math.log/1 on complex, real part

Average Error: 32.1 → 7.1

Time: 2.3s

Precision: binary64

Cost: 13512

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -1.16 \cdot 10^{+48}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -4.1 \cdot 10^{-156}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -1.16e+48)
   (log (- re))
   (if (<= re -4.1e-156) (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 (re <= -1.16e+48) {
		tmp = log(-re);
	} else if (re <= -4.1e-156) {
		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 (re <= (-1.16d+48)) then
        tmp = log(-re)
    else if (re <= (-4.1d-156)) 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 (re <= -1.16e+48) {
		tmp = Math.log(-re);
	} else if (re <= -4.1e-156) {
		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 re <= -1.16e+48:
		tmp = math.log(-re)
	elif re <= -4.1e-156:
		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 (re <= -1.16e+48)
		tmp = log(Float64(-re));
	elseif (re <= -4.1e-156)
		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 (re <= -1.16e+48)
		tmp = log(-re);
	elseif (re <= -4.1e-156)
		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[re, -1.16e+48], N[Log[(-re)], $MachinePrecision], If[LessEqual[re, -4.1e-156], 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 re < -1.15999999999999992e48

   1. Initial program 44.3

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

   2. Taylor expanded in re around -inf 6.3

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

   3. Simplified 6.3

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

      [Start] 6.3	\[ \log \left(-1 \cdot re\right) \]
      rational_best.json-simplify-2 [=>] 6.3	\[ \log \color{blue}{\left(re \cdot -1\right)} \]
      rational_best.json-simplify-12 [=>] 6.3	\[ \log \color{blue}{\left(-re\right)} \]

   if -1.15999999999999992e48 < re < -4.1000000000000002e-156

   1. Initial program 12.0

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

   if -4.1000000000000002e-156 < re

   1. Initial program 32.4

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

   2. Taylor expanded in re around 0 4.9

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

4. Final simplification 7.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.16 \cdot 10^{+48}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -4.1 \cdot 10^{-156}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

Alternatives

Alternative 1
Error	10.3
Cost	6924

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ \mathbf{if}\;re \leq -5.7 \cdot 10^{-70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.1 \cdot 10^{-85}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq -8 \cdot 10^{-155}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

Alternative 2
Error	30.2
Cost	6464

\[\log im \]

Reproduce

herbie shell --seed 2023090 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))