Average Error: 47.6 → 14.4
Time: 15.0s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq -0.009077757186457397:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left(-1 + {\left(\frac{i}{n}\right)}^{n}\right)\right)\\ \mathbf{elif}\;i \leq 7.432995259070106 \cdot 10^{-08}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i + \left(i \cdot i\right) \cdot \left(0.5 + i \cdot 0.16666666666666666\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{-1 + {\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)}}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq -0.009077757186457397:\\
\;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left(-1 + {\left(\frac{i}{n}\right)}^{n}\right)\right)\\

\mathbf{elif}\;i \leq 7.432995259070106 \cdot 10^{-08}:\\
\;\;\;\;100 \cdot \left(n \cdot \frac{i + \left(i \cdot i\right) \cdot \left(0.5 + i \cdot 0.16666666666666666\right)}{i}\right)\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \frac{\frac{-1 + {\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)}}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\

\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 -0.009077757186457397)
   (* 100.0 (* (/ n i) (+ -1.0 (pow (/ i n) n))))
   (if (<= i 7.432995259070106e-08)
     (* 100.0 (* n (/ (+ i (* (* i i) (+ 0.5 (* i 0.16666666666666666)))) i)))
     (*
      100.0
      (/
       (/
        (+ -1.0 (pow (+ (/ i n) 1.0) (* n 2.0)))
        (+ 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 <= -0.009077757186457397) {
		tmp = 100.0 * ((n / i) * (-1.0 + pow((i / n), n)));
	} else if (i <= 7.432995259070106e-08) {
		tmp = 100.0 * (n * ((i + ((i * i) * (0.5 + (i * 0.16666666666666666)))) / i));
	} else {
		tmp = 100.0 * (((-1.0 + pow(((i / n) + 1.0), (n * 2.0))) / (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
Target47.5
Herbie14.4
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -0.0090777571864573969

    1. Initial program 28.2

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

    2. Taylor expanded around inf 64.0

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

    3. Simplified19.8

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

    if -0.0090777571864573969 < i < 7.43299525907010599e-8

    1. Initial program 58.0

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

    2. Taylor expanded around 0 26.3

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

    3. Simplified26.3

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(i \cdot i\right) \cdot \left(0.5 + 0.16666666666666666 \cdot i\right)}}{\frac{i}{n}}\]
    4. Using strategy rm

    5. Applied associate-/r/_binary64_13888.9

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

    6. Simplified8.9

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

    if 7.43299525907010599e-8 < i

    1. Initial program 31.6

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

    3. Applied flip--_binary64_141731.6

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

    4. Simplified31.6

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

    5. Simplified31.6

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

  4. Final simplification14.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.009077757186457397:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left(-1 + {\left(\frac{i}{n}\right)}^{n}\right)\right)\\ \mathbf{elif}\;i \leq 7.432995259070106 \cdot 10^{-08}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i + \left(i \cdot i\right) \cdot \left(0.5 + i \cdot 0.16666666666666666\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{-1 + {\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)}}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020354 
(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))))