math.sqrt on complex, real part

Average Error: 38.5 → 6.6

Time: 8.9s

Precision: binary64

Cost: 33348

Math

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

↓

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

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 (fabs (* (* im (sqrt (/ -0.5 re))) (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 * fabs(((im * sqrt((-0.5 / re))) * 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.abs(((im * Math.sqrt((-0.5 / re))) * 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.fabs(((im * math.sqrt((-0.5 / re))) * 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 * abs(Float64(Float64(im * sqrt(Float64(-0.5 / re))) * 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 * abs(((im * sqrt((-0.5 / re))) * 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[Abs[N[(N[(im * N[Sqrt[N[(-0.5 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $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	38.5
Target	33.7
Herbie	6.6

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

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 57.7

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

   2. Taylor expanded in re around -inf 30.6

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

   3. Simplified 30.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{im \cdot im}{\frac{re}{-0.5}}}} \]
      Proof
      (/.f64 (*.f64 im im) (/.f64 re -1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 im 2)) (/.f64 re -1/2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 im 2) -1/2) re)): 1 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (pow.f64 im 2) re) -1/2)): 0 points increase in error, 1 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 -1/2 (/.f64 (pow.f64 im 2) re))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in im around 0 30.6

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

   5. Simplified 26.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(im \cdot \left(im \cdot \frac{-0.5}{re}\right)\right)}} \]
      Proof
      (*.f64 im (*.f64 im (/.f64 -1/2 re))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 im im) (/.f64 -1/2 re))): 42 points increase in error, 22 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 im 2)) (/.f64 -1/2 re)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (pow.f64 im 2) -1/2) re)): 19 points increase in error, 15 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (pow.f64 im 2) re) -1/2)): 0 points increase in error, 1 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 -1/2 (/.f64 (pow.f64 im 2) re))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 0.5

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

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

   1. Initial program 36.0

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

   2. Simplified 7.4

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
      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)))))))): 126 points increase in error, 1 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 6.6

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

Alternatives

Alternative 1
Error	25.6
Cost	13644

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.6238613152027008 \cdot 10^{-183}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -2.420878260300143 \cdot 10^{-303}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.1647153298127593 \cdot 10^{-288}:\\ \;\;\;\;im \cdot \frac{\sqrt{0.5}}{\sqrt{re \cdot -2}}\\ \mathbf{elif}\;im \leq 9.507072184428753 \cdot 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 2
Error	10.6
Cost	13444

\[\begin{array}{l} \mathbf{if}\;re \leq -5.481885247681275 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\frac{im \cdot -0.25}{\frac{re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 3
Error	26.3
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -6.632288262379874 \cdot 10^{-187}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq -2.420878260300143 \cdot 10^{-303}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.1647153298127593 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\frac{im \cdot -0.25}{\frac{re}{im}}}\\ \mathbf{elif}\;im \leq 9.507072184428753 \cdot 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Error	25.8
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.6238613152027008 \cdot 10^{-183}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -2.420878260300143 \cdot 10^{-303}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.1647153298127593 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\frac{im \cdot -0.25}{\frac{re}{im}}}\\ \mathbf{elif}\;im \leq 9.507072184428753 \cdot 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 5
Error	26.7
Cost	7248

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -6.632288262379874 \cdot 10^{-187}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq -2.420878260300143 \cdot 10^{-303}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.1647153298127593 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\frac{im \cdot -0.25}{\frac{re}{im}}}\\ \mathbf{elif}\;im \leq 9.507072184428753 \cdot 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 6
Error	26.5
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -6.632288262379874 \cdot 10^{-187}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 9.507072184428753 \cdot 10^{-147}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 7
Error	36.3
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 1.058604672530162 \cdot 10^{-141}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 8
Error	47.1
Cost	6720

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

Reproduce

herbie shell --seed 2022300 
(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)))))