math.sqrt on complex, real part

Average Error: 38.3 → 7.3

Time: 13.9s

Precision: binary64

Cost: 13444

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -1.046441282610084 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \left|im \cdot \sqrt{\frac{-1}{re}}\right|\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -1.046441282610084e+39)
   (* 0.5 (fabs (* im (sqrt (/ -1.0 re)))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

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 tmp;
	if (re <= -1.046441282610084e+39) {
		tmp = 0.5 * fabs((im * sqrt((-1.0 / re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	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 tmp;
	if (re <= -1.046441282610084e+39) {
		tmp = 0.5 * Math.abs((im * Math.sqrt((-1.0 / re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))

↓

def code(re, im):
	tmp = 0
	if re <= -1.046441282610084e+39:
		tmp = 0.5 * math.fabs((im * math.sqrt((-1.0 / re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	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)
	tmp = 0.0
	if (re <= -1.046441282610084e+39)
		tmp = Float64(0.5 * abs(Float64(im * sqrt(Float64(-1.0 / re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return tmp
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
end

↓

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.046441282610084e+39)
		tmp = 0.5 * abs((im * sqrt((-1.0 / re))));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	end
	tmp_2 = 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_] := If[LessEqual[re, -1.046441282610084e+39], N[(0.5 * N[Abs[N[(im * N[Sqrt[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Target

Original	38.3
Target	33.5
Herbie	7.3

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

Derivation

1. Split input into 2 regimes

2. if re < -1.046441282610084e39

   1. Initial program 58.7

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

   2. Simplified 39.5

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
      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)))))))): 139 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 39.6

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right)}^{0.25}\right)}^{2}} \]

   4. Taylor expanded in re around -inf 26.5

      \[\leadsto 0.5 \cdot \color{blue}{{\left(e^{0.25 \cdot \left(\log \left({im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}\right)}^{2}} \]

   5. Simplified 48.3

      \[\leadsto 0.5 \cdot \color{blue}{{\left(e^{\mathsf{fma}\left(2, \log im, \log \left(\frac{-1}{re}\right)\right)}\right)}^{0.5}} \]
      Proof
      (pow.f64 (exp.f64 (fma.f64 2 (log.f64 im) (log.f64 (/.f64 -1 re)))) 1/2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (exp.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (log.f64 im)) (log.f64 (/.f64 -1 re))))) 1/2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (exp.f64 (+.f64 (Rewrite<= log-pow_binary64 (log.f64 (pow.f64 im 2))) (log.f64 (/.f64 -1 re)))) 1/2): 7 points increase in error, 40 points decrease in error
      (pow.f64 (exp.f64 (+.f64 (log.f64 (pow.f64 im 2)) (log.f64 (/.f64 -1 re)))) (Rewrite<= metadata-eval (*.f64 2 1/4))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (exp.f64 (+.f64 (log.f64 (pow.f64 im 2)) (log.f64 (/.f64 -1 re)))) 1/4) (pow.f64 (exp.f64 (+.f64 (log.f64 (pow.f64 im 2)) (log.f64 (/.f64 -1 re)))) 1/4))): 7 points increase in error, 12 points decrease in error
      (*.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (+.f64 (log.f64 (pow.f64 im 2)) (log.f64 (/.f64 -1 re))) 1/4))) (pow.f64 (exp.f64 (+.f64 (log.f64 (pow.f64 im 2)) (log.f64 (/.f64 -1 re)))) 1/4)): 1 points increase in error, 2 points decrease in error
      (*.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/4 (+.f64 (log.f64 (pow.f64 im 2)) (log.f64 (/.f64 -1 re)))))) (pow.f64 (exp.f64 (+.f64 (log.f64 (pow.f64 im 2)) (log.f64 (/.f64 -1 re)))) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (exp.f64 (*.f64 1/4 (+.f64 (log.f64 (pow.f64 im 2)) (log.f64 (/.f64 -1 re))))) (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (+.f64 (log.f64 (pow.f64 im 2)) (log.f64 (/.f64 -1 re))) 1/4)))): 1 points increase in error, 14 points decrease in error
      (*.f64 (exp.f64 (*.f64 1/4 (+.f64 (log.f64 (pow.f64 im 2)) (log.f64 (/.f64 -1 re))))) (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/4 (+.f64 (log.f64 (pow.f64 im 2)) (log.f64 (/.f64 -1 re))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unpow2_binary64 (pow.f64 (exp.f64 (*.f64 1/4 (+.f64 (log.f64 (pow.f64 im 2)) (log.f64 (/.f64 -1 re))))) 2)): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 40.2

      \[\leadsto 0.5 \cdot {\color{blue}{\left({\left({\left(\frac{-1}{re} \cdot \left(im \cdot im\right)\right)}^{1.5}\right)}^{0.6666666666666666}\right)}}^{0.5} \]

   7. Applied egg-rr 12.9

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

   if -1.046441282610084e39 < re

   1. Initial program 32.8

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

   2. Simplified 5.8

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
      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)))))))): 139 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. Recombined 2 regimes into one program.

4. Final simplification 7.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.046441282610084 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \left|im \cdot \sqrt{\frac{-1}{re}}\right|\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	26.6
Cost	14172

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left|im \cdot \sqrt{\frac{-1}{re}}\right|\\ t_2 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -5.580850603196456 \cdot 10^{+49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -799693.4838634767:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -6.655822479294364 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.1289079449687082 \cdot 10^{-300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 4.0628426208765444 \cdot 10^{-132}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 2.011744924990974 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2.7563346312568003 \cdot 10^{-60}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{re + im} \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 2
Error	26.6
Cost	14040

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left|im \cdot \sqrt{\frac{-1}{re}}\right|\\ t_2 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -5.580850603196456 \cdot 10^{+49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -799693.4838634767:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -6.655822479294364 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.1289079449687082 \cdot 10^{-300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 4.0628426208765444 \cdot 10^{-132}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 2.011744924990974 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2.7563346312568003 \cdot 10^{-60}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Error	26.5
Cost	7640

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -5.580850603196456 \cdot 10^{+49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -799693.4838634767:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -6.259835303791085 \cdot 10^{-191}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.0628426208765444 \cdot 10^{-132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2.011744924990974 \cdot 10^{-62}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;im \leq 2.7563346312568003 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Error	26.8
Cost	7512

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -5.580850603196456 \cdot 10^{+49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -799693.4838634767:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -6.259835303791085 \cdot 10^{-191}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.0628426208765444 \cdot 10^{-132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2.011744924990974 \cdot 10^{-62}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;im \leq 2.7563346312568003 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 5
Error	26.9
Cost	7248

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -5.580850603196456 \cdot 10^{+49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -799693.4838634767:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -6.259835303791085 \cdot 10^{-191}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.7563346312568003 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 6
Error	35.7
Cost	6984

\[\begin{array}{l} \mathbf{if}\;re \leq -1.046441282610084 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \left(4 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 9.010547225204561 \cdot 10^{-83}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 7
Error	43.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 6.395369297180332 \cdot 10^{-307}:\\ \;\;\;\;0.5 \cdot \left(4 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 8
Error	58.3
Cost	6724

\[\begin{array}{l} \mathbf{if}\;im \cdot im \leq 4 \cdot 10^{-305}:\\ \;\;\;\;0.5 \cdot \left(4 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|im\right|\\ \end{array} \]

Alternative 9
Error	59.7
Cost	576

\[0.5 \cdot \left(-1 + \left(1 + \left(im + im\right)\right)\right) \]

Alternative 10
Error	59.6
Cost	448

\[0.5 \cdot \left(4 \cdot \left(im \cdot im\right)\right) \]

Alternative 11
Error	59.4
Cost	196

\[\begin{array}{l} \mathbf{if}\;im \leq 1.011229484841618 \cdot 10^{-259}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternative 12
Error	60.1
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022291 
(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)))))