math.sqrt on complex, real part

Average Error: 38.3 → 6.3

Time: 7.4s

Precision: binary64

Cost: 33348

Math

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

↓

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

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))) 0.0)
   (* 0.5 (* (sqrt (/ -0.5 re)) (fabs (* im (sqrt 2.0)))))
   (* 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 (sqrt((2.0 * (re + sqrt(((re * re) + (im * im)))))) <= 0.0) {
		tmp = 0.5 * (sqrt((-0.5 / re)) * fabs((im * sqrt(2.0))));
	} 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 (Math.sqrt((2.0 * (re + Math.sqrt(((re * re) + (im * im)))))) <= 0.0) {
		tmp = 0.5 * (Math.sqrt((-0.5 / re)) * Math.abs((im * Math.sqrt(2.0))));
	} 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 math.sqrt((2.0 * (re + math.sqrt(((re * re) + (im * im)))))) <= 0.0:
		tmp = 0.5 * (math.sqrt((-0.5 / re)) * math.fabs((im * math.sqrt(2.0))))
	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 (sqrt(Float64(2.0 * Float64(re + sqrt(Float64(Float64(re * re) + Float64(im * im)))))) <= 0.0)
		tmp = Float64(0.5 * Float64(sqrt(Float64(-0.5 / re)) * abs(Float64(im * sqrt(2.0)))));
	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 (sqrt((2.0 * (re + sqrt(((re * re) + (im * im)))))) <= 0.0)
		tmp = 0.5 * (sqrt((-0.5 / re)) * abs((im * sqrt(2.0))));
	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[N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[(N[Sqrt[N[(-0.5 / re), $MachinePrecision]], $MachinePrecision] * N[Abs[N[(im * N[Sqrt[2.0], $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	6.3

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

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 2 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 56.9

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

   2. Simplified 56.9

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
      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)))))))): 124 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. Taylor expanded in re around -inf 30.8

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

   4. Simplified 30.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\left(-0.5 \cdot im\right) \cdot im}{re}}} \]
      Proof
      (/.f64 (*.f64 (*.f64 -1/2 im) im) re): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1/2 (*.f64 im im))) re): 0 points increase in error, 1 points decrease in error
      (/.f64 (*.f64 -1/2 (Rewrite<= unpow2_binary64 (pow.f64 im 2))) re): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1/2 (/.f64 (pow.f64 im 2) re))): 2 points increase in error, 1 points decrease in error

   5. Applied egg-rr 30.9

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

   6. Applied egg-rr 28.9

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \sqrt{2} \cdot \left(\sqrt{\frac{-0.5}{re}} \cdot im\right)\right)} \]

   7. Simplified 28.9

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{-0.5}{re}} \cdot \left(im \cdot \sqrt{2}\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 -1/2 re)) (*.f64 im (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 (/.f64 -1/2 re)) im) (sqrt.f64 2))): 18 points increase in error, 16 points decrease in error
      (Rewrite=> *-commutative_binary64 (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (/.f64 -1/2 re)) im))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-lft-identity_binary64 (+.f64 0 (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (/.f64 -1/2 re)) im)))): 0 points increase in error, 0 points decrease in error

   8. Applied egg-rr 27.6

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

   9. Simplified 0.5

      \[\leadsto 0.5 \cdot \left(\sqrt{\frac{-0.5}{re}} \cdot \color{blue}{\left|\sqrt{2} \cdot im\right|}\right) \]
      Proof
      (fabs.f64 (*.f64 (sqrt.f64 2) im)): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (Rewrite=> *-commutative_binary64 (*.f64 im (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= rem-sqrt-square_binary64 (sqrt.f64 (*.f64 (*.f64 im (sqrt.f64 2)) (*.f64 im (sqrt.f64 2))))): 111 points increase in error, 15 points decrease in error
      (sqrt.f64 (Rewrite=> swap-sqr_binary64 (*.f64 (*.f64 im im) (*.f64 (sqrt.f64 2) (sqrt.f64 2))))): 15 points increase in error, 14 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 im im) (Rewrite=> rem-square-sqrt_binary64 2))): 38 points increase in error, 49 points decrease in error
      (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 (*.f64 im im)))): 0 points increase in error, 0 points decrease in error

   if 0.0 < (sqrt.f64 (*.f64 2 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 35.7

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

   2. Simplified 7.1

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
      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)))))))): 124 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 6.3

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

Alternatives

Alternative 1
Error	6.3
Cost	33348

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

Alternative 2
Error	10.8
Cost	13708

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{if}\;re \leq -1.02 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq -4.6 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.8 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	10.3
Cost	13708

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{if}\;re \leq -1.75 \cdot 10^{+50}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{-0.5}{re}} \cdot \sqrt{im \cdot \left(2 \cdot im\right)}\right)\\ \mathbf{elif}\;re \leq -2.55 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.45 \cdot 10^{-25}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	27.2
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.25 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{-203}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{-178}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.85 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	27.0
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{if}\;im \leq -1.25 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 1.25 \cdot 10^{-203}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 9 \cdot 10^{-176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 8.1 \cdot 10^{-35}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	37.2
Cost	7116

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot im}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;re \leq 2.3 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{-42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 430000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	27.3
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -8.2 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 6.4 \cdot 10^{-34}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 8
Error	47.3
Cost	6720

\[0.5 \cdot \sqrt{2 \cdot im} \]

Reproduce

herbie shell --seed 2022328 
(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)))))