math.sqrt on complex, real part

Average Error: 39.1 → 11.7

Time: 13.3s

Precision: binary64

Cost: 33356

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{if}\;re \leq -4.317793734724897 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot {\left(\sqrt[3]{\sqrt{\frac{-1}{re}}} \cdot {t_0}^{0.16666666666666666}\right)}^{3}\right)\\ \mathbf{elif}\;re \leq -8.591521564765006 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -1.6992380219034422 \cdot 10^{-38}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666} \cdot \sqrt{\sqrt[3]{t_0}}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* im im)))
        (t_1 (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))
   (if (<= re -4.317793734724897e+101)
     (*
      0.5
      (*
       (sqrt 2.0)
       (pow (* (cbrt (sqrt (/ -1.0 re))) (pow t_0 0.16666666666666666)) 3.0)))
     (if (<= re -8.591521564765006e+29)
       t_1
       (if (<= re -1.6992380219034422e-38)
         (*
          0.5
          (*
           (sqrt 2.0)
           (pow
            (* (pow (/ -1.0 re) 0.16666666666666666) (sqrt (cbrt t_0)))
            3.0)))
         t_1)))))

C

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 t_0 = 0.5 * (im * im);
	double t_1 = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	double tmp;
	if (re <= -4.317793734724897e+101) {
		tmp = 0.5 * (sqrt(2.0) * pow((cbrt(sqrt((-1.0 / re))) * pow(t_0, 0.16666666666666666)), 3.0));
	} else if (re <= -8.591521564765006e+29) {
		tmp = t_1;
	} else if (re <= -1.6992380219034422e-38) {
		tmp = 0.5 * (sqrt(2.0) * pow((pow((-1.0 / re), 0.16666666666666666) * sqrt(cbrt(t_0))), 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
}

↓

public static double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double t_1 = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	double tmp;
	if (re <= -4.317793734724897e+101) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.pow((Math.cbrt(Math.sqrt((-1.0 / re))) * Math.pow(t_0, 0.16666666666666666)), 3.0));
	} else if (re <= -8.591521564765006e+29) {
		tmp = t_1;
	} else if (re <= -1.6992380219034422e-38) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.pow((Math.pow((-1.0 / re), 0.16666666666666666) * Math.sqrt(Math.cbrt(t_0))), 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
end

↓

function code(re, im)
	t_0 = Float64(0.5 * Float64(im * im))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))))
	tmp = 0.0
	if (re <= -4.317793734724897e+101)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * (Float64(cbrt(sqrt(Float64(-1.0 / re))) * (t_0 ^ 0.16666666666666666)) ^ 3.0)));
	elseif (re <= -8.591521564765006e+29)
		tmp = t_1;
	elseif (re <= -1.6992380219034422e-38)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * (Float64((Float64(-1.0 / re) ^ 0.16666666666666666) * sqrt(cbrt(t_0))) ^ 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -4.317793734724897e+101], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[(N[Power[N[Sqrt[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[t$95$0, 0.16666666666666666], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -8.591521564765006e+29], t$95$1, If[LessEqual[re, -1.6992380219034422e-38], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[(N[Power[N[(-1.0 / re), $MachinePrecision], 0.16666666666666666], $MachinePrecision] * N[Sqrt[N[Power[t$95$0, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Target

Original	39.1
Target	34.3
Herbie	11.7

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

Derivation

1. Split input into 3 regimes

2. if re < -4.31779373472489717e101

   1. Initial program 61.4

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

   2. Simplified 41.4

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
      Proof
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (+.f64 re (hypot.f64 re im))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (+.f64 re (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))))))): 136 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 41.6

      \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2}}\right)}^{3}} \]

   4. Taylor expanded in re around -inf 24.8

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot {\left(e^{0.16666666666666666 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left(0.5 \cdot {im}^{2}\right)\right)}\right)}^{3}\right) \cdot {1}^{0.3333333333333333}\right)} \]

   5. Simplified 24.6

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666} \cdot {\left(\left(im \cdot im\right) \cdot 0.5\right)}^{0.16666666666666666}\right)}^{3}\right)} \]
      Proof
      (*.f64 (sqrt.f64 2) (pow.f64 (*.f64 (pow.f64 (/.f64 -1 re) 1/6) (pow.f64 (*.f64 (*.f64 im im) 1/2) 1/6)) 3)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (pow.f64 (*.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (/.f64 -1 re)) 1/6))) (pow.f64 (*.f64 (*.f64 im im) 1/2) 1/6)) 3)): 28 points increase in error, 34 points decrease in error
      (*.f64 (sqrt.f64 2) (pow.f64 (*.f64 (exp.f64 (*.f64 (log.f64 (/.f64 -1 re)) 1/6)) (pow.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 im 2)) 1/2) 1/6)) 3)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (pow.f64 (*.f64 (exp.f64 (*.f64 (log.f64 (/.f64 -1 re)) 1/6)) (pow.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 (pow.f64 im 2))) 1/6)) 3)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (pow.f64 (*.f64 (exp.f64 (*.f64 (log.f64 (/.f64 -1 re)) 1/6)) (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (*.f64 1/2 (pow.f64 im 2))) 1/6)))) 3)): 29 points increase in error, 32 points decrease in error
      (*.f64 (sqrt.f64 2) (pow.f64 (Rewrite<= exp-sum_binary64 (exp.f64 (+.f64 (*.f64 (log.f64 (/.f64 -1 re)) 1/6) (*.f64 (log.f64 (*.f64 1/2 (pow.f64 im 2))) 1/6)))) 3)): 24 points increase in error, 21 points decrease in error
      (*.f64 (sqrt.f64 2) (pow.f64 (exp.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 1/6 (+.f64 (log.f64 (/.f64 -1 re)) (log.f64 (*.f64 1/2 (pow.f64 im 2))))))) 3)): 12 points increase in error, 15 points decrease in error
      (Rewrite<= *-rgt-identity_binary64 (*.f64 (*.f64 (sqrt.f64 2) (pow.f64 (exp.f64 (*.f64 1/6 (+.f64 (log.f64 (/.f64 -1 re)) (log.f64 (*.f64 1/2 (pow.f64 im 2)))))) 3)) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (sqrt.f64 2) (pow.f64 (exp.f64 (*.f64 1/6 (+.f64 (log.f64 (/.f64 -1 re)) (log.f64 (*.f64 1/2 (pow.f64 im 2)))))) 3)) (Rewrite<= pow-base-1_binary64 (pow.f64 1 1/3))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 24.3

      \[\leadsto 0.5 \cdot \left(\sqrt{2} \cdot {\left(\color{blue}{\sqrt[3]{\sqrt{\frac{-1}{re}}}} \cdot {\left(\left(im \cdot im\right) \cdot 0.5\right)}^{0.16666666666666666}\right)}^{3}\right) \]

   if -4.31779373472489717e101 < re < -8.5915215647650058e29 or -1.69923802190344223e-38 < re

   1. Initial program 33.7

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

   2. Simplified 6.1

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
      Proof
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (+.f64 re (hypot.f64 re im))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (+.f64 re (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))))))): 136 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))))): 0 points increase in error, 0 points decrease in error

   if -8.5915215647650058e29 < re < -1.69923802190344223e-38

   1. Initial program 44.6

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

   2. Simplified 27.4

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
      Proof
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (+.f64 re (hypot.f64 re im))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (+.f64 re (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))))))): 136 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 28.1

      \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2}}\right)}^{3}} \]

   4. Taylor expanded in re around -inf 48.2

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot {\left(e^{0.16666666666666666 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left(0.5 \cdot {im}^{2}\right)\right)}\right)}^{3}\right) \cdot {1}^{0.3333333333333333}\right)} \]

   5. Simplified 48.2

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666} \cdot {\left(\left(im \cdot im\right) \cdot 0.5\right)}^{0.16666666666666666}\right)}^{3}\right)} \]
      Proof
      (*.f64 (sqrt.f64 2) (pow.f64 (*.f64 (pow.f64 (/.f64 -1 re) 1/6) (pow.f64 (*.f64 (*.f64 im im) 1/2) 1/6)) 3)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (pow.f64 (*.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (/.f64 -1 re)) 1/6))) (pow.f64 (*.f64 (*.f64 im im) 1/2) 1/6)) 3)): 28 points increase in error, 34 points decrease in error
      (*.f64 (sqrt.f64 2) (pow.f64 (*.f64 (exp.f64 (*.f64 (log.f64 (/.f64 -1 re)) 1/6)) (pow.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 im 2)) 1/2) 1/6)) 3)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (pow.f64 (*.f64 (exp.f64 (*.f64 (log.f64 (/.f64 -1 re)) 1/6)) (pow.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 (pow.f64 im 2))) 1/6)) 3)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (pow.f64 (*.f64 (exp.f64 (*.f64 (log.f64 (/.f64 -1 re)) 1/6)) (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (*.f64 1/2 (pow.f64 im 2))) 1/6)))) 3)): 29 points increase in error, 32 points decrease in error
      (*.f64 (sqrt.f64 2) (pow.f64 (Rewrite<= exp-sum_binary64 (exp.f64 (+.f64 (*.f64 (log.f64 (/.f64 -1 re)) 1/6) (*.f64 (log.f64 (*.f64 1/2 (pow.f64 im 2))) 1/6)))) 3)): 24 points increase in error, 21 points decrease in error
      (*.f64 (sqrt.f64 2) (pow.f64 (exp.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 1/6 (+.f64 (log.f64 (/.f64 -1 re)) (log.f64 (*.f64 1/2 (pow.f64 im 2))))))) 3)): 12 points increase in error, 15 points decrease in error
      (Rewrite<= *-rgt-identity_binary64 (*.f64 (*.f64 (sqrt.f64 2) (pow.f64 (exp.f64 (*.f64 1/6 (+.f64 (log.f64 (/.f64 -1 re)) (log.f64 (*.f64 1/2 (pow.f64 im 2)))))) 3)) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (sqrt.f64 2) (pow.f64 (exp.f64 (*.f64 1/6 (+.f64 (log.f64 (/.f64 -1 re)) (log.f64 (*.f64 1/2 (pow.f64 im 2)))))) 3)) (Rewrite<= pow-base-1_binary64 (pow.f64 1 1/3))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 47.6

      \[\leadsto 0.5 \cdot \left(\sqrt{2} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666} \cdot \color{blue}{\sqrt{\sqrt[3]{\left(im \cdot im\right) \cdot 0.5}}}\right)}^{3}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 11.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.317793734724897 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot {\left(\sqrt[3]{\sqrt{\frac{-1}{re}}} \cdot {\left(0.5 \cdot \left(im \cdot im\right)\right)}^{0.16666666666666666}\right)}^{3}\right)\\ \mathbf{elif}\;re \leq -8.591521564765006 \cdot 10^{+29}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{elif}\;re \leq -1.6992380219034422 \cdot 10^{-38}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666} \cdot \sqrt{\sqrt[3]{0.5 \cdot \left(im \cdot im\right)}}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.7
Cost	33092

\[\begin{array}{l} t_0 := {\left(0.5 \cdot \left(im \cdot im\right)\right)}^{0.16666666666666666}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{if}\;re \leq -4.317793734724897 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot {\left(\sqrt[3]{\sqrt{\frac{-1}{re}}} \cdot t_0\right)}^{3}\right)\\ \mathbf{elif}\;re \leq -8.591521564765006 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -1.6992380219034422 \cdot 10^{-38}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot {\left(t_0 \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	10.0
Cost	27400

\[\begin{array}{l} t_0 := re + \sqrt{im \cdot im + re \cdot re}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-301}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{-im}{re}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	11.8
Cost	27020

\[\begin{array}{l} t_0 := 0.5 \cdot \left(\sqrt{2} \cdot {\left({\left(0.5 \cdot \left(im \cdot im\right)\right)}^{0.16666666666666666} \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right)\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{if}\;re \leq -4.317793734724897 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -8.591521564765006 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -1.6992380219034422 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	27.6
Cost	7640

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.44300434402997 \cdot 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -4.4609744313465935 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -3.351982922660714 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.7929388547969235 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.511204576013951 \cdot 10^{-268}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 2.4266711991997106 \cdot 10^{-196}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 5
Error	27.5
Cost	7640

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{if}\;im \leq -1.44300434402997 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -4.4609744313465935 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -3.351982922660714 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2.7929388547969235 \cdot 10^{-305}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.511204576013951 \cdot 10^{-268}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 2.4266711991997106 \cdot 10^{-196}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 6
Error	28.1
Cost	7512

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.44300434402997 \cdot 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -4.4609744313465935 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -3.351982922660714 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.7929388547969235 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.511204576013951 \cdot 10^{-268}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 3.3460103925569644 \cdot 10^{-189}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 7
Error	30.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;re \leq -3.4753067000748696 \cdot 10^{-58}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq 1.774028530533559 \cdot 10^{-38}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 8
Error	32.1
Cost	6984

\[\begin{array}{l} \mathbf{if}\;re \leq -5.380513839019052 \cdot 10^{-87}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 2.993897023552852 \cdot 10^{-140}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 9
Error	36.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 2.993897023552852 \cdot 10^{-140}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 10
Error	47.5
Cost	6720

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

Reproduce

herbie shell --seed 2022311 
(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)))))