math.log10 on complex, real part

Average Error: 32.4 → 7.6

Time: 4.1s

Precision: binary64

Cost: 20040

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

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]

↓

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

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -1.16e+48)
   (/ (log (- re)) (log 10.0))
   (if (<= re -4.1e-156)
     (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0))
     (/ (log im) (log 10.0)))))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

↓

double code(double re, double im) {
	double tmp;
	if (re <= -1.16e+48) {
		tmp = log(-re) / log(10.0);
	} else if (re <= -4.1e-156) {
		tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
	} else {
		tmp = log(im) / log(10.0);
	}
	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)))) / log(10.0d0)
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) / log(10.0d0)
    else if (re <= (-4.1d-156)) then
        tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
    else
        tmp = log(im) / log(10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
}

↓

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.16e+48) {
		tmp = Math.log(-re) / Math.log(10.0);
	} else if (re <= -4.1e-156) {
		tmp = Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
	} else {
		tmp = Math.log(im) / Math.log(10.0);
	}
	return tmp;
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

↓

def code(re, im):
	tmp = 0
	if re <= -1.16e+48:
		tmp = math.log(-re) / math.log(10.0)
	elif re <= -4.1e-156:
		tmp = math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)
	else:
		tmp = math.log(im) / math.log(10.0)
	return tmp

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

↓

function code(re, im)
	tmp = 0.0
	if (re <= -1.16e+48)
		tmp = Float64(log(Float64(-re)) / log(10.0));
	elseif (re <= -4.1e-156)
		tmp = Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0));
	else
		tmp = Float64(log(im) / log(10.0));
	end
	return tmp
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

↓

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.16e+48)
		tmp = log(-re) / log(10.0);
	elseif (re <= -4.1e-156)
		tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
	else
		tmp = log(im) / log(10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

↓

code[re_, im_] := If[LessEqual[re, -1.16e+48], N[(N[Log[(-re)], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -4.1e-156], N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision], N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.15999999999999992e48

   1. Initial program 44.5

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]

   2. Taylor expanded in re around -inf 6.9

      \[\leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)}}{\log 10} \]

   3. Simplified 6.9

      \[\leadsto \frac{\log \color{blue}{\left(-re\right)}}{\log 10} \]
      Proof

      [Start] 6.9	\[ \frac{\log \left(-1 \cdot re\right)}{\log 10} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 6.9	\[ \frac{\log \color{blue}{\left(re \cdot -1\right)}}{\log 10} \]
      rational_best_oopsla_all_46_json_45_simplify-94 [<=] 6.9	\[ \frac{\log \color{blue}{\left(-re\right)}}{\log 10} \]

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

   1. Initial program 12.5

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]

   if -4.1000000000000002e-156 < re

   1. Initial program 32.7

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]

   2. Taylor expanded in re around 0 5.4

      \[\leadsto \frac{\log \color{blue}{im}}{\log 10} \]
3. Recombined 3 regimes into one program.

4. Final simplification 7.6

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

Alternatives

Alternative 1
Error	10.5
Cost	13188

\[\begin{array}{l} \mathbf{if}\;re \leq -5.7 \cdot 10^{-70}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 2
Error	30.5
Cost	12992

\[\frac{\log im}{\log 10} \]

Reproduce

herbie shell --seed 2023090 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))