Average Error: 39.2 → 17.7
Time: 11.0s
Precision: binary64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -3.306189866184851 \cdot 10^{+67}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\left(-re\right) - re}}{\left|im\right|}}\\ \mathbf{elif}\;re \leq 6.858325427122592 \cdot 10^{-295}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{\sqrt{2}}{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \left(\left|im\right| \cdot \sqrt{\sqrt{2}}\right)\right)\\ \mathbf{elif}\;re \leq 1.0440481246371513 \cdot 10^{+130}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -3.306189866184851 \cdot 10^{+67}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\left(-re\right) - re}}{\left|im\right|}}\\

\mathbf{elif}\;re \leq 6.858325427122592 \cdot 10^{-295}:\\
\;\;\;\;0.5 \cdot \left(\sqrt{\frac{\sqrt{2}}{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \left(\left|im\right| \cdot \sqrt{\sqrt{2}}\right)\right)\\

\mathbf{elif}\;re \leq 1.0440481246371513 \cdot 10^{+130}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\

\end{array}
(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -3.306189866184851e+67)
   (* 0.5 (/ (sqrt 2.0) (/ (sqrt (- (- re) re)) (fabs im))))
   (if (<= re 6.858325427122592e-295)
     (*
      0.5
      (*
       (sqrt (/ (sqrt 2.0) (- (sqrt (+ (* re re) (* im im))) re)))
       (* (fabs im) (sqrt (sqrt 2.0)))))
     (if (<= re 1.0440481246371513e+130)
       (* 0.5 (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))))
       (* 0.5 (* 2.0 (sqrt re)))))))
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 <= -3.306189866184851e+67) {
		tmp = 0.5 * (sqrt(2.0) / (sqrt(-re - re) / fabs(im)));
	} else if (re <= 6.858325427122592e-295) {
		tmp = 0.5 * (sqrt(sqrt(2.0) / (sqrt((re * re) + (im * im)) - re)) * (fabs(im) * sqrt(sqrt(2.0))));
	} else if (re <= 1.0440481246371513e+130) {
		tmp = 0.5 * sqrt(2.0 * (re + sqrt((re * re) + (im * im))));
	} else {
		tmp = 0.5 * (2.0 * sqrt(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

Original39.2
Target34.0
Herbie17.7
\[\begin{array}{l} \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < -3.3061898661848511e67

    1. Initial program 59.3

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied flip-+_binary64_209859.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    4. Applied associate-*r/_binary64_206659.3

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    5. Applied sqrt-div_binary64_214159.3

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    6. Simplified41.4

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
    7. Using strategy rm

    8. Applied sqrt-prod_binary64_214041.4

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{im \cdot im}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

    9. Applied associate-/l*_binary64_206941.4

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}{\sqrt{im \cdot im}}}}\]

    10. Simplified37.6

      \[\leadsto 0.5 \cdot \frac{\sqrt{2}}{\color{blue}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}{\left|im\right|}}}\]

    11. Taylor expanded around -inf 12.7

      \[\leadsto 0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{-1 \cdot re} - re}}{\left|im\right|}}\]

    12. Simplified12.7

      \[\leadsto 0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\left(-re\right)} - re}}{\left|im\right|}}\]

    if -3.3061898661848511e67 < re < 6.8583254271225918e-295

    1. Initial program 38.3

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied flip-+_binary64_209838.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    4. Applied associate-*r/_binary64_206638.1

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    5. Applied sqrt-div_binary64_214138.3

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    6. Simplified31.8

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
    7. Using strategy rm

    8. Applied sqrt-prod_binary64_214031.9

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{im \cdot im}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

    9. Applied associate-/l*_binary64_206931.9

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}{\sqrt{im \cdot im}}}}\]

    10. Simplified22.0

      \[\leadsto 0.5 \cdot \frac{\sqrt{2}}{\color{blue}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}{\left|im\right|}}}\]
    11. Using strategy rm

    12. Applied div-inv_binary64_212122.1

      \[\leadsto 0.5 \cdot \frac{\sqrt{2}}{\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re} \cdot \frac{1}{\left|im\right|}}}\]

    13. Applied add-sqr-sqrt_binary64_214622.1

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re} \cdot \frac{1}{\left|im\right|}}\]

    14. Applied times-frac_binary64_213022.0

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\sqrt{\sqrt{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \frac{\sqrt{\sqrt{2}}}{\frac{1}{\left|im\right|}}\right)}\]

    15. Simplified22.0

      \[\leadsto 0.5 \cdot \left(\frac{\sqrt{\sqrt{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \color{blue}{\left(\left|im\right| \cdot \sqrt{\sqrt{2}}\right)}\right)\]
    16. Using strategy rm

    17. Applied sqrt-undiv_binary64_214521.9

      \[\leadsto 0.5 \cdot \left(\color{blue}{\sqrt{\frac{\sqrt{2}}{\sqrt{re \cdot re + im \cdot im} - re}}} \cdot \left(\left|im\right| \cdot \sqrt{\sqrt{2}}\right)\right)\]

    if 6.8583254271225918e-295 < re < 1.04404812463715129e130

    1. Initial program 20.6

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

    if 1.04404812463715129e130 < re

    1. Initial program 57.6

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

    2. Taylor expanded around 0 9.6

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)}\]

    3. Simplified8.5

      \[\leadsto 0.5 \cdot \color{blue}{\left(2 \cdot \sqrt{re}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.306189866184851 \cdot 10^{+67}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\left(-re\right) - re}}{\left|im\right|}}\\ \mathbf{elif}\;re \leq 6.858325427122592 \cdot 10^{-295}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{\sqrt{2}}{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \left(\left|im\right| \cdot \sqrt{\sqrt{2}}\right)\right)\\ \mathbf{elif}\;re \leq 1.0440481246371513 \cdot 10^{+130}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021060 
(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)))))