math.log/1 on complex, real part

Average Error: 32.0 → 7.4

Time: 2.5s

Precision: binary64

Cost: 13512

\[ \begin{array}{c}[re, im] = \mathsf{sort}([re, im])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \leq 1.05 \cdot 10^{-165}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+98}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (log (sqrt (+ (* re re) (* im im)))))

↓

(FPCore (re im)
 :precision binary64
 (if (<= im 1.05e-165)
   (log (- re))
   (if (<= im 4.3e+98) (log (sqrt (+ (* re re) (* im im)))) (log im))))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im))));
}

↓

double code(double re, double im) {
	double tmp;
	if (im <= 1.05e-165) {
		tmp = log(-re);
	} else if (im <= 4.3e+98) {
		tmp = log(sqrt(((re * re) + (im * im))));
	} else {
		tmp = log(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im))))
end function

↓

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.05d-165) then
        tmp = log(-re)
    else if (im <= 4.3d+98) then
        tmp = log(sqrt(((re * re) + (im * im))))
    else
        tmp = log(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im))));
}

↓

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.05e-165) {
		tmp = Math.log(-re);
	} else if (im <= 4.3e+98) {
		tmp = Math.log(Math.sqrt(((re * re) + (im * im))));
	} else {
		tmp = Math.log(im);
	}
	return tmp;
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im))))

↓

def code(re, im):
	tmp = 0
	if im <= 1.05e-165:
		tmp = math.log(-re)
	elif im <= 4.3e+98:
		tmp = math.log(math.sqrt(((re * re) + (im * im))))
	else:
		tmp = math.log(im)
	return tmp

Julia

function code(re, im)
	return log(sqrt(Float64(Float64(re * re) + Float64(im * im))))
end

↓

function code(re, im)
	tmp = 0.0
	if (im <= 1.05e-165)
		tmp = log(Float64(-re));
	elseif (im <= 4.3e+98)
		tmp = log(sqrt(Float64(Float64(re * re) + Float64(im * im))));
	else
		tmp = log(im);
	end
	return tmp
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im))));
end

↓

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.05e-165)
		tmp = log(-re);
	elseif (im <= 4.3e+98)
		tmp = log(sqrt(((re * re) + (im * im))));
	else
		tmp = log(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

↓

code[re_, im_] := If[LessEqual[im, 1.05e-165], N[Log[(-re)], $MachinePrecision], If[LessEqual[im, 4.3e+98], N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Log[im], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.04999999999999997e-165

   1. Initial program 32.8

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

   2. Taylor expanded in re around -inf 4.9

      \[\leadsto \log \color{blue}{\left(-1 \cdot re\right)} \]

   3. Simplified 4.9

      \[\leadsto \log \color{blue}{\left(-re\right)} \]
      Proof

      [Start] 4.9	\[ \log \left(-1 \cdot re\right) \]
      rational.json-simplify-2 [=>] 4.9	\[ \log \color{blue}{\left(re \cdot -1\right)} \]
      rational.json-simplify-8 [<=] 4.9	\[ \log \color{blue}{\left(-re\right)} \]

   if 1.04999999999999997e-165 < im < 4.3000000000000001e98

   1. Initial program 12.5

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

   if 4.3000000000000001e98 < im

   1. Initial program 49.9

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

   2. Taylor expanded in re around 0 5.4

      \[\leadsto \log \color{blue}{im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.05 \cdot 10^{-165}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+98}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

Alternatives

Alternative 1
Error	10.0
Cost	6660

\[\begin{array}{l} \mathbf{if}\;im \leq 7.5 \cdot 10^{-97}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

Alternative 2
Error	30.8
Cost	6464

\[\log im \]

Reproduce

herbie shell --seed 2023077 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))