math.sqrt on complex, real part

Average Error: 39.1 → 28.2

Time: 8.8s

Precision: binary64

Cost: 14424

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{-\frac{{im}^{2}}{re}}\\ \mathbf{if}\;re \leq -1.48 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -7.4 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \sqrt{-2 \cdot im}\\ \mathbf{elif}\;re \leq -0.74:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -5.2 \cdot 10^{-169}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im}\right)\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{-290}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+57}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (- (/ (pow im 2.0) re))))))
   (if (<= re -1.48e+53)
     t_0
     (if (<= re -7.4e+25)
       (* 0.5 (sqrt (* -2.0 im)))
       (if (<= re -0.74)
         t_0
         (if (<= re -5.2e-169)
           (* 0.5 (* (sqrt 2.0) (sqrt im)))
           (if (<= re 2.2e-290)
             (* 0.5 (sqrt (* 2.0 (- re im))))
             (if (<= re 3.6e+57)
               (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re))))
               (* 0.5 (sqrt (* 2.0 (+ re re))))))))))))

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 t_0 = 0.5 * sqrt(-(pow(im, 2.0) / re));
	double tmp;
	if (re <= -1.48e+53) {
		tmp = t_0;
	} else if (re <= -7.4e+25) {
		tmp = 0.5 * sqrt((-2.0 * im));
	} else if (re <= -0.74) {
		tmp = t_0;
	} else if (re <= -5.2e-169) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(im));
	} else if (re <= 2.2e-290) {
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	} else if (re <= 3.6e+57) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + re)));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt(-((im ** 2.0d0) / re))
    if (re <= (-1.48d+53)) then
        tmp = t_0
    else if (re <= (-7.4d+25)) then
        tmp = 0.5d0 * sqrt(((-2.0d0) * im))
    else if (re <= (-0.74d0)) then
        tmp = t_0
    else if (re <= (-5.2d-169)) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(im))
    else if (re <= 2.2d-290) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - im)))
    else if (re <= 3.6d+57) then
        tmp = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + re)))
    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 t_0 = 0.5 * Math.sqrt(-(Math.pow(im, 2.0) / re));
	double tmp;
	if (re <= -1.48e+53) {
		tmp = t_0;
	} else if (re <= -7.4e+25) {
		tmp = 0.5 * Math.sqrt((-2.0 * im));
	} else if (re <= -0.74) {
		tmp = t_0;
	} else if (re <= -5.2e-169) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(im));
	} else if (re <= 2.2e-290) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - im)));
	} else if (re <= 3.6e+57) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + re)));
	}
	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):
	t_0 = 0.5 * math.sqrt(-(math.pow(im, 2.0) / re))
	tmp = 0
	if re <= -1.48e+53:
		tmp = t_0
	elif re <= -7.4e+25:
		tmp = 0.5 * math.sqrt((-2.0 * im))
	elif re <= -0.74:
		tmp = t_0
	elif re <= -5.2e-169:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(im))
	elif re <= 2.2e-290:
		tmp = 0.5 * math.sqrt((2.0 * (re - im)))
	elif re <= 3.6e+57:
		tmp = 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + re)))
	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)
	t_0 = Float64(0.5 * sqrt(Float64(-Float64((im ^ 2.0) / re))))
	tmp = 0.0
	if (re <= -1.48e+53)
		tmp = t_0;
	elseif (re <= -7.4e+25)
		tmp = Float64(0.5 * sqrt(Float64(-2.0 * im)));
	elseif (re <= -0.74)
		tmp = t_0;
	elseif (re <= -5.2e-169)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(im)));
	elseif (re <= 2.2e-290)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))));
	elseif (re <= 3.6e+57)
		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 + re))));
	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)
	t_0 = 0.5 * sqrt(-((im ^ 2.0) / re));
	tmp = 0.0;
	if (re <= -1.48e+53)
		tmp = t_0;
	elseif (re <= -7.4e+25)
		tmp = 0.5 * sqrt((-2.0 * im));
	elseif (re <= -0.74)
		tmp = t_0;
	elseif (re <= -5.2e-169)
		tmp = 0.5 * (sqrt(2.0) * sqrt(im));
	elseif (re <= 2.2e-290)
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	elseif (re <= 3.6e+57)
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
	else
		tmp = 0.5 * sqrt((2.0 * (re + re)));
	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_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[(-N[(N[Power[im, 2.0], $MachinePrecision] / re), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.48e+53], t$95$0, If[LessEqual[re, -7.4e+25], N[(0.5 * N[Sqrt[N[(-2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -0.74], t$95$0, If[LessEqual[re, -5.2e-169], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.2e-290], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.6e+57], 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 + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Target

Original	39.1
Target	34.1
Herbie	28.2

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

Derivation

1. Split input into 6 regimes

2. if re < -1.48e53 or -7.3999999999999998e25 < re < -0.73999999999999999

   1. Initial program 57.9

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

   2. Taylor expanded in re around -inf 33.9

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

   3. Simplified 33.9

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

      [Start] 33.9	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 33.9	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]

   4. Taylor expanded in im around 0 33.9

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]

   5. Simplified 33.9

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

      [Start] 33.9	\[ 0.5 \cdot \sqrt{-1 \cdot \frac{{im}^{2}}{re}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 33.9	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re} \cdot -1}} \]
      rational_best_oopsla_all_46_json_45_simplify-94 [<=] 33.9	\[ 0.5 \cdot \sqrt{\color{blue}{-\frac{{im}^{2}}{re}}} \]

   if -1.48e53 < re < -7.3999999999999998e25

   1. Initial program 49.8

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

   2. Taylor expanded in im around -inf 49.8

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

   3. Simplified 49.8

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

      [Start] 49.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(-0.5 \cdot \frac{{re}^{2}}{im} + -1 \cdot im\right) + re\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 49.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-1 \cdot im + -0.5 \cdot \frac{{re}^{2}}{im}\right)} + re\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 49.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\color{blue}{im \cdot -1} + -0.5 \cdot \frac{{re}^{2}}{im}\right) + re\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 49.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\color{blue}{\left(-im\right)} + -0.5 \cdot \frac{{re}^{2}}{im}\right) + re\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 49.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\left(-im\right) + \color{blue}{\frac{{re}^{2}}{im} \cdot -0.5}\right) + re\right)} \]

   4. Applied egg-rr 49.8

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

   5. Simplified 49.8

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

      [Start] 49.8	\[ 0.5 \cdot \left(\sqrt{2 \cdot \left(re - im\right) - \frac{{re}^{2}}{im}} + 0\right) \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 49.8	\[ 0.5 \cdot \color{blue}{\sqrt{2 \cdot \left(re - im\right) - \frac{{re}^{2}}{im}}} \]

   6. Taylor expanded in re around 0 49.1

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-2 \cdot im}} \]

   if -0.73999999999999999 < re < -5.20000000000000028e-169

   1. Initial program 39.2

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

   2. Taylor expanded in re around 0 43.3

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

   if -5.20000000000000028e-169 < re < 2.2000000000000001e-290

   1. Initial program 30.2

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

   2. Taylor expanded in im around -inf 36.2

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

   3. Simplified 36.2

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

      [Start] 36.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot im + re\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 36.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im \cdot -1} + re\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 36.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-im\right)} + re\right)} \]

   4. Applied egg-rr 36.2

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

   5. Simplified 36.2

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

      [Start] 36.2	\[ 0.5 \cdot \left(\sqrt{-2 \cdot \left(im - re\right)} + 0\right) \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 36.2	\[ 0.5 \cdot \color{blue}{\sqrt{-2 \cdot \left(im - re\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-87 [=>] 36.2	\[ 0.5 \cdot \sqrt{\color{blue}{\left(--2\right) \cdot \left(re - im\right)}} \]
      metadata-eval [=>] 36.2	\[ 0.5 \cdot \sqrt{\color{blue}{2} \cdot \left(re - im\right)} \]

   if 2.2000000000000001e-290 < re < 3.6000000000000002e57

   1. Initial program 21.9

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

   if 3.6000000000000002e57 < re

   1. Initial program 45.2

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

   2. Taylor expanded in re around inf 13.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{re} + re\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 28.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.48 \cdot 10^{+53}:\\ \;\;\;\;0.5 \cdot \sqrt{-\frac{{im}^{2}}{re}}\\ \mathbf{elif}\;re \leq -7.4 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \sqrt{-2 \cdot im}\\ \mathbf{elif}\;re \leq -0.74:\\ \;\;\;\;0.5 \cdot \sqrt{-\frac{{im}^{2}}{re}}\\ \mathbf{elif}\;re \leq -5.2 \cdot 10^{-169}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im}\right)\\ \mathbf{elif}\;re \leq 2.2 \cdot 10^{-290}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+57}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	32.0
Cost	13648

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{-\frac{{im}^{2}}{re}}\\ \mathbf{if}\;re \leq -7.2 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -2.1 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot \sqrt{-2 \cdot im}\\ \mathbf{elif}\;re \leq -0.48:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.7 \cdot 10^{-169}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im}\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-271}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;re \leq 7 \cdot 10^{-34}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array} \]

Alternative 2
Error	26.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -8 \cdot 10^{-106}:\\ \;\;\;\;0.5 \cdot \sqrt{-2 \cdot im}\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Error	25.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -2.8 \cdot 10^{-107}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Error	29.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;im \leq -1.6 \cdot 10^{-266}:\\ \;\;\;\;0.5 \cdot \sqrt{-2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 5
Error	47.5
Cost	6720

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

Reproduce

herbie shell --seed 2023090 
(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)))))