math.sqrt on complex, real part

Average Error: 38.1 → 8.6

Time: 10.8s

Precision: binary64

Cost: 13974

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -5.3 \cdot 10^{+85} \lor \neg \left(re \leq -7.6 \cdot 10^{+69}\right) \land \left(re \leq -1.02 \cdot 10^{-19} \lor \neg \left(re \leq -1.25 \cdot 10^{-32}\right) \land re \leq -5.8 \cdot 10^{-119}\right):\\ \;\;\;\;0.5 \cdot \left|im \cdot \sqrt{\frac{-1}{re}}\right|\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (or (<= re -5.3e+85)
         (and (not (<= re -7.6e+69))
              (or (<= re -1.02e-19)
                  (and (not (<= re -1.25e-32)) (<= re -5.8e-119)))))
   (* 0.5 (fabs (* im (sqrt (/ -1.0 re)))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

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 <= -5.3e+85) || (!(re <= -7.6e+69) && ((re <= -1.02e-19) || (!(re <= -1.25e-32) && (re <= -5.8e-119))))) {
		tmp = 0.5 * fabs((im * sqrt((-1.0 / re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	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 <= -5.3e+85) || (!(re <= -7.6e+69) && ((re <= -1.02e-19) || (!(re <= -1.25e-32) && (re <= -5.8e-119))))) {
		tmp = 0.5 * Math.abs((im * Math.sqrt((-1.0 / re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	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 <= -5.3e+85) or (not (re <= -7.6e+69) and ((re <= -1.02e-19) or (not (re <= -1.25e-32) and (re <= -5.8e-119)))):
		tmp = 0.5 * math.fabs((im * math.sqrt((-1.0 / re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	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 <= -5.3e+85) || (!(re <= -7.6e+69) && ((re <= -1.02e-19) || (!(re <= -1.25e-32) && (re <= -5.8e-119)))))
		tmp = Float64(0.5 * abs(Float64(im * sqrt(Float64(-1.0 / re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	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 <= -5.3e+85) || (~((re <= -7.6e+69)) && ((re <= -1.02e-19) || (~((re <= -1.25e-32)) && (re <= -5.8e-119)))))
		tmp = 0.5 * abs((im * sqrt((-1.0 / re))));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	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[Or[LessEqual[re, -5.3e+85], And[N[Not[LessEqual[re, -7.6e+69]], $MachinePrecision], Or[LessEqual[re, -1.02e-19], And[N[Not[LessEqual[re, -1.25e-32]], $MachinePrecision], LessEqual[re, -5.8e-119]]]]], N[(0.5 * N[Abs[N[(im * N[Sqrt[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Target

Original	38.1
Target	33.0
Herbie	8.6

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

Derivation

1. Split input into 2 regimes

2. if re < -5.2999999999999999e85 or -7.60000000000000055e69 < re < -1.02000000000000004e-19 or -1.25e-32 < re < -5.8e-119

   1. Initial program 52.5

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

   2. Taylor expanded in re around -inf 55.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-1 \cdot re + -0.5 \cdot \frac{{im}^{2}}{re}\right)} + re\right)} \]

   3. Simplified 55.3

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

      [Start] 55.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(-1 \cdot re + -0.5 \cdot \frac{{im}^{2}}{re}\right) + re\right)} \]
      +-commutative [=>] 55.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re} + -1 \cdot re\right)} + re\right)} \]
      mul-1-neg [=>] 55.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(-0.5 \cdot \frac{{im}^{2}}{re} + \color{blue}{\left(-re\right)}\right) + re\right)} \]
      unsub-neg [=>] 55.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re} - re\right)} + re\right)} \]
      associate-*r/ [=>] 55.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\color{blue}{\frac{-0.5 \cdot {im}^{2}}{re}} - re\right) + re\right)} \]
      associate-/l* [=>] 55.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\color{blue}{\frac{-0.5}{\frac{re}{{im}^{2}}}} - re\right) + re\right)} \]
      unpow2 [=>] 55.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(\frac{-0.5}{\frac{re}{\color{blue}{im \cdot im}}} - re\right) + re\right)} \]

   4. Applied egg-rr 20.9

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

   5. Applied egg-rr 52.1

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

   6. Simplified 20.7

      \[\leadsto 0.5 \cdot \left|\color{blue}{im \cdot \sqrt{\frac{-1}{re}}}\right| \]
      Proof

      [Start] 52.1	\[ 0.5 \cdot \left|e^{\mathsf{log1p}\left(im \cdot \sqrt{\frac{-1}{re}}\right)} - 1\right| \]
      expm1-def [=>] 24.7	\[ 0.5 \cdot \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(im \cdot \sqrt{\frac{-1}{re}}\right)\right)}\right| \]
      expm1-log1p [=>] 20.7	\[ 0.5 \cdot \left|\color{blue}{im \cdot \sqrt{\frac{-1}{re}}}\right| \]

   if -5.2999999999999999e85 < re < -7.60000000000000055e69 or -1.02000000000000004e-19 < re < -1.25e-32 or -5.8e-119 < re

   1. Initial program 31.1

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

   2. Simplified 2.7

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

      [Start] 31.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 31.1	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 2.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 8.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.3 \cdot 10^{+85} \lor \neg \left(re \leq -7.6 \cdot 10^{+69}\right) \land \left(re \leq -1.02 \cdot 10^{-19} \lor \neg \left(re \leq -1.25 \cdot 10^{-32}\right) \land re \leq -5.8 \cdot 10^{-119}\right):\\ \;\;\;\;0.5 \cdot \left|im \cdot \sqrt{\frac{-1}{re}}\right|\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	25.6
Cost	13380

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{if}\;re \leq -2.3 \cdot 10^{-126}:\\ \;\;\;\;0.5 \cdot \left|im \cdot \sqrt{\frac{-1}{re}}\right|\\ \mathbf{elif}\;re \leq 3.5 \cdot 10^{-281}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{-223}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 2 \cdot 10^{-190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2 \cdot 10^{-120}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq 1.18 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 2
Error	25.8
Cost	7509

\[\begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{-132}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 1.25 \cdot 10^{-254}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{elif}\;im \leq 8 \cdot 10^{-200} \lor \neg \left(im \leq 1.5 \cdot 10^{-50}\right) \land im \leq 1.4 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Error	25.5
Cost	7509

\[\begin{array}{l} \mathbf{if}\;im \leq -6.2 \cdot 10^{-136}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{-253}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{elif}\;im \leq 7.4 \cdot 10^{-200} \lor \neg \left(im \leq 2.6 \cdot 10^{-50}\right) \land im \leq 1.4 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Error	26.0
Cost	7444

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{im}{\sqrt{-re}}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.45 \cdot 10^{-133}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 5 \cdot 10^{-254}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{-194}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 3.4 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 5.9 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 5
Error	25.7
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -6.2 \cdot 10^{-137}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-110}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 6
Error	46.2
Cost	6852

\[\begin{array}{l} \mathbf{if}\;im \leq -5 \cdot 10^{-311}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 7
Error	36.2
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 1.26 \cdot 10^{-120}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 8
Error	47.1
Cost	6720

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

Reproduce

herbie shell --seed 2023012 
(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)))))