math.log/1 on complex, real part

Average Error: 31.8 → 6.7

Time: 8.5s

Precision: binary64

Cost: 13512

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -4.4 \cdot 10^{+77}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -1.02 \cdot 10^{-157}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -4.4e+77)
   (log (- re))
   (if (<= re -1.02e-157) (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 <= -4.4e+77) {
		tmp = log(-re);
	} else if (re <= -1.02e-157) {
		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 <= (-4.4d+77)) then
        tmp = log(-re)
    else if (re <= (-1.02d-157)) 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 <= -4.4e+77) {
		tmp = Math.log(-re);
	} else if (re <= -1.02e-157) {
		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 <= -4.4e+77:
		tmp = math.log(-re)
	elif re <= -1.02e-157:
		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 <= -4.4e+77)
		tmp = log(Float64(-re));
	elseif (re <= -1.02e-157)
		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 <= -4.4e+77)
		tmp = log(-re);
	elseif (re <= -1.02e-157)
		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, -4.4e+77], N[Log[(-re)], $MachinePrecision], If[LessEqual[re, -1.02e-157], 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 < -4.4000000000000001e77

   1. Initial program 47.4

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

   2. Taylor expanded in re around -inf 5.7

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

   3. Simplified 5.7

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

      [Start] 5.7	\[ \log \left(-1 \cdot re\right) \]
      rational_best-simplify-1 [=>] 5.7	\[ \log \color{blue}{\left(re \cdot -1\right)} \]
      rational_best-simplify-11 [<=] 5.7	\[ \log \color{blue}{\left(-re\right)} \]

   if -4.4000000000000001e77 < re < -1.0200000000000001e-157

   1. Initial program 10.9

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

   if -1.0200000000000001e-157 < re

   1. Initial program 32.7

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

   2. Taylor expanded in re around 0 4.8

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

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.4 \cdot 10^{+77}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -1.02 \cdot 10^{-157}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

Alternatives

Alternative 1
Error	10.8
Cost	6924

\[\begin{array}{l} t_0 := \log \left(-re\right)\\ \mathbf{if}\;re \leq -5.3 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -3.9 \cdot 10^{-116}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq -7.1 \cdot 10^{-177}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array} \]

Alternative 2
Error	30.5
Cost	6464

\[\log im \]

Reproduce

herbie shell --seed 2023099 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))