Average Error: 9.0 → 9.0
Time: 1.4s
Precision: binary64
\[\sqrt{0.5 \cdot \left(1 + \frac{r}{\sqrt{\frac{\left(4 \cdot p\right) \cdot p}{q \cdot q} + \left(1 - \frac{r}{q}\right) \cdot \left(1 - \frac{r}{q}\right)}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + \frac{r}{\sqrt{\frac{\left(4 \cdot p\right) \cdot p}{q \cdot q} + \left(1 - \frac{r}{q}\right) \cdot \left(1 - \frac{r}{q}\right)}}\right)}\]
\sqrt{0.5 \cdot \left(1 + \frac{r}{\sqrt{\frac{\left(4 \cdot p\right) \cdot p}{q \cdot q} + \left(1 - \frac{r}{q}\right) \cdot \left(1 - \frac{r}{q}\right)}}\right)}
↓
\sqrt{0.5 \cdot \left(1 + \frac{r}{\sqrt{\frac{\left(4 \cdot p\right) \cdot p}{q \cdot q} + \left(1 - \frac{r}{q}\right) \cdot \left(1 - \frac{r}{q}\right)}}\right)}
double code(double r, double p, double q) {
	return ((double) sqrt(((double) (0.5 * ((double) (1.0 + ((double) (r / ((double) sqrt(((double) (((double) (((double) (((double) (4.0 * p)) * p)) / ((double) (q * q)))) + ((double) (((double) (1.0 - ((double) (r / q)))) * ((double) (1.0 - ((double) (r / q))))))))))))))))));
}
↓
double code(double r, double p, double q) {
	return ((double) sqrt(((double) (0.5 * ((double) (1.0 + ((double) (r / ((double) sqrt(((double) (((double) (((double) (((double) (4.0 * p)) * p)) / ((double) (q * q)))) + ((double) (((double) (1.0 - ((double) (r / q)))) * ((double) (1.0 - ((double) (r / q))))))))))))))))));
}

Error

Bits error versus r

Bits error versus p

Bits error versus q

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 9.0

    \[\sqrt{0.5 \cdot \left(1 + \frac{r}{\sqrt{\frac{\left(4 \cdot p\right) \cdot p}{q \cdot q} + \left(1 - \frac{r}{q}\right) \cdot \left(1 - \frac{r}{q}\right)}}\right)}\]

  2. Final simplification9.0

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{r}{\sqrt{\frac{\left(4 \cdot p\right) \cdot p}{q \cdot q} + \left(1 - \frac{r}{q}\right) \cdot \left(1 - \frac{r}{q}\right)}}\right)}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (r p q)
  :name "(sqrt (* 0.5 (+ 1 (/ r (sqrt (+ (/ (* (* 4 p) p) (* q q)) (* (- 1 (/ r q)) (- 1 (/ r q)))))))))"
  :precision binary64
  (sqrt (* 0.5 (+ 1.0 (/ r (sqrt (+ (/ (* (* 4.0 p) p) (* q q)) (* (- 1.0 (/ r q)) (- 1.0 (/ r q))))))))))