / (i / n));
	} else if (i <= 0.2208532522868418) {
		tmp = 100.0 * (((0.16666666666666666 * (n * (i * i))) + (n + ((0.5 * (i * n)) + (0.3333333333333333 * ((i * i) / n))))) - (0.5 * (i + (i * i))));
	} else if ((i <= 1.4546833800995257e+152) || !(i <= 6.596829234964693e+202)) {
		tmp = ((((pow(n, 3.0) * pow(log(i), 2.0)) / i) * 50.0) + ((100.0 * (((log(i) * (n * n)) / i) + (pow(n, 3.0) / (i * i)))) + (50.0 * ((pow(n, 3.0) * pow(log(n), 2.0)) / i)))) - (100.0 * (((log(i) * (pow(n, 3.0) * log(n))) / i) + (((n * n) * log(n)) / i)));
	} else {
		tmp = 10.0 * (10.0 * ((-1.0 + pow(((i / n) + 1.0), n)) / (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

Original47.6
Target47.1
Herbie11.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \]

Derivation

  1. Split input into 4 regimes

  2. if i < -1.42525208760425811e-4

    1. Initial program 27.9

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

    2. Taylor expanded around inf 11.6

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

    3. Simplified11.6

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

    if -1.42525208760425811e-4 < i < 0.2208532522868418

    1. Initial program 58.2

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

    2. Taylor expanded around 0 9.3

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

    3. Simplified9.3

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

    if 0.2208532522868418 < i < 1.4546833800995257e152 or 6.5968292349646927e202 < i

    1. Initial program 32.2

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

    2. Taylor expanded around 0 20.6

      \[\leadsto \color{blue}{\left(50 \cdot \frac{{\log i}^{2} \cdot {n}^{3}}{i} + \left(100 \cdot \frac{{n}^{2} \cdot \log i}{i} + \left(100 \cdot \frac{{n}^{3}}{{i}^{2}} + 50 \cdot \frac{{n}^{3} \cdot {\log n}^{2}}{i}\right)\right)\right) - \left(100 \cdot \frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i} + 100 \cdot \frac{{n}^{2} \cdot \log n}{i}\right)} \]

    3. Simplified20.6

      \[\leadsto \color{blue}{\left(\frac{{n}^{3} \cdot {\log i}^{2}}{i} \cdot 50 + \left(100 \cdot \left(\frac{\left(n \cdot n\right) \cdot \log i}{i} + \frac{{n}^{3}}{i \cdot i}\right) + \frac{{n}^{3} \cdot {\log n}^{2}}{i} \cdot 50\right)\right) - 100 \cdot \left(\frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i} + \frac{\left(n \cdot n\right) \cdot \log n}{i}\right)} \]

    if 1.4546833800995257e152 < i < 6.5968292349646927e202

    1. Initial program 29.9

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

    3. Applied add-sqr-sqrt_binary6429.9

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

    4. Applied associate-*l*_binary6429.9

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

    5. Simplified29.9

      \[\leadsto \sqrt{100} \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 10\right)} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.0001425252087604258:\\ \;\;\;\;100 \cdot \frac{e^{i} + -1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.2208532522868418:\\ \;\;\;\;100 \cdot \left(\left(0.16666666666666666 \cdot \left(n \cdot \left(i \cdot i\right)\right) + \left(n + \left(0.5 \cdot \left(i \cdot n\right) + 0.3333333333333333 \cdot \frac{i \cdot i}{n}\right)\right)\right) - 0.5 \cdot \left(i + i \cdot i\right)\right)\\ \mathbf{elif}\;i \leq 1.4546833800995257 \cdot 10^{+152} \lor \neg \left(i \leq 6.596829234964693 \cdot 10^{+202}\right):\\ \;\;\;\;\left(\frac{{n}^{3} \cdot {\log i}^{2}}{i} \cdot 50 + \left(100 \cdot \left(\frac{\log i \cdot \left(n \cdot n\right)}{i} + \frac{{n}^{3}}{i \cdot i}\right) + 50 \cdot \frac{{n}^{3} \cdot {\log n}^{2}}{i}\right)\right) - 100 \cdot \left(\frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i} + \frac{\left(n \cdot n\right) \cdot \log n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;10 \cdot \left(10 \cdot \frac{-1 + {\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021205 
(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))))