?

Average Error: 32.3 → 7.6
Time: 2.0s
Precision: binary64
Cost: 20040

?

\[ \begin{array}{c}[re, im] = \mathsf{sort}([re, im])\\ \end{array} \]
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.45 \cdot 10^{+110}:\\ \;\;\;\;\frac{\log \left(re \cdot -1\right)}{\log 10}\\ \mathbf{elif}\;re \leq -3.5 \cdot 10^{-154}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]
(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -1.45e+110)
   (/ (log (* re -1.0)) (log 10.0))
   (if (<= re -3.5e-154)
     (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0))
     (/ (log im) (log 10.0)))))
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 <= -1.45e+110) {
		tmp = log((re * -1.0)) / log(10.0);
	} else if (re <= -3.5e-154) {
		tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
	} else {
		tmp = log(im) / log(10.0);
	}
	return tmp;
}

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 <= (-1.45d+110)) then
        tmp = log((re * (-1.0d0))) / log(10.0d0)
    else if (re <= (-3.5d-154)) then
        tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
    else
        tmp = log(im) / log(10.0d0)
    end if
    code = tmp
end function

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 <= -1.45e+110) {
		tmp = Math.log((re * -1.0)) / Math.log(10.0);
	} else if (re <= -3.5e-154) {
		tmp = Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
	} else {
		tmp = Math.log(im) / Math.log(10.0);
	}
	return tmp;
}

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 <= -1.45e+110:
		tmp = math.log((re * -1.0)) / math.log(10.0)
	elif re <= -3.5e-154:
		tmp = math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)
	else:
		tmp = math.log(im) / math.log(10.0)
	return tmp

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 <= -1.45e+110)
		tmp = Float64(log(Float64(re * -1.0)) / log(10.0));
	elseif (re <= -3.5e-154)
		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

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 <= -1.45e+110)
		tmp = log((re * -1.0)) / log(10.0);
	elseif (re <= -3.5e-154)
		tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
	else
		tmp = log(im) / log(10.0);
	end
	tmp_2 = tmp;
end

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, -1.45e+110], N[(N[Log[N[(re * -1.0), $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -3.5e-154], 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]]]

\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.45 \cdot 10^{+110}:\\
\;\;\;\;\frac{\log \left(re \cdot -1\right)}{\log 10}\\

\mathbf{elif}\;re \leq -3.5 \cdot 10^{-154}:\\
\;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log im}{\log 10}\\


\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Split input into 3 regimes

  2. if re < -1.45e110

    1. Initial program 53.8

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

    2. Taylor expanded in re around -inf 5.5

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

    3. Simplified5.5

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

      [Start]5.5

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

      rational.json-simplify-2 [=>]5.5

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

    if -1.45e110 < re < -3.5000000000000001e-154

    1. Initial program 12.0

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

    if -3.5000000000000001e-154 < re

    1. Initial program 33.1

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

    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 simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.45 \cdot 10^{+110}:\\ \;\;\;\;\frac{\log \left(re \cdot -1\right)}{\log 10}\\ \mathbf{elif}\;re \leq -3.5 \cdot 10^{-154}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternatives

Alternative 1
Error10.9
Cost13252
\[\begin{array}{l} \mathbf{if}\;re \leq -6 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(re \cdot -1\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]
Alternative 2
Error30.6
Cost12992
\[\frac{\log im}{\log 10} \]

Error

Reproduce?

herbie shell --seed 2023053 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))