math.sqrt on complex, real part

Average Error: 39.5% → 99.5%

Time: 13.0s

Precision: binary64

Cost: 26372.00

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{-296}:\\ \;\;\;\;0.5 \cdot \left|\frac{\sqrt{2} \cdot im}{\sqrt{\mathsf{hypot}\left(re, im\right) - re}}\right|\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e-296)
   (* 0.5 (fabs (/ (* (sqrt 2.0) im) (sqrt (- (hypot re im) re)))))
   (* 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 (re <= -1.8e-296) {
		tmp = 0.5 * fabs(((sqrt(2.0) * im) / sqrt((hypot(re, im) - re))));
	} 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 (re <= -1.8e-296) {
		tmp = 0.5 * Math.abs(((Math.sqrt(2.0) * im) / Math.sqrt((Math.hypot(re, im) - re))));
	} 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 re <= -1.8e-296:
		tmp = 0.5 * math.fabs(((math.sqrt(2.0) * im) / math.sqrt((math.hypot(re, im) - re))))
	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 (re <= -1.8e-296)
		tmp = Float64(0.5 * abs(Float64(Float64(sqrt(2.0) * im) / sqrt(Float64(hypot(re, im) - re)))));
	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 (re <= -1.8e-296)
		tmp = 0.5 * abs(((sqrt(2.0) * im) / sqrt((hypot(re, im) - re))));
	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[re, -1.8e-296], N[(0.5 * N[Abs[N[(N[(N[Sqrt[2.0], $MachinePrecision] * im), $MachinePrecision] / N[Sqrt[N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $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	39.5%
Target	47.5%
Herbie	99.5%

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

Derivation

1. Split input into 2 regimes

2. if re < -1.7999999999999999e-296

   1. Initial program 27.0

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

   2. Applied egg-rr 51.1

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

   3. Applied egg-rr 82.9

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

   4. Applied egg-rr 99.2

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

   if -1.7999999999999999e-296 < re

   1. Initial program 51.3

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

   2. Simplified 99.9

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

      [Start] 51.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 51.3	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 99.9	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{-296}:\\ \;\;\;\;0.5 \cdot \left|\frac{\sqrt{2} \cdot im}{\sqrt{\mathsf{hypot}\left(re, im\right) - re}}\right|\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	87.5%
Cost	13709.00

\[\begin{array}{l} \mathbf{if}\;re \leq -5.4 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left|im \cdot \sqrt{\frac{-1}{re}}\right|\\ \mathbf{elif}\;re \leq -4 \cdot 10^{-55} \lor \neg \left(re \leq -1.35 \cdot 10^{-87}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|im \cdot \frac{1}{\sqrt{-re}}\right|\\ \end{array} \]

Alternative 2
Error	91.6%
Cost	13700.00

\[\begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{-295}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\mathsf{hypot}\left(re, im\right) - re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 3
Error	59.9%
Cost	13380.00

\[\begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{-124}:\\ \;\;\;\;0.5 \cdot \left|im \cdot \sqrt{\frac{-1}{re}}\right|\\ \mathbf{elif}\;re \leq -1.85 \cdot 10^{-199}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{re}{\frac{im}{re}} \cdot -0.5 + \left(re - im\right)\right)}\\ \mathbf{elif}\;re \leq -4.5 \cdot 10^{-255}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{-204}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;re \leq 1.16 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 4
Error	52.3%
Cost	7772.00

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ t_1 := 0.5 \cdot \sqrt{im \cdot \frac{im}{-re}}\\ \mathbf{if}\;re \leq -9 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -4.6 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -4.3 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -4.2 \cdot 10^{-197}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.25 \cdot 10^{-254}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{-197}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 5
Error	53.1%
Cost	7508.00

\[\begin{array}{l} \mathbf{if}\;re \leq -230000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{-re}}\\ \mathbf{elif}\;re \leq -7 \cdot 10^{-200}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\left(-re\right) - im}\right)}\\ \mathbf{elif}\;re \leq -3.8 \cdot 10^{-258}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 3.1 \cdot 10^{-197}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;re \leq 3 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 6
Error	60.3%
Cost	7376.00

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -3.6 \cdot 10^{-48}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{-268}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 3 \cdot 10^{-151}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{-107}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 7
Error	60.2%
Cost	7112.00

\[\begin{array}{l} \mathbf{if}\;im \leq -1.06 \cdot 10^{-48}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{-122}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 8
Error	60.6%
Cost	7112.00

\[\begin{array}{l} \mathbf{if}\;im \leq -3.4 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{-113}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 9
Error	59.7%
Cost	6984.00

\[\begin{array}{l} \mathbf{if}\;im \leq -1.65 \cdot 10^{-49}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{-112}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 10
Error	52.3%
Cost	6852.00

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

Alternative 11
Error	26.2%
Cost	6720.00

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

Reproduce

herbie shell --seed 2023098 
(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)))))