Average Error: 5.9 → 0.3
Time: 2.4s
Precision: binary64
\[\frac{x}{\left(\left(\left(x \cdot R\right) \cdot R\right) \cdot R\right) \cdot \sqrt{1 - {\left(\frac{x}{R}\right)}^{2}}}\]
↓
\[\frac{\frac{1}{\sqrt{1 - {\left(\frac{x}{R}\right)}^{2}}}}{{R}^{3}}\]
\frac{x}{\left(\left(\left(x \cdot R\right) \cdot R\right) \cdot R\right) \cdot \sqrt{1 - {\left(\frac{x}{R}\right)}^{2}}}
↓
\frac{\frac{1}{\sqrt{1 - {\left(\frac{x}{R}\right)}^{2}}}}{{R}^{3}}
double code(double x, double R) {
	return ((double) (x / ((double) (((double) (((double) (((double) (x * R)) * R)) * R)) * ((double) sqrt(((double) (1.0 - ((double) pow(((double) (x / R)), 2.0))))))))));
}
↓
double code(double x, double R) {
	return ((double) (((double) (1.0 / ((double) sqrt(((double) (1.0 - ((double) pow(((double) (x / R)), 2.0)))))))) / ((double) pow(R, 3.0))));
}

Error

Bits error versus x

Bits error versus R

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 5.9

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

  2. Simplified0.3

    \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{1 - {\left(\frac{x}{R}\right)}^{2}}}}{{R}^{3}}}\]

  3. Final simplification0.3

    \[\leadsto \frac{\frac{1}{\sqrt{1 - {\left(\frac{x}{R}\right)}^{2}}}}{{R}^{3}}\]

Reproduce

herbie shell --seed 2020152 
(FPCore (x R)
  :name "(/ x (* (* (* (* x R) R) R) (sqrt (- 1 (pow (/ x R) 2)))))"
  :precision binary64
  (/ x (* (* (* (* x R) R) R) (sqrt (- 1.0 (pow (/ x R) 2.0))))))