math.log/1 on complex, real part

Average Error: 31.2 → 6.9

Time: 2.3s

Precision: binary64

Cost: 13512

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \leq 1.7 \cdot 10^{-175}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+84}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= im 1.7e-175)
   (log (- re))
   (if (<= im 4.5e+84) (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 <= 1.7e-175) {
		tmp = log(-re);
	} else if (im <= 4.5e+84) {
		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 <= 1.7d-175) then
        tmp = log(-re)
    else if (im <= 4.5d+84) 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 <= 1.7e-175) {
		tmp = Math.log(-re);
	} else if (im <= 4.5e+84) {
		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 <= 1.7e-175:
		tmp = math.log(-re)
	elif im <= 4.5e+84:
		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 <= 1.7e-175)
		tmp = log(Float64(-re));
	elseif (im <= 4.5e+84)
		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 <= 1.7e-175)
		tmp = log(-re);
	elseif (im <= 4.5e+84)
		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, 1.7e-175], N[Log[(-re)], $MachinePrecision], If[LessEqual[im, 4.5e+84], 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 < 1.7e-175

   1. Initial program 31.7

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

   2. Taylor expanded in re around -inf 3.9

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

   3. Simplified 3.9

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

      [Start] 3.9	\[ \log \left(-1 \cdot re\right) \]
      rational.json-simplify-2 [=>] 3.9	\[ \log \color{blue}{\left(re \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 3.9	\[ \log \color{blue}{\left(-re\right)} \]

   if 1.7e-175 < im < 4.4999999999999997e84

   1. Initial program 12.6

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

   if 4.4999999999999997e84 < im

   1. Initial program 47.7

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

   2. Taylor expanded in re around 0 4.7

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

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.7 \cdot 10^{-175}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+84}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

Alternatives

Alternative 1
Error	11.6
Cost	6924

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ \mathbf{if}\;re \leq -1.2 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.45 \cdot 10^{+27}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq -3.6 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

Alternative 2
Error	29.6
Cost	6464

\[\log im \]

Reproduce

herbie shell --seed 2023065 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))