math.sqrt on complex, real part

Percentage Accurate: 42.9% → 83.0%

Time: 10.2s

Precision: binary64

Cost: 13444

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{re} \cdot \left(-im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -7.2e+46)
   (* 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 <= -7.2e+46) {
		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 <= -7.2e+46) {
		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 <= -7.2e+46:
		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 <= -7.2e+46)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im / re) * Float64(-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 <= -7.2e+46)
		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, -7.2e+46], N[(0.5 * N[Sqrt[N[(N[(im / re), $MachinePrecision] * (-im)), $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	42.9%
Target	49.8%
Herbie	83.0%

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

Derivation

1. Split input into 2 regimes

2. if re < -7.1999999999999997e46

   1. Initial program 7.7%

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

   2. Simplified 26.5%

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

      [Start] 7.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 7.7	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 26.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 60.0%

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

   4. Simplified 60.0%

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

      [Start] 60.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      *-commutative [=>] 60.0	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]
      unpow2 [=>] 60.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5\right)} \]

   5. Applied egg-rr 68.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \sqrt{\left(\frac{im}{re} \cdot im\right) \cdot -1}\right)} \]
      Step-by-step derivation

      [Start] 60.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)} \]
      add-log-exp [=>] 16.4	\[ 0.5 \cdot \color{blue}{\log \left(e^{\sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}}\right)} \]
      *-un-lft-identity [=>] 16.4	\[ 0.5 \cdot \log \color{blue}{\left(1 \cdot e^{\sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}}\right)} \]
      log-prod [=>] 16.4	\[ 0.5 \cdot \color{blue}{\left(\log 1 + \log \left(e^{\sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}}\right)\right)} \]
      metadata-eval [=>] 16.4	\[ 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}}\right)\right) \]
      add-log-exp [<=] 60.0	\[ 0.5 \cdot \left(0 + \color{blue}{\sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}}\right) \]
      *-commutative [=>] 60.0	\[ 0.5 \cdot \left(0 + \sqrt{\color{blue}{\left(\frac{im \cdot im}{re} \cdot -0.5\right) \cdot 2}}\right) \]
      associate-*l* [=>] 60.0	\[ 0.5 \cdot \left(0 + \sqrt{\color{blue}{\frac{im \cdot im}{re} \cdot \left(-0.5 \cdot 2\right)}}\right) \]
      associate-/l* [=>] 68.3	\[ 0.5 \cdot \left(0 + \sqrt{\color{blue}{\frac{im}{\frac{re}{im}}} \cdot \left(-0.5 \cdot 2\right)}\right) \]
      associate-/r/ [=>] 68.3	\[ 0.5 \cdot \left(0 + \sqrt{\color{blue}{\left(\frac{im}{re} \cdot im\right)} \cdot \left(-0.5 \cdot 2\right)}\right) \]
      metadata-eval [=>] 68.3	\[ 0.5 \cdot \left(0 + \sqrt{\left(\frac{im}{re} \cdot im\right) \cdot \color{blue}{-1}}\right) \]

   6. Simplified 68.3%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{\frac{im}{re} \cdot \left(-im\right)}} \]
      Step-by-step derivation

      [Start] 68.3	\[ 0.5 \cdot \left(0 + \sqrt{\left(\frac{im}{re} \cdot im\right) \cdot -1}\right) \]
      +-lft-identity [=>] 68.3	\[ 0.5 \cdot \color{blue}{\sqrt{\left(\frac{im}{re} \cdot im\right) \cdot -1}} \]
      associate-*l* [=>] 68.3	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{im}{re} \cdot \left(im \cdot -1\right)}} \]
      *-commutative [<=] 68.3	\[ 0.5 \cdot \sqrt{\frac{im}{re} \cdot \color{blue}{\left(-1 \cdot im\right)}} \]
      neg-mul-1 [<=] 68.3	\[ 0.5 \cdot \sqrt{\frac{im}{re} \cdot \color{blue}{\left(-im\right)}} \]

   if -7.1999999999999997e46 < re

   1. Initial program 47.3%

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

   2. Simplified 93.6%

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

      [Start] 47.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 47.3	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 93.6	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{re} \cdot \left(-im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	59.3%
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.45 \cdot 10^{-138}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{-190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{-168}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{re} \cdot \left(-im\right)}\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 2
Accuracy	59.0%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -2.55 \cdot 10^{-135}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \left(-2\right)}\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Accuracy	59.7%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -5.3 \cdot 10^{-139}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Accuracy	30.3%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -2.1 \cdot 10^{-200}:\\ \;\;\;\;265720.5\\ \mathbf{elif}\;im \leq 10^{-297}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 5
Accuracy	58.7%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -5.1 \cdot 10^{-141}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \left(-2\right)}\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{-14}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 6
Accuracy	41.9%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{-13}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 7
Accuracy	8.7%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+47}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;4.885243819643157 \cdot 10^{-22}\\ \end{array} \]

Alternative 8
Accuracy	8.8%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -8 \cdot 10^{+49}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;4.923200210024256 \cdot 10^{-15}\\ \end{array} \]

Alternative 9
Accuracy	8.9%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -2.1 \cdot 10^{+48}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1.071673525377229 \cdot 10^{-5}\\ \end{array} \]

Alternative 10
Accuracy	8.9%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -4 \cdot 10^{+50}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.0078125\\ \end{array} \]

Alternative 11
Accuracy	8.9%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{+48}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.28935185185185186\\ \end{array} \]

Alternative 12
Accuracy	8.9%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -5.2 \cdot 10^{+49}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1.1851851851851851\\ \end{array} \]

Alternative 13
Accuracy	8.9%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -6.4 \cdot 10^{+47}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1.6875\\ \end{array} \]

Alternative 14
Accuracy	8.9%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+50}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;19.2216796875\\ \end{array} \]

Alternative 15
Accuracy	9.0%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{+48}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;265720.5\\ \end{array} \]

Alternative 16
Accuracy	9.0%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -4.9 \cdot 10^{+48}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;3812798742493.5\\ \end{array} \]

Alternative 17
Accuracy	6.0%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023161 
(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)))))