Average Error: 16.4 → 16.4
Time: 2.0s
Precision: binary64
\[\sqrt{0.5 + \frac{1 - \frac{r}{q}}{2 \cdot \sqrt{{\left(\frac{p}{q}\right)}^{2} + {\left(1 - \frac{r}{q}\right)}^{2}}}}\]
↓
\[\sqrt{0.5 + \frac{1 - \frac{r}{q}}{2 \cdot \sqrt{{\left(\frac{p}{q}\right)}^{2} + {\left(1 - \frac{r}{q}\right)}^{2}}}}\]
\sqrt{0.5 + \frac{1 - \frac{r}{q}}{2 \cdot \sqrt{{\left(\frac{p}{q}\right)}^{2} + {\left(1 - \frac{r}{q}\right)}^{2}}}}
↓
\sqrt{0.5 + \frac{1 - \frac{r}{q}}{2 \cdot \sqrt{{\left(\frac{p}{q}\right)}^{2} + {\left(1 - \frac{r}{q}\right)}^{2}}}}
double code(double r, double q, double p) {
	return ((double) sqrt(((double) (0.5 + ((double) (((double) (1.0 - ((double) (r / q)))) / ((double) (2.0 * ((double) sqrt(((double) (((double) pow(((double) (p / q)), 2.0)) + ((double) pow(((double) (1.0 - ((double) (r / q)))), 2.0))))))))))))));
}
↓
double code(double r, double q, double p) {
	return ((double) sqrt(((double) (0.5 + ((double) (((double) (1.0 - ((double) (r / q)))) / ((double) (2.0 * ((double) sqrt(((double) (((double) pow(((double) (p / q)), 2.0)) + ((double) pow(((double) (1.0 - ((double) (r / q)))), 2.0))))))))))))));
}

Error

Bits error versus r

Bits error versus q

Bits error versus p

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 16.4

    \[\sqrt{0.5 + \frac{1 - \frac{r}{q}}{2 \cdot \sqrt{{\left(\frac{p}{q}\right)}^{2} + {\left(1 - \frac{r}{q}\right)}^{2}}}}\]

  2. Final simplification16.4

    \[\leadsto \sqrt{0.5 + \frac{1 - \frac{r}{q}}{2 \cdot \sqrt{{\left(\frac{p}{q}\right)}^{2} + {\left(1 - \frac{r}{q}\right)}^{2}}}}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (r q p)
  :name "(sqrt (+ 0.5 (/ (- 1 (/ r q)) (* 2 (sqrt (+ (pow (/ p q) 2) (pow (- 1 (/ r q)) 2)))))))"
  :precision binary64
  (sqrt (+ 0.5 (/ (- 1.0 (/ r q)) (* 2.0 (sqrt (+ (pow (/ p q) 2.0) (pow (- 1.0 (/ r q)) 2.0))))))))