math.sqrt on complex, real part

Average Error: 61.09% → 17.41%

Time: 8.6s

Precision: binary64

Cost: 13709

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -9 \cdot 10^{+62} \lor \neg \left(re \leq -1 \cdot 10^{+23}\right) \land re \leq -195:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (or (<= re -9e+62) (and (not (<= re -1e+23)) (<= re -195.0)))
   (* 0.5 (sqrt (* (- im) (/ 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 <= -9e+62) || (!(re <= -1e+23) && (re <= -195.0))) {
		tmp = 0.5 * sqrt((-im * (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 <= -9e+62) || (!(re <= -1e+23) && (re <= -195.0))) {
		tmp = 0.5 * Math.sqrt((-im * (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 <= -9e+62) or (not (re <= -1e+23) and (re <= -195.0)):
		tmp = 0.5 * math.sqrt((-im * (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 <= -9e+62) || (!(re <= -1e+23) && (re <= -195.0)))
		tmp = Float64(0.5 * sqrt(Float64(Float64(-im) * Float64(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 <= -9e+62) || (~((re <= -1e+23)) && (re <= -195.0)))
		tmp = 0.5 * sqrt((-im * (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[Or[LessEqual[re, -9e+62], And[N[Not[LessEqual[re, -1e+23]], $MachinePrecision], LessEqual[re, -195.0]]], N[(0.5 * N[Sqrt[N[((-im) * N[(im / re), $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	61.09%
Target	53.21%
Herbie	17.41%

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

Derivation

1. Split input into 2 regimes

2. if re < -8.99999999999999997e62 or -9.9999999999999992e22 < re < -195

   1. Initial program 91.17

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

   2. Simplified 59.43

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

      [Start] 91.17	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 91.17	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 59.43	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 51.93

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

   4. Simplified 44.21

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

      [Start] 51.93	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      *-commutative [=>] 51.93	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]
      unpow2 [=>] 51.93	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5\right)} \]
      associate-/l* [=>] 44.21	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\frac{im}{\frac{re}{im}}} \cdot -0.5\right)} \]

   5. Taylor expanded in im around 0 51.9

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

   6. Simplified 44.19

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

      [Start] 51.9	\[ 0.5 \cdot \sqrt{-1 \cdot \frac{{im}^{2}}{re}} \]
      mul-1-neg [=>] 51.9	\[ 0.5 \cdot \sqrt{\color{blue}{-\frac{{im}^{2}}{re}}} \]
      unpow2 [=>] 51.9	\[ 0.5 \cdot \sqrt{-\frac{\color{blue}{im \cdot im}}{re}} \]
      associate-*r/ [<=] 44.19	\[ 0.5 \cdot \sqrt{-\color{blue}{im \cdot \frac{im}{re}}} \]
      distribute-rgt-neg-in [=>] 44.19	\[ 0.5 \cdot \sqrt{\color{blue}{im \cdot \left(-\frac{im}{re}\right)}} \]
      distribute-frac-neg [<=] 44.19	\[ 0.5 \cdot \sqrt{im \cdot \color{blue}{\frac{-im}{re}}} \]

   if -8.99999999999999997e62 < re < -9.9999999999999992e22 or -195 < re

   1. Initial program 52.25

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

   2. Simplified 9.53

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

      [Start] 52.25	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 52.25	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 9.53	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 17.41

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9 \cdot 10^{+62} \lor \neg \left(re \leq -1 \cdot 10^{+23}\right) \land re \leq -195:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	49.46%
Cost	7888

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{if}\;re \leq -7.2 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -9.5 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;re \leq -6.8 \cdot 10^{-165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{-271}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(\frac{re}{\frac{im}{re}} \cdot -0.5 - im\right)\right)}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 2
Error	49.49%
Cost	7508

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{if}\;re \leq -7.5 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.1 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;re \leq -1.5 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 4 \cdot 10^{-270}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 3
Error	40.76%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -1.7 \cdot 10^{-106}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 1.46 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Error	40.28%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -6.8 \cdot 10^{-106}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 1.46 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 5
Error	41.09%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -4.2 \cdot 10^{-106}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 1.46 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 6
Error	57.71%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;im \leq 1.46 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 7
Error	73.36%
Cost	6720

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

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)))))