Average Error: 38.7 → 17.0
Time: 7.0s
Precision: binary64
\[\]
\[\]
↓
\[\]
↓
double code(double re, double im) {
	return 0.5 * sqrt(2.0 * (sqrt((re * re) + (im * im)) - re));
}

↓
double code(double re, double im) {
	double tmp;
	if (re <= -3.0495999745604484e+144) {
		tmp = 0.5 * sqrt(2.0 * (re * -2.0));
	} else if (re <= -2.4561711144511768e-77) {
		tmp = 0.5 * sqrt(2.0 * (sqrt((re * re) + (im * im)) - re));
	} else if (re <= 5.628209854884318e-65) {
		tmp = 0.5 * sqrt(2.0 * (im - re));
	} else if (re <= 1.3361830242464705e+154) {
		tmp = 0.5 * (sqrt(2.0 * (im * im)) / sqrt(re + sqrt((re * re) + (im * im))));
	} else {
		tmp = 0.5 * sqrt(2.0 * ((im * im) / (re + re)));
	}
	return tmp;
}

Error
Bits error versus re
Bits error versus im
Try it out
Your Program's Arguments
Results
Enter valid numbers for all inputs
Derivation
  1. Split input into 5 regimes
  2. if  re  < -3.0495999745604484e144
    1. Initial program 61.5
      \[\]
    2. Taylor expanded around -inf 9.4
      \[\leadsto \]
    3. Simplified9.4
      \[\leadsto \]
    if -3.0495999745604484e144 <  re  < -2.4561711144511768e-77
    1. Initial program 15.7
      \[\]
    if -2.4561711144511768e-77 <  re  < 5.6282098548843183e-65
    1. Initial program 29.0
      \[\]
    2. Taylor expanded around 0 11.1
      \[\leadsto \]
    if 5.6282098548843183e-65 <  re  < 1.3361830242464705e154
    1. Initial program 48.4
      \[\]
    2. Using strategy rm
    3. Applied flip--48.4
      \[\leadsto \]
    4. Simplified31.7
      \[\leadsto \]
    5. Simplified31.7
      \[\leadsto \]
    6. Using strategy rm
    7. Applied associate-*r/31.7
      \[\leadsto \]
    8. Applied sqrt-div28.9
      \[\leadsto \]
    9. Simplified28.9
      \[\leadsto \]
    if 1.3361830242464705e154 <  re 
    1. Initial program 64.0
      \[\]
    2. Using strategy rm
    3. Applied flip--64.0
      \[\leadsto \]
    4. Simplified49.9
      \[\leadsto \]
    5. Simplified49.9
      \[\leadsto \]
    6. Taylor expanded around inf 29.9
      \[\leadsto \]
  3. Recombined 5 regimes into one program.
  4. Final simplification17.0
    \[\leadsto \]
Reproduce
herbie shell --seed 2020338 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))