math.sqrt on complex, imaginary part, im greater than 0 branch

Average Error: 38.7 → 6.2

Time: 7.0s

Precision: binary64

Cost: 20356

\[im > 0\]

Math

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

↓

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

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (* 0.5 (/ im (sqrt re)))
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))))

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(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = 0.5 * (im / sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	}
	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(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	}
	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(((re * re) + (im * im))) - re) <= 0.0:
		tmp = 0.5 * (im / math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	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 (Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re) <= 0.0)
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	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(((re * re) + (im * im))) - re) <= 0.0)
		tmp = 0.5 * (im / sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	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[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 58.7

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

   2. Simplified 52.7

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
      Proof
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (-.f64 (hypot.f64 re im) re)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (-.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) re)))): 139 points increase in error, 0 points decrease in error

   3. Taylor expanded in re around inf 35.0

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

   4. Simplified 35.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{im \cdot im}{re}\right)}} \]
      Proof
      (*.f64 1/2 (/.f64 (*.f64 im im) re)): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 im 2)) re)): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 54.3

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

   6. Simplified 5.2

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
      Proof
      (/.f64 im (sqrt.f64 re)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 (/.f64 im (sqrt.f64 re))) (sqrt.f64 (/.f64 im (sqrt.f64 re))))): 30 points increase in error, 23 points decrease in error
      (Rewrite<= fabs-sqr_binary64 (fabs.f64 (*.f64 (sqrt.f64 (/.f64 im (sqrt.f64 re))) (sqrt.f64 (/.f64 im (sqrt.f64 re)))))): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (Rewrite=> rem-square-sqrt_binary64 (/.f64 im (sqrt.f64 re)))): 23 points increase in error, 30 points decrease in error
      (Rewrite<= rem-sqrt-square_binary64 (sqrt.f64 (*.f64 (/.f64 im (sqrt.f64 re)) (/.f64 im (sqrt.f64 re))))): 53 points increase in error, 0 points decrease in error
      (sqrt.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 im im) (*.f64 (sqrt.f64 re) (sqrt.f64 re))))): 18 points increase in error, 5 points decrease in error
      (sqrt.f64 (/.f64 (*.f64 im im) (Rewrite=> rem-square-sqrt_binary64 re))): 6 points increase in error, 48 points decrease in error
      (sqrt.f64 (Rewrite=> associate-/l*_binary64 (/.f64 im (/.f64 re im)))): 12 points increase in error, 15 points decrease in error
      (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 im (/.f64 re im)) 1/2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-rgt-identity_binary64 (+.f64 (pow.f64 (/.f64 im (/.f64 re im)) 1/2) 0)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> unpow1/2_binary64 (sqrt.f64 (/.f64 im (/.f64 re im)))) 0): 0 points increase in error, 0 points decrease in error
      (+.f64 (sqrt.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 im im) re))) 0): 15 points increase in error, 12 points decrease in error
      (+.f64 (sqrt.f64 (/.f64 (*.f64 im im) (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 re) (sqrt.f64 re))))) 0): 48 points increase in error, 6 points decrease in error
      (+.f64 (sqrt.f64 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 im (sqrt.f64 re)) (/.f64 im (sqrt.f64 re))))) 0): 5 points increase in error, 18 points decrease in error
      (+.f64 (Rewrite=> rem-sqrt-square_binary64 (fabs.f64 (/.f64 im (sqrt.f64 re)))) 0): 0 points increase in error, 53 points decrease in error
      (+.f64 (fabs.f64 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 (/.f64 im (sqrt.f64 re))) (sqrt.f64 (/.f64 im (sqrt.f64 re)))))) 0): 30 points increase in error, 23 points decrease in error
      (+.f64 (Rewrite=> fabs-sqr_binary64 (*.f64 (sqrt.f64 (/.f64 im (sqrt.f64 re))) (sqrt.f64 (/.f64 im (sqrt.f64 re))))) 0): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> rem-square-sqrt_binary64 (/.f64 im (sqrt.f64 re))) 0): 23 points increase in error, 30 points decrease in error
      (+.f64 (/.f64 im (sqrt.f64 re)) (Rewrite<= metadata-eval (-.f64 1 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 im (sqrt.f64 re)) 1) 1)): 58 points increase in error, 6 points decrease in error
      (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (/.f64 im (sqrt.f64 re)))) 1): 0 points increase in error, 0 points decrease in error

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 35.1

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

   2. Simplified 6.3

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
      Proof
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (-.f64 (hypot.f64 re im) re)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (-.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))) re)))): 139 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 6.2

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

Alternatives

Alternative 1
Error	15.5
Cost	13380

\[\begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{-14}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{re \cdot -2} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{+19}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 2
Error	15.5
Cost	7112

\[\begin{array}{l} \mathbf{if}\;re \leq -1.6 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+19}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 3
Error	15.8
Cost	7048

\[\begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{-28}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+19}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 4
Error	15.8
Cost	6984

\[\begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-22}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+19}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 5
Error	23.3
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq -1.6 \cdot 10^{-22}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 6
Error	30.2
Cost	6720

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

Reproduce

herbie shell --seed 2022329 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))