math.sqrt on complex, real part

Average Error: 38.5 → 25.0

Time: 14.9s

Precision: binary64

Cost: 14160

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \leq -8.8 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-im\right) + re\right)}\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{-156}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{-128}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{-\frac{1}{re}}\right)\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+96}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= im -8.8e-35)
   (* 0.5 (sqrt (* 2.0 (+ (- im) re))))
   (if (<= im 2.3e-156)
     (* 0.5 (* 2.0 (sqrt re)))
     (if (<= im 1.6e-128)
       (* 0.5 (* im (sqrt (- (/ 1.0 re)))))
       (if (<= im 3.1e+96)
         (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re))))
         (* 0.5 (sqrt (* 2.0 (+ 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 (im <= -8.8e-35) {
		tmp = 0.5 * sqrt((2.0 * (-im + re)));
	} else if (im <= 2.3e-156) {
		tmp = 0.5 * (2.0 * sqrt(re));
	} else if (im <= 1.6e-128) {
		tmp = 0.5 * (im * sqrt(-(1.0 / re)));
	} else if (im <= 3.1e+96) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
end function

↓

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-8.8d-35)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (-im + re)))
    else if (im <= 2.3d-156) then
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    else if (im <= 1.6d-128) then
        tmp = 0.5d0 * (im * sqrt(-(1.0d0 / re)))
    else if (im <= 3.1d+96) then
        tmp = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    end if
    code = tmp
end function

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 (im <= -8.8e-35) {
		tmp = 0.5 * Math.sqrt((2.0 * (-im + re)));
	} else if (im <= 2.3e-156) {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	} else if (im <= 1.6e-128) {
		tmp = 0.5 * (im * Math.sqrt(-(1.0 / re)));
	} else if (im <= 3.1e+96) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (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 im <= -8.8e-35:
		tmp = 0.5 * math.sqrt((2.0 * (-im + re)))
	elif im <= 2.3e-156:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	elif im <= 1.6e-128:
		tmp = 0.5 * (im * math.sqrt(-(1.0 / re)))
	elif im <= 3.1e+96:
		tmp = 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (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 (im <= -8.8e-35)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(-im) + re))));
	elseif (im <= 2.3e-156)
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	elseif (im <= 1.6e-128)
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(-Float64(1.0 / re)))));
	elseif (im <= 3.1e+96)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(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 (im <= -8.8e-35)
		tmp = 0.5 * sqrt((2.0 * (-im + re)));
	elseif (im <= 2.3e-156)
		tmp = 0.5 * (2.0 * sqrt(re));
	elseif (im <= 1.6e-128)
		tmp = 0.5 * (im * sqrt(-(1.0 / re)));
	elseif (im <= 3.1e+96)
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
	else
		tmp = 0.5 * sqrt((2.0 * (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[im, -8.8e-35], N[(0.5 * N[Sqrt[N[(2.0 * N[((-im) + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.3e-156], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.6e-128], N[(0.5 * N[(im * N[Sqrt[(-N[(1.0 / re), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.1e+96], 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], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	38.5
Target	33.5
Herbie	25.0

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

Derivation

1. Split input into 5 regimes

2. if im < -8.79999999999999975e-35

   1. Initial program 39.8

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

   2. Taylor expanded in im around -inf 15.8

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

   3. Simplified 15.8

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

      [Start] 15.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot im + re\right)} \]
      rational.json-simplify-2 [=>] 15.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im \cdot -1} + re\right)} \]
      rational.json-simplify-9 [=>] 15.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-im\right)} + re\right)} \]

   if -8.79999999999999975e-35 < im < 2.3e-156

   1. Initial program 39.9

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

   2. Taylor expanded in im around 0 38.8

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

   3. Simplified 38.4

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

      [Start] 38.8	\[ 0.5 \cdot \left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right) \]
      exponential.json-simplify-24 [=>] 38.4	\[ 0.5 \cdot \left(\color{blue}{\sqrt{{2}^{2}}} \cdot \sqrt{re}\right) \]
      metadata-eval [=>] 38.4	\[ 0.5 \cdot \left(\sqrt{\color{blue}{4}} \cdot \sqrt{re}\right) \]
      metadata-eval [=>] 38.4	\[ 0.5 \cdot \left(\color{blue}{2} \cdot \sqrt{re}\right) \]

   if 2.3e-156 < im < 1.5999999999999999e-128

   1. Initial program 27.4

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

   2. Taylor expanded in re around -inf 54.2

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

   3. Simplified 54.2

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

      [Start] 54.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      rational.json-simplify-2 [=>] 54.2	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]

   4. Taylor expanded in im around 0 64.0

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

   5. Simplified 41.8

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

      [Start] 64.0	\[ 0.5 \cdot \left(\left(\sqrt{-1} \cdot im\right) \cdot \sqrt{\frac{1}{re}}\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ 0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot \left(\sqrt{-1} \cdot im\right)\right)} \]
      rational.json-simplify-2 [=>] 64.0	\[ 0.5 \cdot \left(\sqrt{\frac{1}{re}} \cdot \color{blue}{\left(im \cdot \sqrt{-1}\right)}\right) \]
      rational.json-simplify-43 [=>] 64.0	\[ 0.5 \cdot \color{blue}{\left(im \cdot \left(\sqrt{-1} \cdot \sqrt{\frac{1}{re}}\right)\right)} \]
      exponential.json-simplify-20 [=>] 41.8	\[ 0.5 \cdot \left(im \cdot \color{blue}{\sqrt{\frac{1}{re} \cdot -1}}\right) \]
      rational.json-simplify-9 [=>] 41.8	\[ 0.5 \cdot \left(im \cdot \sqrt{\color{blue}{-\frac{1}{re}}}\right) \]

   if 1.5999999999999999e-128 < im < 3.0999999999999998e96

   1. Initial program 24.2

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

   if 3.0999999999999998e96 < im

   1. Initial program 50.8

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

   2. Taylor expanded in re around 0 10.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + im\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 25.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8.8 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-im\right) + re\right)}\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{-156}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{-128}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{-\frac{1}{re}}\right)\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+96}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	26.6
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -8.8 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-im\right) + re\right)}\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.6 \cdot 10^{-128}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{-\frac{1}{re}}\right)\\ \mathbf{elif}\;im \leq 3.4 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 2
Error	26.7
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -8.8 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 3.4 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Error	26.5
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -8.8 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-im\right) + re\right)}\\ \mathbf{elif}\;im \leq 8 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Error	27.0
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -9.2 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{-33}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 5
Error	30.6
Cost	6852

\[\begin{array}{l} \mathbf{if}\;im \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 6
Error	47.5
Cost	6720

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

Reproduce

herbie shell --seed 2023077 
(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)))))