Average Error: 5.5 → 6.2
Time: 2.4s
Precision: binary64
\[\frac{x}{\left(\left(\left(s \cdot R\right) \cdot R\right) \cdot R\right) \cdot \sqrt{1 - {\left(\frac{x}{R}\right)}^{2}}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.9377723723314305 \cdot 10^{-56} \lor \neg \left(x \le -1.3579511830668391 \cdot 10^{-243}\right):\\ \;\;\;\;\frac{x}{\left(\left(\left(s \cdot R\right) \cdot R\right) \cdot R\right) \cdot \sqrt{1 - {\left(\frac{x}{R}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{s}}{{R}^{3} \cdot \sqrt{1 - {\left(\frac{x}{R}\right)}^{2}}}\\ \end{array}\]
\frac{x}{\left(\left(\left(s \cdot R\right) \cdot R\right) \cdot R\right) \cdot \sqrt{1 - {\left(\frac{x}{R}\right)}^{2}}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.9377723723314305 \cdot 10^{-56} \lor \neg \left(x \le -1.3579511830668391 \cdot 10^{-243}\right):\\
\;\;\;\;\frac{x}{\left(\left(\left(s \cdot R\right) \cdot R\right) \cdot R\right) \cdot \sqrt{1 - {\left(\frac{x}{R}\right)}^{2}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{s}}{{R}^{3} \cdot \sqrt{1 - {\left(\frac{x}{R}\right)}^{2}}}\\

\end{array}
double code(double x, double s, double R) {
	return ((double) (x / ((double) (((double) (((double) (((double) (s * R)) * R)) * R)) * ((double) sqrt(((double) (1.0 - ((double) pow(((double) (x / R)), 2.0))))))))));
}
↓
double code(double x, double s, double R) {
	double VAR;
	if (((x <= -3.9377723723314305e-56) || !(x <= -1.357951183066839e-243))) {
		VAR = ((double) (x / ((double) (((double) (((double) (((double) (s * R)) * R)) * R)) * ((double) sqrt(((double) (1.0 - ((double) pow(((double) (x / R)), 2.0))))))))));
	} else {
		VAR = ((double) (((double) (x / s)) / ((double) (((double) pow(R, 3.0)) * ((double) sqrt(((double) (1.0 - ((double) pow(((double) (x / R)), 2.0))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus s

Bits error versus R

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -3.9377723723314305e-56 or -1.3579511830668391e-243 < x

    1. Initial program 5.7

      \[\frac{x}{\left(\left(\left(s \cdot R\right) \cdot R\right) \cdot R\right) \cdot \sqrt{1 - {\left(\frac{x}{R}\right)}^{2}}}\]

    if -3.9377723723314305e-56 < x < -1.3579511830668391e-243

    1. Initial program 4.8

      \[\frac{x}{\left(\left(\left(s \cdot R\right) \cdot R\right) \cdot R\right) \cdot \sqrt{1 - {\left(\frac{x}{R}\right)}^{2}}}\]

    2. Simplified7.9

      \[\leadsto \color{blue}{\frac{\frac{x}{s}}{{R}^{3} \cdot \sqrt{1 - {\left(\frac{x}{R}\right)}^{2}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.9377723723314305 \cdot 10^{-56} \lor \neg \left(x \le -1.3579511830668391 \cdot 10^{-243}\right):\\ \;\;\;\;\frac{x}{\left(\left(\left(s \cdot R\right) \cdot R\right) \cdot R\right) \cdot \sqrt{1 - {\left(\frac{x}{R}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{s}}{{R}^{3} \cdot \sqrt{1 - {\left(\frac{x}{R}\right)}^{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (x s R)
  :name "(/ x (* (* (* (* s R) R) R) (sqrt (- 1 (pow (/ x R) 2)))))"
  :precision binary64
  (/ x (* (* (* (* s R) R) R) (sqrt (- 1.0 (pow (/ x R) 2.0))))))