math.sqrt on complex, real part

Average Error: 38.6 → 26.2

Time: 10.3s

Precision: binary64

Cost: 7556

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \leq -1.7 \cdot 10^{-67}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\left(-im\right) + re \cdot \left(-0.5 \cdot \frac{re}{im}\right)\right) + re\right)}\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{-194}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= im -1.7e-67)
   (* 0.5 (sqrt (* 2.0 (+ (+ (- im) (* re (* -0.5 (/ re im)))) re))))
   (if (<= im 2.45e-194)
     (* 0.5 (sqrt (* 2.0 (+ re 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 <= -1.7e-67) {
		tmp = 0.5 * sqrt((2.0 * ((-im + (re * (-0.5 * (re / im)))) + re)));
	} else if (im <= 2.45e-194) {
		tmp = 0.5 * sqrt((2.0 * (re + 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 <= (-1.7d-67)) then
        tmp = 0.5d0 * sqrt((2.0d0 * ((-im + (re * ((-0.5d0) * (re / im)))) + re)))
    else if (im <= 2.45d-194) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + 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 <= -1.7e-67) {
		tmp = 0.5 * Math.sqrt((2.0 * ((-im + (re * (-0.5 * (re / im)))) + re)));
	} else if (im <= 2.45e-194) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + 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 <= -1.7e-67:
		tmp = 0.5 * math.sqrt((2.0 * ((-im + (re * (-0.5 * (re / im)))) + re)))
	elif im <= 2.45e-194:
		tmp = 0.5 * math.sqrt((2.0 * (re + 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 <= -1.7e-67)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(Float64(-im) + Float64(re * Float64(-0.5 * Float64(re / im)))) + re))));
	elseif (im <= 2.45e-194)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + 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 <= -1.7e-67)
		tmp = 0.5 * sqrt((2.0 * ((-im + (re * (-0.5 * (re / im)))) + re)));
	elseif (im <= 2.45e-194)
		tmp = 0.5 * sqrt((2.0 * (re + 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, -1.7e-67], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[((-im) + N[(re * N[(-0.5 * N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.45e-194], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	38.6
Target	33.4
Herbie	26.2

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

Derivation

1. Split input into 3 regimes

2. if im < -1.70000000000000005e-67

   1. Initial program 39.0

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

   2. Applied egg-rr 39.0

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

   3. Applied egg-rr 39.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{re \cdot \left(re \cdot 4\right) + \left(im \cdot im + re \cdot \left(re \cdot -3\right)\right)}} + re\right)} \]

   4. Simplified 39.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{im \cdot im + re \cdot \left(4 \cdot re + re \cdot -3\right)}} + re\right)} \]
      Proof

      [Start] 39.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot \left(re \cdot 4\right) + \left(im \cdot im + re \cdot \left(re \cdot -3\right)\right)} + re\right)} \]
      rational.json-simplify-11 [=>] 39.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{im \cdot im + \left(re \cdot \left(re \cdot 4\right) + re \cdot \left(re \cdot -3\right)\right)}} + re\right)} \]
      rational.json-simplify-39 [=>] 39.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + \left(\color{blue}{\left(re \cdot 4\right) \cdot re} + re \cdot \left(re \cdot -3\right)\right)} + re\right)} \]
      rational.json-simplify-35 [=>] 39.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + \color{blue}{re \cdot \left(re \cdot 4 + re \cdot -3\right)}} + re\right)} \]
      rational.json-simplify-39 [=>] 39.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot \left(\color{blue}{4 \cdot re} + re \cdot -3\right)} + re\right)} \]

   5. Applied egg-rr 39.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot \color{blue}{\left(\left(re \cdot 4 - re\right) - \left(re + re\right)\right)}} + re\right)} \]

   6. Taylor expanded in im around -inf 22.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-0.5 \cdot \frac{\left(4 \cdot re - \left(re + 2 \cdot re\right)\right) \cdot re}{im} + -1 \cdot im\right)} + re\right)} \]

   7. Simplified 19.2

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

      [Start] 22.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(-0.5 \cdot \frac{\left(4 \cdot re - \left(re + 2 \cdot re\right)\right) \cdot re}{im} + -1 \cdot im\right) + re\right)} \]
      rational.json-simplify-41 [=>] 22.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-1 \cdot im + -0.5 \cdot \frac{\left(4 \cdot re - \left(re + 2 \cdot re\right)\right) \cdot re}{im}\right)} + re\right)} \]
      rational.json-simplify-39 [=>] 22.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\color{blue}{im \cdot -1} + -0.5 \cdot \frac{\left(4 \cdot re - \left(re + 2 \cdot re\right)\right) \cdot re}{im}\right) + re\right)} \]
      rational.json-simplify-72 [=>] 22.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\color{blue}{\left(-im\right)} + -0.5 \cdot \frac{\left(4 \cdot re - \left(re + 2 \cdot re\right)\right) \cdot re}{im}\right) + re\right)} \]
      rational.json-simplify-19 [=>] 19.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\left(-im\right) + -0.5 \cdot \color{blue}{\left(re \cdot \frac{4 \cdot re - \left(re + 2 \cdot re\right)}{im}\right)}\right) + re\right)} \]
      rational.json-simplify-3 [=>] 19.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\left(-im\right) + \color{blue}{re \cdot \left(-0.5 \cdot \frac{4 \cdot re - \left(re + 2 \cdot re\right)}{im}\right)}\right) + re\right)} \]
      rational.json-simplify-39 [<=] 19.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\left(-im\right) + re \cdot \left(-0.5 \cdot \frac{\color{blue}{re \cdot 4} - \left(re + 2 \cdot re\right)}{im}\right)\right) + re\right)} \]
      metadata-eval [<=] 19.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\left(-im\right) + re \cdot \left(-0.5 \cdot \frac{re \cdot \color{blue}{\frac{1}{0.25}} - \left(re + 2 \cdot re\right)}{im}\right)\right) + re\right)} \]
      rational.json-simplify-19 [<=] 19.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\left(-im\right) + re \cdot \left(-0.5 \cdot \frac{\color{blue}{\frac{1 \cdot re}{0.25}} - \left(re + 2 \cdot re\right)}{im}\right)\right) + re\right)} \]
      rational.json-simplify-39 [<=] 19.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\left(-im\right) + re \cdot \left(-0.5 \cdot \frac{\frac{\color{blue}{re \cdot 1}}{0.25} - \left(re + 2 \cdot re\right)}{im}\right)\right) + re\right)} \]
      rational.json-simplify-63 [=>] 19.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\left(-im\right) + re \cdot \left(-0.5 \cdot \frac{\frac{\color{blue}{re}}{0.25} - \left(re + 2 \cdot re\right)}{im}\right)\right) + re\right)} \]

   if -1.70000000000000005e-67 < im < 2.45000000000000002e-194

   1. Initial program 39.5

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

   2. Taylor expanded in re around inf 37.0

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

   if 2.45000000000000002e-194 < im

   1. Initial program 37.7

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

   2. Taylor expanded in re around 0 24.0

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

4. Final simplification 26.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.7 \cdot 10^{-67}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\left(-im\right) + re \cdot \left(-0.5 \cdot \frac{re}{im}\right)\right) + re\right)}\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{-194}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	25.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -3.3 \cdot 10^{-67}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{-194}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 2
Error	36.5
Cost	6980

\[\begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{-194}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Error	44.7
Cost	6848

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

Reproduce

herbie shell --seed 2023066 
(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)))))