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

Average Error: 37.8 → 8.6

Time: 9.1s

Precision: binary64

Cost: 13708

\[im > 0\]

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{if}\;re \leq 3.355254974716523 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 9.401840013691732 \cdot 10^{-11}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 3.207152800022526 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]

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 (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))))
   (if (<= re 3.355254974716523e-105)
     t_0
     (if (<= re 9.401840013691732e-11)
       (* im (/ 0.5 (sqrt re)))
       (if (<= re 3.207152800022526e+83) t_0 (/ (* 0.5 im) (sqrt 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 t_0 = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	double tmp;
	if (re <= 3.355254974716523e-105) {
		tmp = t_0;
	} else if (re <= 9.401840013691732e-11) {
		tmp = im * (0.5 / sqrt(re));
	} else if (re <= 3.207152800022526e+83) {
		tmp = t_0;
	} else {
		tmp = (0.5 * im) / sqrt(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 t_0 = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	double tmp;
	if (re <= 3.355254974716523e-105) {
		tmp = t_0;
	} else if (re <= 9.401840013691732e-11) {
		tmp = im * (0.5 / Math.sqrt(re));
	} else if (re <= 3.207152800022526e+83) {
		tmp = t_0;
	} else {
		tmp = (0.5 * im) / Math.sqrt(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):
	t_0 = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	tmp = 0
	if re <= 3.355254974716523e-105:
		tmp = t_0
	elif re <= 9.401840013691732e-11:
		tmp = im * (0.5 / math.sqrt(re))
	elif re <= 3.207152800022526e+83:
		tmp = t_0
	else:
		tmp = (0.5 * im) / math.sqrt(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)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))))
	tmp = 0.0
	if (re <= 3.355254974716523e-105)
		tmp = t_0;
	elseif (re <= 9.401840013691732e-11)
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	elseif (re <= 3.207152800022526e+83)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(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)
	t_0 = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	tmp = 0.0;
	if (re <= 3.355254974716523e-105)
		tmp = t_0;
	elseif (re <= 9.401840013691732e-11)
		tmp = im * (0.5 / sqrt(re));
	elseif (re <= 3.207152800022526e+83)
		tmp = t_0;
	else
		tmp = (0.5 * im) / sqrt(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_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 3.355254974716523e-105], t$95$0, If[LessEqual[re, 9.401840013691732e-11], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.207152800022526e+83], t$95$0, N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < 3.35525497471652297e-105 or 9.4018400136917322e-11 < re < 3.20715280002252573e83

   1. Initial program 32.4

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

   2. Simplified 5.4

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

   if 3.35525497471652297e-105 < re < 9.4018400136917322e-11

   1. Initial program 38.8

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

   2. Simplified 25.2

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

   3. Taylor expanded in re around inf 52.2

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

   4. Applied egg-rr 34.9

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

   5. Applied egg-rr 35.0

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

   6. Applied egg-rr 35.0

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

   if 3.20715280002252573e83 < re

   1. Initial program 60.2

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

   2. Simplified 40.0

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

   3. Taylor expanded in re around inf 31.4

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

   4. Applied egg-rr 11.4

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

   5. Applied egg-rr 33.9

      \[\leadsto \color{blue}{{\left({\left(\frac{im}{\sqrt{re}} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Applied egg-rr 11.2

      \[\leadsto \color{blue}{\frac{im \cdot 0.5}{\sqrt{re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 8.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.355254974716523 \cdot 10^{-105}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{elif}\;re \leq 9.401840013691732 \cdot 10^{-11}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 3.207152800022526 \cdot 10^{+83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.7
Cost	7248

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot im}\\ t_1 := \frac{0.5 \cdot im}{\sqrt{re}}\\ \mathbf{if}\;re \leq -19698.416591660265:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.8657912924586768 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 9.401840013691732 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 3.207152800022526 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	15.7
Cost	7248

\[\begin{array}{l} \mathbf{if}\;re \leq -19698.416591660265:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3.355254974716523 \cdot 10^{-105}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 9.401840013691732 \cdot 10^{-11}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 3.207152800022526 \cdot 10^{+83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]

Alternative 3
Error	23.9
Cost	7116

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot im}\\ t_1 := \frac{0.5 \cdot im}{\sqrt{re}}\\ \mathbf{if}\;re \leq 1.8657912924586768 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 9.401840013691732 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq 3.207152800022526 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	47.6
Cost	6720

\[im \cdot \frac{0.5}{\sqrt{re}} \]

Alternative 5
Error	47.6
Cost	6720

\[\frac{0.5 \cdot im}{\sqrt{re}} \]

Reproduce

herbie shell --seed 2022295 
(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)))))