math.sqrt on complex, real part

Average Error: 38.8 → 11.6

Time: 10.0s

Precision: binary64

Cost: 26440

Math

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

↓

\[\begin{array}{l} t_0 := re + \mathsf{hypot}\left(re, im\right)\\ \mathbf{if}\;re \leq -1.2176412365628154 \cdot 10^{+156}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq -9.421138506343906 \cdot 10^{+85}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot {\left(\sqrt{t_0}\right)}^{2}}\\ \mathbf{elif}\;re \leq -1.9492330657275904 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot t_0}\\ \end{array} \]

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 (+ re (hypot re im))))
   (if (<= re -1.2176412365628154e+156)
     (* 0.5 (sqrt (* (- im) (/ im re))))
     (if (<= re -9.421138506343906e+85)
       (* 0.5 (sqrt (* 2.0 (pow (sqrt t_0) 2.0))))
       (if (<= re -1.9492330657275904e+24)
         (* 0.5 (/ im (sqrt (- re))))
         (* 0.5 (sqrt (* 2.0 t_0))))))))

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 = re + hypot(re, im);
	double tmp;
	if (re <= -1.2176412365628154e+156) {
		tmp = 0.5 * sqrt((-im * (im / re)));
	} else if (re <= -9.421138506343906e+85) {
		tmp = 0.5 * sqrt((2.0 * pow(sqrt(t_0), 2.0)));
	} else if (re <= -1.9492330657275904e+24) {
		tmp = 0.5 * (im / sqrt(-re));
	} else {
		tmp = 0.5 * sqrt((2.0 * t_0));
	}
	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 = re + Math.hypot(re, im);
	double tmp;
	if (re <= -1.2176412365628154e+156) {
		tmp = 0.5 * Math.sqrt((-im * (im / re)));
	} else if (re <= -9.421138506343906e+85) {
		tmp = 0.5 * Math.sqrt((2.0 * Math.pow(Math.sqrt(t_0), 2.0)));
	} else if (re <= -1.9492330657275904e+24) {
		tmp = 0.5 * (im / Math.sqrt(-re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * t_0));
	}
	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 = re + math.hypot(re, im)
	tmp = 0
	if re <= -1.2176412365628154e+156:
		tmp = 0.5 * math.sqrt((-im * (im / re)))
	elif re <= -9.421138506343906e+85:
		tmp = 0.5 * math.sqrt((2.0 * math.pow(math.sqrt(t_0), 2.0)))
	elif re <= -1.9492330657275904e+24:
		tmp = 0.5 * (im / math.sqrt(-re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * t_0))
	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(re + hypot(re, im))
	tmp = 0.0
	if (re <= -1.2176412365628154e+156)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-im) * Float64(im / re))));
	elseif (re <= -9.421138506343906e+85)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * (sqrt(t_0) ^ 2.0))));
	elseif (re <= -1.9492330657275904e+24)
		tmp = Float64(0.5 * Float64(im / sqrt(Float64(-re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * t_0)));
	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 = re + hypot(re, im);
	tmp = 0.0;
	if (re <= -1.2176412365628154e+156)
		tmp = 0.5 * sqrt((-im * (im / re)));
	elseif (re <= -9.421138506343906e+85)
		tmp = 0.5 * sqrt((2.0 * (sqrt(t_0) ^ 2.0)));
	elseif (re <= -1.9492330657275904e+24)
		tmp = 0.5 * (im / sqrt(-re));
	else
		tmp = 0.5 * sqrt((2.0 * t_0));
	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[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.2176412365628154e+156], N[(0.5 * N[Sqrt[N[((-im) * N[(im / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -9.421138506343906e+85], N[(0.5 * N[Sqrt[N[(2.0 * N[Power[N[Sqrt[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.9492330657275904e+24], N[(0.5 * N[(im / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	38.8
Target	33.8
Herbie	11.6

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

Derivation

1. Split input into 4 regimes

2. if re < -1.2176412365628154e156

   1. Initial program 64.0

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

   2. Simplified 42.7

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
      Proof
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (+.f64 re (hypot.f64 re im))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (+.f64 re (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))))))): 139 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in re around -inf 32.3

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

   4. Applied egg-rr 32.3

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

   5. Taylor expanded in im around 0 32.3

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

   6. Simplified 23.8

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot \frac{-im}{re}}} \]
      Proof
      (*.f64 im (/.f64 (neg.f64 im) re)): 0 points increase in error, 0 points decrease in error
      (*.f64 im (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 im re)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 im (/.f64 im re)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 im im) re))): 54 points increase in error, 16 points decrease in error
      (neg.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 im 2)) re)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (pow.f64 im 2) re))): 0 points increase in error, 0 points decrease in error

   if -1.2176412365628154e156 < re < -9.4211385063439058e85

   1. Initial program 54.1

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

   2. Simplified 39.0

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
      Proof
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (+.f64 re (hypot.f64 re im))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (+.f64 re (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))))))): 139 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 39.1

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

   if -9.4211385063439058e85 < re < -1.9492330657275904e24

   1. Initial program 48.2

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

   2. Simplified 31.5

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
      Proof
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (+.f64 re (hypot.f64 re im))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (+.f64 re (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))))))): 139 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in re around -inf 40.7

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

   4. Applied egg-rr 40.7

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

   5. Applied egg-rr 42.5

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

   if -1.9492330657275904e24 < re

   1. Initial program 32.8

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

   2. Simplified 5.3

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
      Proof
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (+.f64 re (hypot.f64 re im))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (+.f64 re (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))))))): 139 points increase in error, 0 points decrease in error
      (*.f64 1/2 (sqrt.f64 (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))))): 0 points increase in error, 0 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 11.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.2176412365628154 \cdot 10^{+156}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq -9.421138506343906 \cdot 10^{+85}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot {\left(\sqrt{re + \mathsf{hypot}\left(re, im\right)}\right)}^{2}}\\ \mathbf{elif}\;re \leq -1.9492330657275904 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.6
Cost	13708

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{if}\;re \leq -1.2176412365628154 \cdot 10^{+156}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq -9.421138506343906 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.9492330657275904 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	26.4
Cost	13576

\[\begin{array}{l} \mathbf{if}\;im \leq -2.8478078209064497 \cdot 10^{-71}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -1.747329922203876 \cdot 10^{-249}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{-im}}{\sqrt{\frac{re}{im}}}\\ \mathbf{elif}\;im \leq 1.569306940882843 \cdot 10^{-192}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot 2\right)}\\ \mathbf{elif}\;im \leq 1.2141225401825247 \cdot 10^{-91}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Error	26.7
Cost	7640

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(re \cdot 2\right)}\\ \mathbf{if}\;im \leq -2.1822495746485088 \cdot 10^{-69}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -1.8575299460274046 \cdot 10^{-185}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -8.355528269550826 \cdot 10^{-227}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq -1.6065140864874557 \cdot 10^{-254}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{re}}\\ \mathbf{elif}\;im \leq 1.569306940882843 \cdot 10^{-192}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.2141225401825247 \cdot 10^{-91}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Error	26.7
Cost	7508

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(re \cdot 2\right)}\\ \mathbf{if}\;im \leq -2.1822495746485088 \cdot 10^{-69}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -1.8575299460274046 \cdot 10^{-185}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.6065140864874557 \cdot 10^{-254}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{\frac{re}{im}}}\\ \mathbf{elif}\;im \leq 1.569306940882843 \cdot 10^{-192}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.2141225401825247 \cdot 10^{-91}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 5
Error	28.5
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ t_1 := 0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{if}\;im \leq -8.355528269550826 \cdot 10^{-227}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 6.401439717272362 \cdot 10^{-270}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.569306940882843 \cdot 10^{-192}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.2141225401825247 \cdot 10^{-91}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	26.2
Cost	7244

\[\begin{array}{l} \mathbf{if}\;im \leq -2.1822495746485088 \cdot 10^{-69}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 1.569306940882843 \cdot 10^{-192}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot 2\right)}\\ \mathbf{elif}\;im \leq 1.2141225401825247 \cdot 10^{-91}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 7
Error	27.9
Cost	7048

\[\begin{array}{l} \mathbf{if}\;im \leq -8.355528269550826 \cdot 10^{-227}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 1.2141225401825247 \cdot 10^{-91}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 8
Error	42.6
Cost	6916

\[\begin{array}{l} \mathbf{if}\;re \leq -7.52858483313813 \cdot 10^{-105}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 9
Error	62.9
Cost	6720

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

Alternative 10
Error	62.9
Cost	6720

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

Alternative 11
Error	47.0
Cost	6720

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

Reproduce

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