math.sqrt on complex, real part

Average Error: 38.3 → 10.6

Time: 9.4s

Precision: binary64

Cost: 13444

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{\frac{re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -5.5e+72)
   (* 0.5 (sqrt (/ (- im) (/ re im))))
   (* 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 <= -5.5e+72) {
		tmp = 0.5 * sqrt((-im / (re / im)));
	} 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 <= -5.5e+72) {
		tmp = 0.5 * Math.sqrt((-im / (re / im)));
	} 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 <= -5.5e+72:
		tmp = 0.5 * math.sqrt((-im / (re / im)))
	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 <= -5.5e+72)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-im) / Float64(re / im))));
	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 <= -5.5e+72)
		tmp = 0.5 * sqrt((-im / (re / im)));
	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, -5.5e+72], N[(0.5 * N[Sqrt[N[((-im) / N[(re / im), $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	38.3
Target	33.2
Herbie	10.6

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

Derivation

1. Split input into 2 regimes

2. if re < -5.5e72

   1. Initial program 59.9

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

   2. Simplified 40.3

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

      [Start] 59.9	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 59.9	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 40.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 31.6

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

   4. Simplified 31.6

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

      [Start] 31.6	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      *-commutative [=>] 31.6	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]
      unpow2 [=>] 31.6	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5\right)} \]

   5. Applied egg-rr 31.6

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

   6. Taylor expanded in im around 0 31.6

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

   7. Simplified 25.7

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

      [Start] 31.6	\[ 0.5 \cdot \sqrt{-1 \cdot \frac{{im}^{2}}{re}} \]
      associate-*r/ [=>] 31.6	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{-1 \cdot {im}^{2}}{re}}} \]
      unpow2 [=>] 31.6	\[ 0.5 \cdot \sqrt{\frac{-1 \cdot \color{blue}{\left(im \cdot im\right)}}{re}} \]
      associate-*r* [=>] 31.6	\[ 0.5 \cdot \sqrt{\frac{\color{blue}{\left(-1 \cdot im\right) \cdot im}}{re}} \]
      neg-mul-1 [<=] 31.6	\[ 0.5 \cdot \sqrt{\frac{\color{blue}{\left(-im\right)} \cdot im}{re}} \]
      associate-/l* [=>] 25.7	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{-im}{\frac{re}{im}}}} \]

   if -5.5e72 < re

   1. Initial program 33.2

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

   2. Simplified 7.0

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

      [Start] 33.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 33.2	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 7.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{\frac{re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	26.3
Cost	7640

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -2.9 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.42 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -7.6 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.1 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 3.7 \cdot 10^{-243}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 3 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 2
Error	25.9
Cost	7640

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{if}\;im \leq -8.8 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -3.3 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -8.4 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -6.2 \cdot 10^{-308}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 8.4 \cdot 10^{-246}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Error	25.9
Cost	7640

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{if}\;im \leq -2.2 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -4.5 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -3.3 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -6 \cdot 10^{-309}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{-247}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{\sqrt{-re}}{im}}\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Error	26.5
Cost	7512

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -5.8 \cdot 10^{-51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -4 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -3.6 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.7 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.85 \cdot 10^{-252}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 5
Error	26.5
Cost	7248

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -5 \cdot 10^{-51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.05 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -5.2 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 6
Error	37.0
Cost	6852

\[\begin{array}{l} \mathbf{if}\;im \leq 2.9 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 7
Error	47.2
Cost	6720

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

Reproduce

herbie shell --seed 2023038 
(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)))))