Average Error: 37.9 → 22.8
Time: 4.8s
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 <= -1.0515333860087074e+143) {
		tmp = 0.5 * (sqrt((im * im) * 2.0) / sqrt(re * -2.0));
	} else if (re <= 8.161050826261294e-169) {
		tmp = 0.5 * ((fabs(im) / sqrt(sqrt(sqrt((im * im) + (re * re)) - re))) * (sqrt(2.0) / sqrt(sqrt(sqrt((im * im) + (re * re)) - re))));
	} else {
		tmp = 0.5 * sqrt(2.0 * (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
Target
Original37.9
Target33.1
Herbie22.8
\[\]
Derivation
  1. Split input into 3 regimes
  2. if  re  < -1.05153338600870742e143
    1. Initial program 63.2
      \[\]
    2. Using strategy rm
    3. Applied flip-+63.2
      \[\leadsto \]
    4. Applied associate-*r/63.2
      \[\leadsto \]
    5. Applied sqrt-div63.2
      \[\leadsto \]
    6. Simplified49.1
      \[\leadsto \]
    7. Taylor expanded around -inf 19.4
      \[\leadsto \]
    8. Simplified19.4
      \[\leadsto \]
    if -1.05153338600870742e143 <  re  < 8.1610508262612942e-169
    1. Initial program 36.9
      \[\]
    2. Using strategy rm
    3. Applied flip-+37.2
      \[\leadsto \]
    4. Applied associate-*r/37.2
      \[\leadsto \]
    5. Applied sqrt-div37.5
      \[\leadsto \]
    6. Simplified30.4
      \[\leadsto \]
    7. Using strategy rm
    8. Applied add-sqr-sqrt30.4
      \[\leadsto \]
    9. Applied sqrt-prod30.6
      \[\leadsto \]
    10. Applied sqrt-prod30.5
      \[\leadsto \]
    11. Applied times-frac30.5
      \[\leadsto \]
    12. Simplified23.1
      \[\leadsto \]
    13. Simplified23.1
      \[\leadsto \]
    if 8.1610508262612942e-169 <  re 
    1. Initial program 31.1
      \[\]
    2. Taylor expanded around inf 23.4
      \[\leadsto \]
  3. Recombined 3 regimes into one program.
  4. Final simplification22.8
    \[\leadsto \]
Reproduce
herbie shell --seed 2020338 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))