Average Error: 47.6 → 12.0
Time: 12.4s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq 346.1220968706167:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;i \leq 5.111910710628546 \cdot 10^{+112}:\\ \;\;\;\;100 \cdot \frac{{n}^{2} \cdot \left(\log i - \log n\right)}{i}\\ \mathbf{elif}\;i \leq 8.043070052233076 \cdot 10^{+233}:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq 346.1220968706167:\\
\;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\

\mathbf{elif}\;i \leq 5.111910710628546 \cdot 10^{+112}:\\
\;\;\;\;100 \cdot \frac{{n}^{2} \cdot \left(\log i - \log n\right)}{i}\\

\mathbf{elif}\;i \leq 8.043070052233076 \cdot 10^{+233}:\\
\;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \frac{0}{\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 346.1220968706167)
   (* (* n 100.0) (/ (expm1 i) i))
   (if (<= i 5.111910710628546e+112)
     (* 100.0 (/ (* (pow n 2.0) (- (log i) (log n))) i))
     (if (<= i 8.043070052233076e+233)
       (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))
       (* 100.0 (/ 0.0 (/ 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 <= 346.1220968706167) {
		tmp = (n * 100.0) * (expm1(i) / i);
	} else if (i <= 5.111910710628546e+112) {
		tmp = 100.0 * ((pow(n, 2.0) * (log(i) - log(n))) / i);
	} else if (i <= 8.043070052233076e+233) {
		tmp = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	} else {
		tmp = 100.0 * (0.0 / (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

Original47.6
Target47.0
Herbie12.0
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \]

Derivation

  1. Split input into 4 regimes

  2. if i < 346.1220968706167

    1. Initial program 50.3

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

    2. Taylor expanded in n around inf 44.7

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

    3. Simplified13.7

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]

    4. Applied *-un-lft-identity_binary6413.7

      \[\leadsto 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{\color{blue}{1 \cdot i}} \]

    5. Applied times-frac_binary649.9

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{n}{1} \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]

    6. Applied associate-*r*_binary6410.0

      \[\leadsto \color{blue}{\left(100 \cdot \frac{n}{1}\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}} \]

    7. Simplified10.0

      \[\leadsto \color{blue}{\left(n \cdot 100\right)} \cdot \frac{\mathsf{expm1}\left(i\right)}{i} \]

    if 346.1220968706167 < i < 5.1119107106285459e112

    1. Initial program 31.3

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

    2. Taylor expanded in n around 0 8.8

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

    if 5.1119107106285459e112 < i < 8.04307005223307566e233

    1. Initial program 29.6

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

    if 8.04307005223307566e233 < i

    1. Initial program 33.5

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

    2. Taylor expanded in i around 0 29.6

      \[\leadsto 100 \cdot \frac{\color{blue}{1} - 1}{\frac{i}{n}} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification12.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 346.1220968706167:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;i \leq 5.111910710628546 \cdot 10^{+112}:\\ \;\;\;\;100 \cdot \frac{{n}^{2} \cdot \left(\log i - \log n\right)}{i}\\ \mathbf{elif}\;i \leq 8.043070052233076 \cdot 10^{+233}:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(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))))