math.log/2 on complex, real part

Percentage Accurate: 51.3% → 99.4%

Time: 12.5s

Alternatives: 3

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (/
  (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))

C

double code(double re, double im, double base) {
	return ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0d0)) / ((log(base) * log(base)) + (0.0d0 * 0.0d0))
end function

Java

public static double code(double re, double im, double base) {
	return ((Math.log(Math.sqrt(((re * re) + (im * im)))) * Math.log(base)) + (Math.atan2(im, re) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}

Python

def code(re, im, base):
	return ((math.log(math.sqrt(((re * re) + (im * im)))) * math.log(base)) + (math.atan2(im, re) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

Julia

function code(re, im, base)
	return Float64(Float64(Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * log(base)) + Float64(atan(im, re) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(N[ArcTan[im / re], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (/
  (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))

C

double code(double re, double im, double base) {
	return ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0d0)) / ((log(base) * log(base)) + (0.0d0 * 0.0d0))
end function

Java

public static double code(double re, double im, double base) {
	return ((Math.log(Math.sqrt(((re * re) + (im * im)))) * Math.log(base)) + (Math.atan2(im, re) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}

Python

def code(re, im, base):
	return ((math.log(math.sqrt(((re * re) + (im * im)))) * math.log(base)) + (math.atan2(im, re) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

Julia

function code(re, im, base)
	return Float64(Float64(Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * log(base)) + Float64(atan(im, re) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(N[ArcTan[im / re], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log base} \end{array} \]

FPCore

(FPCore (re im base) :precision binary64 (/ (log (hypot re im)) (log base)))

C

double code(double re, double im, double base) {
	return log(hypot(re, im)) / log(base);
}

Java

public static double code(double re, double im, double base) {
	return Math.log(Math.hypot(re, im)) / Math.log(base);
}

Python

def code(re, im, base):
	return math.log(math.hypot(re, im)) / math.log(base)

Julia

function code(re, im, base)
	return Float64(log(hypot(re, im)) / log(base))
end

MATLAB

function tmp = code(re, im, base)
	tmp = log(hypot(re, im)) / log(base);
end

Wolfram

code[re_, im_, base_] := N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.8%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Step-by-step derivation

   1. mul0-rgt 46.8%

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

   2. +-rgt-identity 46.8%

      \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]

   3. metadata-eval 46.8%

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]

   4. +-rgt-identity 46.8%

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{\log base \cdot \log base}} \]

   5. times-frac 46.8%

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base} \cdot \frac{\log base}{\log base}} \]

   6. *-inverses 46.8%

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base} \cdot \color{blue}{1} \]

   7. *-rgt-identity 46.8%

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}} \]

   8. hypot-def 99.5%

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log base} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log base}} \]

4. Final simplification 99.5%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log base} \]

Alternative 2: 44.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{-49}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (if (<= re -3.1e-49) (/ (log (- re)) (log base)) (/ (log im) (log base))))

C

double code(double re, double im, double base) {
	double tmp;
	if (re <= -3.1e-49) {
		tmp = log(-re) / log(base);
	} else {
		tmp = log(im) / log(base);
	}
	return tmp;
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    real(8) :: tmp
    if (re <= (-3.1d-49)) then
        tmp = log(-re) / log(base)
    else
        tmp = log(im) / log(base)
    end if
    code = tmp
end function

Java

public static double code(double re, double im, double base) {
	double tmp;
	if (re <= -3.1e-49) {
		tmp = Math.log(-re) / Math.log(base);
	} else {
		tmp = Math.log(im) / Math.log(base);
	}
	return tmp;
}

Python

def code(re, im, base):
	tmp = 0
	if re <= -3.1e-49:
		tmp = math.log(-re) / math.log(base)
	else:
		tmp = math.log(im) / math.log(base)
	return tmp

Julia

function code(re, im, base)
	tmp = 0.0
	if (re <= -3.1e-49)
		tmp = Float64(log(Float64(-re)) / log(base));
	else
		tmp = Float64(log(im) / log(base));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im, base)
	tmp = 0.0;
	if (re <= -3.1e-49)
		tmp = log(-re) / log(base);
	else
		tmp = log(im) / log(base);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_, base_] := If[LessEqual[re, -3.1e-49], N[(N[Log[(-re)], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision], N[(N[Log[im], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.1e-49

   1. Initial program 33.7%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   2. Step-by-step derivation

      1. mul0-rgt 33.7%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

      2. +-rgt-identity 33.7%

         \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]

      3. metadata-eval 33.7%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]

      4. +-rgt-identity 33.7%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{\log base \cdot \log base}} \]

      5. times-frac 33.9%

         \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base} \cdot \frac{\log base}{\log base}} \]

      6. *-inverses 33.9%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base} \cdot \color{blue}{1} \]

      7. *-rgt-identity 33.9%

         \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}} \]

      8. hypot-def 99.5%

         \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log base} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log base}} \]
   4. Step-by-step derivation

      1. clear-num 99.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{\log base}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}} \]

      2. associate-/r/ 99.4%

         \[\leadsto \color{blue}{\frac{1}{\log base} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)} \]

   5. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{1}{\log base} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)} \]
   6. Step-by-step derivation

      1. associate-/r/ 99.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{\log base}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}} \]

   7. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\log base}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}} \]

   8. Taylor expanded in re around -inf 80.6%

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{\log base}{\log \left(\frac{-1}{re}\right)}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 80.6%

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

      2. neg-mul-1 80.6%

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

   10. Simplified 80.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{-\log base}{\log \left(\frac{-1}{re}\right)}}} \]
   11. Step-by-step derivation

       1. frac-2neg 80.6%

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

       2. metadata-eval 80.6%

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

       3. frac-2neg 80.6%

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

       4. metadata-eval 80.6%

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

       5. distribute-frac-neg 80.6%

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

       6. remove-double-neg 80.6%

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

       7. frac-2neg 80.6%

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

       8. distribute-frac-neg 80.6%

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

       9. remove-double-neg 80.6%

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

       10. associate-/r/ 80.7%

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

       11. neg-log 80.7%

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

       12. clear-num 80.7%

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

       13. div-inv 80.7%

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

       14. metadata-eval 80.7%

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

   12. Applied egg-rr 80.7%

       \[\leadsto \color{blue}{\frac{1}{\log base} \cdot \log \left(re \cdot -1\right)} \]
   13. Step-by-step derivation

       1. associate-*l/ 80.8%

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

       2. *-lft-identity 80.8%

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

       3. *-commutative 80.8%

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

       4. mul-1-neg 80.8%

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

   14. Simplified 80.8%

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

   if -3.1e-49 < re

   1. Initial program 51.3%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   2. Step-by-step derivation

      1. mul0-rgt 51.3%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

      2. +-rgt-identity 51.3%

         \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]

      3. metadata-eval 51.3%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]

      4. +-rgt-identity 51.3%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{\log base \cdot \log base}} \]

      5. times-frac 51.3%

         \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base} \cdot \frac{\log base}{\log base}} \]

      6. *-inverses 51.3%

         \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base} \cdot \color{blue}{1} \]

      7. *-rgt-identity 51.3%

         \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}} \]

      8. hypot-def 99.5%

         \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log base} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log base}} \]

   4. Taylor expanded in re around 0 31.9%

      \[\leadsto \color{blue}{\frac{\log im}{\log base}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{-49}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 3: 27.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{\log im}{\log base} \end{array} \]

FPCore

(FPCore (re im base) :precision binary64 (/ (log im) (log base)))

C

double code(double re, double im, double base) {
	return log(im) / log(base);
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = log(im) / log(base)
end function

Java

public static double code(double re, double im, double base) {
	return Math.log(im) / Math.log(base);
}

Python

def code(re, im, base):
	return math.log(im) / math.log(base)

Julia

function code(re, im, base)
	return Float64(log(im) / log(base))
end

MATLAB

function tmp = code(re, im, base)
	tmp = log(im) / log(base);
end

Wolfram

code[re_, im_, base_] := N[(N[Log[im], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.8%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Step-by-step derivation

   1. mul0-rgt 46.8%

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

   2. +-rgt-identity 46.8%

      \[\leadsto \frac{\color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base + 0 \cdot 0} \]

   3. metadata-eval 46.8%

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base + \color{blue}{0}} \]

   4. +-rgt-identity 46.8%

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\color{blue}{\log base \cdot \log base}} \]

   5. times-frac 46.8%

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base} \cdot \frac{\log base}{\log base}} \]

   6. *-inverses 46.8%

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base} \cdot \color{blue}{1} \]

   7. *-rgt-identity 46.8%

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}} \]

   8. hypot-def 99.5%

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log base} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log base}} \]

4. Taylor expanded in re around 0 27.0%

   \[\leadsto \color{blue}{\frac{\log im}{\log base}} \]

5. Final simplification 27.0%

   \[\leadsto \frac{\log im}{\log base} \]

Reproduce

herbie shell --seed 2023182 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))