math.log/2 on complex, real part

Percentage Accurate: 51.6% → 99.4%

Time: 9.9s

Alternatives: 3

Speedup: 2.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 51.6% 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 56.0%

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

   1. mul0-rgt 56.0%

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

   2. +-rgt-identity 56.0%

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

   3. metadata-eval 56.0%

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

   4. +-rgt-identity 56.0%

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

   5. times-frac 56.1%

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

   6. *-inverses 56.1%

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

   7. *-rgt-identity 56.1%

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

   8. hypot-def 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 2: 42.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.05 \cdot 10^{-36} \lor \neg \left(re \leq -3.9 \cdot 10^{-84}\right) \land re \leq -3.9 \cdot 10^{-178}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (if (or (<= re -1.05e-36) (and (not (<= re -3.9e-84)) (<= re -3.9e-178)))
   (/ (log (- re)) (log base))
   (/ (log im) (log base))))

C

double code(double re, double im, double base) {
	double tmp;
	if ((re <= -1.05e-36) || (!(re <= -3.9e-84) && (re <= -3.9e-178))) {
		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 <= (-1.05d-36)) .or. (.not. (re <= (-3.9d-84))) .and. (re <= (-3.9d-178))) 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 <= -1.05e-36) || (!(re <= -3.9e-84) && (re <= -3.9e-178))) {
		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 <= -1.05e-36) or (not (re <= -3.9e-84) and (re <= -3.9e-178)):
		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 <= -1.05e-36) || (!(re <= -3.9e-84) && (re <= -3.9e-178)))
		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 <= -1.05e-36) || (~((re <= -3.9e-84)) && (re <= -3.9e-178)))
		tmp = log(-re) / log(base);
	else
		tmp = log(im) / log(base);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_, base_] := If[Or[LessEqual[re, -1.05e-36], And[N[Not[LessEqual[re, -3.9e-84]], $MachinePrecision], LessEqual[re, -3.9e-178]]], 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 < -1.04999999999999995e-36 or -3.90000000000000023e-84 < re < -3.90000000000000025e-178

   1. Initial program 51.0%

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

      1. mul0-rgt 51.0%

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

      2. +-rgt-identity 51.0%

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

      3. metadata-eval 51.0%

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

      4. +-rgt-identity 51.0%

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

      5. times-frac 51.0%

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

      6. *-inverses 51.0%

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

      7. *-rgt-identity 51.0%

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

      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. pow1 99.5%

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

      2. metadata-eval 99.5%

         \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{{\log base}^{\color{blue}{\left(2 - 1\right)}}} \]

      3. pow-div 99.5%

         \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{\frac{{\log base}^{2}}{{\log base}^{1}}}} \]

      4. pow2 99.5%

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

      5. pow1 99.5%

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

      6. associate-/r/ 99.2%

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

      7. pow2 99.2%

         \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{{\log base}^{2}}} \cdot \log base \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{{\log base}^{2}} \cdot \log base} \]

   6. Taylor expanded in re around -inf 69.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\log \left(\frac{-1}{re}\right)}{{\log base}^{2}}\right)} \cdot \log base \]
   7. Step-by-step derivation

      1. associate-*r/ 69.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot \log \left(\frac{-1}{re}\right)}{{\log base}^{2}}} \cdot \log base \]

      2. mul-1-neg 69.3%

         \[\leadsto \frac{\color{blue}{-\log \left(\frac{-1}{re}\right)}}{{\log base}^{2}} \cdot \log base \]

   8. Simplified 69.3%

      \[\leadsto \color{blue}{\frac{-\log \left(\frac{-1}{re}\right)}{{\log base}^{2}}} \cdot \log base \]
   9. Step-by-step derivation

      1. expm1-log1p-u 69.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\log \left(\frac{-1}{re}\right)}{{\log base}^{2}}\right)\right)} \cdot \log base \]

      2. expm1-udef 62.6%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-\log \left(\frac{-1}{re}\right)}{{\log base}^{2}}\right)} - 1\right)} \cdot \log base \]

      3. div-inv 62.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(-\log \left(\frac{-1}{re}\right)\right) \cdot \frac{1}{{\log base}^{2}}}\right)} - 1\right) \cdot \log base \]

      4. neg-log 62.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\log \left(\frac{1}{\frac{-1}{re}}\right)} \cdot \frac{1}{{\log base}^{2}}\right)} - 1\right) \cdot \log base \]

      5. frac-2neg 62.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\log \left(\frac{1}{\color{blue}{\frac{--1}{-re}}}\right) \cdot \frac{1}{{\log base}^{2}}\right)} - 1\right) \cdot \log base \]

      6. metadata-eval 62.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\log \left(\frac{1}{\frac{\color{blue}{1}}{-re}}\right) \cdot \frac{1}{{\log base}^{2}}\right)} - 1\right) \cdot \log base \]

      7. remove-double-div 62.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\log \color{blue}{\left(-re\right)} \cdot \frac{1}{{\log base}^{2}}\right)} - 1\right) \cdot \log base \]

      8. pow-flip 62.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\log \left(-re\right) \cdot \color{blue}{{\log base}^{\left(-2\right)}}\right)} - 1\right) \cdot \log base \]

      9. metadata-eval 62.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\log \left(-re\right) \cdot {\log base}^{\color{blue}{-2}}\right)} - 1\right) \cdot \log base \]

   10. Applied egg-rr 62.6%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(-re\right) \cdot {\log base}^{-2}\right)} - 1\right)} \cdot \log base \]
   11. Step-by-step derivation

       1. expm1-def 69.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(-re\right) \cdot {\log base}^{-2}\right)\right)} \cdot \log base \]

       2. expm1-log1p 69.4%

          \[\leadsto \color{blue}{\left(\log \left(-re\right) \cdot {\log base}^{-2}\right)} \cdot \log base \]

       3. *-commutative 69.4%

          \[\leadsto \color{blue}{\left({\log base}^{-2} \cdot \log \left(-re\right)\right)} \cdot \log base \]

   12. Simplified 69.4%

       \[\leadsto \color{blue}{\left({\log base}^{-2} \cdot \log \left(-re\right)\right)} \cdot \log base \]

   13. Taylor expanded in base around 0 69.5%

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

   if -1.04999999999999995e-36 < re < -3.90000000000000023e-84 or -3.90000000000000025e-178 < re

   1. Initial program 57.8%

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

      1. mul0-rgt 57.8%

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

      2. +-rgt-identity 57.8%

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

      3. metadata-eval 57.8%

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

      4. +-rgt-identity 57.8%

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

      5. times-frac 57.9%

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

      6. *-inverses 57.9%

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

      7. *-rgt-identity 57.9%

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

      8. hypot-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in re around 0 38.0%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.05 \cdot 10^{-36} \lor \neg \left(re \leq -3.9 \cdot 10^{-84}\right) \land re \leq -3.9 \cdot 10^{-178}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 3: 26.3% 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 56.0%

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

   1. mul0-rgt 56.0%

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

   2. +-rgt-identity 56.0%

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

   3. metadata-eval 56.0%

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

   4. +-rgt-identity 56.0%

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

   5. times-frac 56.1%

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

   6. *-inverses 56.1%

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

   7. *-rgt-identity 56.1%

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

   8. hypot-def 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in re around 0 32.0%

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

5. Final simplification 32.0%

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

Reproduce

herbie shell --seed 2023178 
(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))))