math.sqrt on complex, real part

Average Error: 38.7 → 9.8

Time: 9.6s

Precision: binary64

Cost: 26884

Math

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

↓

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

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 (pow (* (- im) (/ im re)) 0.5))
   (* 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 * pow((-im * (im / re)), 0.5);
	} 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.pow((-im * (im / re)), 0.5);
	} 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.pow((-im * (im / re)), 0.5)
	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 * (Float64(Float64(-im) * Float64(im / re)) ^ 0.5));
	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 * ((-im * (im / re)) ^ 0.5);
	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[Power[N[((-im) * N[(im / re), $MachinePrecision]), $MachinePrecision], 0.5], $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	38.7
Target	33.5
Herbie	9.8

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

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 57.2

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

   2. Simplified 57.2

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

   3. Taylor expanded in re around -inf 29.4

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

   4. Simplified 25.6

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

   5. Applied egg-rr 31.7

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(\frac{im \cdot im}{re} \cdot -1\right)}^{0.16666666666666666}\right)}^{3}} \]

   6. Taylor expanded in im around 0 34.7

      \[\leadsto 0.5 \cdot \color{blue}{{\left(e^{0.16666666666666666 \cdot \left(2 \cdot \log im + \log \left(\frac{-1}{re}\right)\right)}\right)}^{3}} \]

   7. Simplified 25.6

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

   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 7.6

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

4. Final simplification 9.8

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

Alternatives

Alternative 1
Error	25.8
Cost	7756

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.4373072726950491 \cdot 10^{-70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -1.417005299423253 \cdot 10^{-174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -5.659947122078499 \cdot 10^{-188}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(\frac{re}{\frac{im}{re}} \cdot -0.5 - im\right)\right)}\\ \mathbf{elif}\;im \leq 7.052414792736788 \cdot 10^{-232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.9391364952293654 \cdot 10^{-76}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 2
Error	26.2
Cost	7508

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.4373072726950491 \cdot 10^{-70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.417005299423253 \cdot 10^{-174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -5.659947122078499 \cdot 10^{-188}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 7.052414792736788 \cdot 10^{-232}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2.9391364952293654 \cdot 10^{-76}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 3
Error	26.0
Cost	7508

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.4373072726950491 \cdot 10^{-70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.417005299423253 \cdot 10^{-174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -5.659947122078499 \cdot 10^{-188}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 7.052414792736788 \cdot 10^{-232}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2.9391364952293654 \cdot 10^{-76}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Error	25.7
Cost	7508

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{if}\;im \leq -1.4373072726950491 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.417005299423253 \cdot 10^{-174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -5.659947122078499 \cdot 10^{-188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 7.052414792736788 \cdot 10^{-232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.9391364952293654 \cdot 10^{-76}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 5
Error	26.4
Cost	7248

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.4373072726950491 \cdot 10^{-70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.417005299423253 \cdot 10^{-174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -5.659947122078499 \cdot 10^{-188}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.1723606900133284 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 6
Error	37.2
Cost	6852

\[\begin{array}{l} \mathbf{if}\;im \leq 1.1723606900133284 \cdot 10^{-150}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 7
Error	47.6
Cost	6720

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

Reproduce

herbie shell --seed 2022284 
(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)))))