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

Average Error: 39.3 → 7.8

Time: 8.3s

Precision: binary64

Cost: 13444

\[im > 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re 1.7e-6)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* 0.5 (* im (pow re -0.5)))))

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 <= 1.7e-6) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	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 <= 1.7e-6) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	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 <= 1.7e-6:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	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 <= 1.7e-6)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	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 <= 1.7e-6)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	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, 1.7e-6], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.70000000000000003e-6

   1. Initial program 33.3

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

   2. Simplified 4.9

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
      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)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))))): 0 points increase in error, 6 points decrease in error
      (*.f64 1/2 (sqrt.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 2 (neg.f64 -1)) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))))): 7 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 (*.f64 2 (Rewrite=> metadata-eval 1)) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))): 0 points increase in error, 7 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 (Rewrite=> metadata-eval 2) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))): 7 points increase in error, 0 points decrease in error

   if 1.70000000000000003e-6 < re

   1. Initial program 56.7

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

   2. Simplified 37.4

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]
      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)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))))): 0 points increase in error, 6 points decrease in error
      (*.f64 1/2 (sqrt.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 2 (neg.f64 -1)) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))))): 7 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 (*.f64 2 (Rewrite=> metadata-eval 1)) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))): 0 points increase in error, 7 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 (Rewrite=> metadata-eval 2) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))): 7 points increase in error, 0 points decrease in error

   3. Taylor expanded in re around inf 35.8

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

   4. Simplified 35.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{0.5}{re} \cdot \left(im \cdot im\right)\right)}} \]
      Proof
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (*.f64 (/.f64 1/2 re) (*.f64 im im))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (*.f64 (/.f64 1/2 re) (Rewrite<= unpow2_binary64 (pow.f64 im 2)))))): 5 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (Rewrite<= associate-/r/_binary64 (/.f64 1/2 (/.f64 re (pow.f64 im 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 1/2 (pow.f64 im 2)) re))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (pow.f64 im 2) re)))))): 0 points increase in error, 5 points decrease in error

   5. Applied egg-rr 35.8

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

   6. Applied egg-rr 46.1

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]

   7. Simplified 16.4

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
      Proof
      (*.f64 1/2 (/.f64 im (sqrt.f64 re))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (/.f64 im (sqrt.f64 re)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (/.f64 im (sqrt.f64 re)))) 1))): 0 points increase in error, 0 points decrease in error

   8. Applied egg-rr 16.4

      \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{-0.5} \cdot im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	16.1
Cost	7048

\[\begin{array}{l} \mathbf{if}\;re \leq -6.8 \cdot 10^{-67}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.8 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 2
Error	23.0
Cost	6916

\[\begin{array}{l} \mathbf{if}\;re \leq 1.5 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 3
Error	23.0
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 4
Error	30.3
Cost	6720

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

Reproduce

herbie shell --seed 2022340 
(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)))))