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

Average Error: 38.8 → 6.5

Time: 6.3s

Precision: binary64

Cost: 27400

\[im > 0\]

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im} - re\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-287}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ (* re re) (* im im))) re))
        (t_1 (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))))
   (if (<= t_0 -2e-287) t_1 (if (<= t_0 0.0) (* 0.5 (/ im (sqrt re))) t_1))))

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 t_0 = sqrt(((re * re) + (im * im))) - re;
	double t_1 = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	double tmp;
	if (t_0 <= -2e-287) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = 0.5 * (im / sqrt(re));
	} else {
		tmp = t_1;
	}
	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 t_0 = Math.sqrt(((re * re) + (im * im))) - re;
	double t_1 = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	double tmp;
	if (t_0 <= -2e-287) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else {
		tmp = t_1;
	}
	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):
	t_0 = math.sqrt(((re * re) + (im * im))) - re
	t_1 = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	tmp = 0
	if t_0 <= -2e-287:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = 0.5 * (im / math.sqrt(re))
	else:
		tmp = t_1
	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)
	t_0 = Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re)
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))))
	tmp = 0.0
	if (t_0 <= -2e-287)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	else
		tmp = t_1;
	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)
	t_0 = sqrt(((re * re) + (im * im))) - re;
	t_1 = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	tmp = 0.0;
	if (t_0 <= -2e-287)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = 0.5 * (im / sqrt(re));
	else
		tmp = t_1;
	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_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-287], t$95$1, If[LessEqual[t$95$0, 0.0], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < -2.00000000000000004e-287 or 0.0 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 36.1

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

   2. Simplified 7.2

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
      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)))): 113 points increase in error, 0 points decrease in error

   if -2.00000000000000004e-287 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 57.3

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

   2. Simplified 55.3

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
      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)))): 113 points increase in error, 0 points decrease in error

   3. Taylor expanded in re around inf 30.4

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

   4. Simplified 26.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{im}{\frac{re}{im}}\right)}} \]
      Proof
      (*.f64 1/2 (/.f64 im (/.f64 re im))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 im im) re))): 45 points increase in error, 25 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. Taylor expanded in im around 0 2.0

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

   6. Applied egg-rr 52.7

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

   7. Simplified 2.0

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
      Proof
      (/.f64 im (sqrt.f64 re)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-rgt-identity_binary64 (+.f64 (/.f64 im (sqrt.f64 re)) 0)): 0 points increase in error, 0 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)): 63 points increase in error, 0 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
3. Recombined 2 regimes into one program.

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq -2 \cdot 10^{-287}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.2
Cost	7512

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot 2}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ t_2 := 0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{if}\;re \leq -4.6 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -2 \cdot 10^{-47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -9 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{-91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{-46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	15.7
Cost	7512

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ t_2 := 0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{if}\;re \leq -2.2 \cdot 10^{+30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -1.06 \cdot 10^{-47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -4.1 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{-44}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+23}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	23.3
Cost	7116

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot 2}\\ t_1 := 0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{if}\;re \leq 2.9 \cdot 10^{-91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	30.8
Cost	6720

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

Reproduce

herbie shell --seed 2022339 
(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)))))