math.log/2 on complex, real part

Average Error: 31.9 → 17.6

Time: 19.7s

Precision: binary64

Cost: 33484

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(\sqrt{{re}^{2} + {im}^{2}}\right)\\ \mathbf{if}\;im \leq -1.4 \cdot 10^{+92}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log base}\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{-291}:\\ \;\;\;\;\frac{0.5}{\left(\log base \cdot 0.5\right) \cdot \frac{1}{t_0}}\\ \mathbf{elif}\;im \leq 10^{-224}:\\ \;\;\;\;\frac{\frac{1}{\log base}}{\log base} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \left(-\log base\right) + \tan^{-1}_* \frac{im}{re} \cdot 0\right)\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+55}:\\ \;\;\;\;\frac{t_0}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

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))))

↓

(FPCore (re im base)
 :precision binary64
 (let* ((t_0 (log (sqrt (+ (pow re 2.0) (pow im 2.0))))))
   (if (<= im -1.4e+92)
     (/ (log (- im)) (log base))
     (if (<= im 2.5e-291)
       (/ 0.5 (* (* (log base) 0.5) (/ 1.0 t_0)))
       (if (<= im 1e-224)
         (*
          (/ (/ 1.0 (log base)) (log base))
          (+ (* (log (/ -1.0 re)) (- (log base))) (* (atan2 im re) 0.0)))
         (if (<= im 4.8e+55) (/ t_0 (log base)) (/ (log im) (log base))))))))

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));
}

↓

double code(double re, double im, double base) {
	double t_0 = log(sqrt((pow(re, 2.0) + pow(im, 2.0))));
	double tmp;
	if (im <= -1.4e+92) {
		tmp = log(-im) / log(base);
	} else if (im <= 2.5e-291) {
		tmp = 0.5 / ((log(base) * 0.5) * (1.0 / t_0));
	} else if (im <= 1e-224) {
		tmp = ((1.0 / log(base)) / log(base)) * ((log((-1.0 / re)) * -log(base)) + (atan2(im, re) * 0.0));
	} else if (im <= 4.8e+55) {
		tmp = t_0 / 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
    code = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0d0)) / ((log(base) * log(base)) + (0.0d0 * 0.0d0))
end function

↓

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(sqrt(((re ** 2.0d0) + (im ** 2.0d0))))
    if (im <= (-1.4d+92)) then
        tmp = log(-im) / log(base)
    else if (im <= 2.5d-291) then
        tmp = 0.5d0 / ((log(base) * 0.5d0) * (1.0d0 / t_0))
    else if (im <= 1d-224) then
        tmp = ((1.0d0 / log(base)) / log(base)) * ((log(((-1.0d0) / re)) * -log(base)) + (atan2(im, re) * 0.0d0))
    else if (im <= 4.8d+55) then
        tmp = t_0 / 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) {
	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));
}

↓

public static double code(double re, double im, double base) {
	double t_0 = Math.log(Math.sqrt((Math.pow(re, 2.0) + Math.pow(im, 2.0))));
	double tmp;
	if (im <= -1.4e+92) {
		tmp = Math.log(-im) / Math.log(base);
	} else if (im <= 2.5e-291) {
		tmp = 0.5 / ((Math.log(base) * 0.5) * (1.0 / t_0));
	} else if (im <= 1e-224) {
		tmp = ((1.0 / Math.log(base)) / Math.log(base)) * ((Math.log((-1.0 / re)) * -Math.log(base)) + (Math.atan2(im, re) * 0.0));
	} else if (im <= 4.8e+55) {
		tmp = t_0 / Math.log(base);
	} else {
		tmp = Math.log(im) / Math.log(base);
	}
	return tmp;
}

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))

↓

def code(re, im, base):
	t_0 = math.log(math.sqrt((math.pow(re, 2.0) + math.pow(im, 2.0))))
	tmp = 0
	if im <= -1.4e+92:
		tmp = math.log(-im) / math.log(base)
	elif im <= 2.5e-291:
		tmp = 0.5 / ((math.log(base) * 0.5) * (1.0 / t_0))
	elif im <= 1e-224:
		tmp = ((1.0 / math.log(base)) / math.log(base)) * ((math.log((-1.0 / re)) * -math.log(base)) + (math.atan2(im, re) * 0.0))
	elif im <= 4.8e+55:
		tmp = t_0 / math.log(base)
	else:
		tmp = math.log(im) / math.log(base)
	return tmp

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

↓

function code(re, im, base)
	t_0 = log(sqrt(Float64((re ^ 2.0) + (im ^ 2.0))))
	tmp = 0.0
	if (im <= -1.4e+92)
		tmp = Float64(log(Float64(-im)) / log(base));
	elseif (im <= 2.5e-291)
		tmp = Float64(0.5 / Float64(Float64(log(base) * 0.5) * Float64(1.0 / t_0)));
	elseif (im <= 1e-224)
		tmp = Float64(Float64(Float64(1.0 / log(base)) / log(base)) * Float64(Float64(log(Float64(-1.0 / re)) * Float64(-log(base))) + Float64(atan(im, re) * 0.0)));
	elseif (im <= 4.8e+55)
		tmp = Float64(t_0 / log(base));
	else
		tmp = Float64(log(im) / log(base));
	end
	return tmp
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

↓

function tmp_2 = code(re, im, base)
	t_0 = log(sqrt(((re ^ 2.0) + (im ^ 2.0))));
	tmp = 0.0;
	if (im <= -1.4e+92)
		tmp = log(-im) / log(base);
	elseif (im <= 2.5e-291)
		tmp = 0.5 / ((log(base) * 0.5) * (1.0 / t_0));
	elseif (im <= 1e-224)
		tmp = ((1.0 / log(base)) / log(base)) * ((log((-1.0 / re)) * -log(base)) + (atan2(im, re) * 0.0));
	elseif (im <= 4.8e+55)
		tmp = t_0 / log(base);
	else
		tmp = log(im) / log(base);
	end
	tmp_2 = tmp;
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]

↓

code[re_, im_, base_] := Block[{t$95$0 = N[Log[N[Sqrt[N[(N[Power[re, 2.0], $MachinePrecision] + N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[im, -1.4e+92], N[(N[Log[(-im)], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.5e-291], N[(0.5 / N[(N[(N[Log[base], $MachinePrecision] * 0.5), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e-224], N[(N[(N[(1.0 / N[Log[base], $MachinePrecision]), $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision] * (-N[Log[base], $MachinePrecision])), $MachinePrecision] + N[(N[ArcTan[im / re], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.8e+55], N[(t$95$0 / N[Log[base], $MachinePrecision]), $MachinePrecision], N[(N[Log[im], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -1.4e92

   1. Initial program 50.2

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

   2. Simplified 50.2

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base}} \]
      Proof

      [Start] 50.2	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
      metadata-eval [=>] 50.2	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + \color{blue}{0}} \]
      rational.json-simplify-65 [=>] 50.2	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\color{blue}{\log base \cdot \log base}} \]
      rational.json-simplify-41 [=>] 50.2	\[ \frac{\color{blue}{\tan^{-1}_* \frac{im}{re} \cdot 0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base} \]

   3. Taylor expanded in base around 0 50.2

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^{2} + {im}^{2}}\right)}{\log base}} \]

   4. Taylor expanded in im around -inf 8.7

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

   5. Simplified 8.7

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

      [Start] 8.7	\[ \frac{\log \left(-1 \cdot im\right)}{\log base} \]
      rational.json-simplify-39 [=>] 8.7	\[ \frac{\log \color{blue}{\left(im \cdot -1\right)}}{\log base} \]
      rational.json-simplify-72 [=>] 8.7	\[ \frac{\log \color{blue}{\left(-im\right)}}{\log base} \]

   if -1.4e92 < im < 2.5000000000000002e-291

   1. Initial program 21.9

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

   2. Simplified 21.9

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base}} \]
      Proof

      [Start] 21.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
      metadata-eval [=>] 21.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + \color{blue}{0}} \]
      rational.json-simplify-65 [=>] 21.9	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\color{blue}{\log base \cdot \log base}} \]
      rational.json-simplify-41 [=>] 21.9	\[ \frac{\color{blue}{\tan^{-1}_* \frac{im}{re} \cdot 0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base} \]

   3. Taylor expanded in base around 0 21.9

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^{2} + {im}^{2}}\right)}{\log base}} \]

   4. Applied egg-rr 21.9

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

   5. Applied egg-rr 22.0

      \[\leadsto \color{blue}{\frac{0.5}{\left(\log base \cdot 0.5\right) \cdot \frac{1}{\log \left(\sqrt{{re}^{2} + {im}^{2}}\right)}}} \]

   if 2.5000000000000002e-291 < im < 1e-224

   1. Initial program 30.6

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

   2. Simplified 30.6

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base}} \]
      Proof

      [Start] 30.6	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
      metadata-eval [=>] 30.6	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + \color{blue}{0}} \]
      rational.json-simplify-65 [=>] 30.6	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\color{blue}{\log base \cdot \log base}} \]
      rational.json-simplify-41 [=>] 30.6	\[ \frac{\color{blue}{\tan^{-1}_* \frac{im}{re} \cdot 0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base} \]

   3. Applied egg-rr 30.7

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot 0 + \color{blue}{\frac{1}{\frac{\frac{1}{\log base}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}}{\log base \cdot \log base} \]

   4. Taylor expanded in re around -inf 34.5

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot 0 + \frac{1}{\color{blue}{\frac{-1}{\log base \cdot \log \left(\frac{-1}{re}\right)}}}}{\log base \cdot \log base} \]

   5. Simplified 34.5

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot 0 + \frac{1}{\color{blue}{\frac{\frac{-1}{\log \left(\frac{-1}{re}\right)}}{\log base}}}}{\log base \cdot \log base} \]
      Proof

      [Start] 34.5	\[ \frac{\tan^{-1}_* \frac{im}{re} \cdot 0 + \frac{1}{\frac{-1}{\log base \cdot \log \left(\frac{-1}{re}\right)}}}{\log base \cdot \log base} \]
      rational.json-simplify-15 [=>] 34.6	\[ \frac{\tan^{-1}_* \frac{im}{re} \cdot 0 + \frac{1}{\color{blue}{\frac{\frac{-1}{\log base}}{\log \left(\frac{-1}{re}\right)}}}}{\log base \cdot \log base} \]
      rational.json-simplify-7 [=>] 34.5	\[ \frac{\tan^{-1}_* \frac{im}{re} \cdot 0 + \frac{1}{\color{blue}{\frac{\frac{-1}{\log \left(\frac{-1}{re}\right)}}{\log base}}}}{\log base \cdot \log base} \]

   6. Applied egg-rr 34.5

      \[\leadsto \color{blue}{\frac{\frac{1}{\log base}}{\log base} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \left(-\log base\right) + \tan^{-1}_* \frac{im}{re} \cdot 0\right)} \]

   if 1e-224 < im < 4.7999999999999998e55

   1. Initial program 20.1

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

   2. Simplified 20.1

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base}} \]
      Proof

      [Start] 20.1	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
      metadata-eval [=>] 20.1	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + \color{blue}{0}} \]
      rational.json-simplify-65 [=>] 20.1	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\color{blue}{\log base \cdot \log base}} \]
      rational.json-simplify-41 [=>] 20.1	\[ \frac{\color{blue}{\tan^{-1}_* \frac{im}{re} \cdot 0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base} \]

   3. Taylor expanded in base around 0 20.1

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^{2} + {im}^{2}}\right)}{\log base}} \]

   if 4.7999999999999998e55 < im

   1. Initial program 45.3

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

   2. Simplified 45.3

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base}} \]
      Proof

      [Start] 45.3	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
      metadata-eval [=>] 45.3	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + \color{blue}{0}} \]
      rational.json-simplify-65 [=>] 45.3	\[ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\color{blue}{\log base \cdot \log base}} \]
      rational.json-simplify-41 [=>] 45.3	\[ \frac{\color{blue}{\tan^{-1}_* \frac{im}{re} \cdot 0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}}{\log base \cdot \log base} \]

   3. Taylor expanded in re around 0 11.0

      \[\leadsto \color{blue}{\frac{\log im}{\log base}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.4 \cdot 10^{+92}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log base}\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{-291}:\\ \;\;\;\;\frac{0.5}{\left(\log base \cdot 0.5\right) \cdot \frac{1}{\log \left(\sqrt{{re}^{2} + {im}^{2}}\right)}}\\ \mathbf{elif}\;im \leq 10^{-224}:\\ \;\;\;\;\frac{\frac{1}{\log base}}{\log base} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \left(-\log base\right) + \tan^{-1}_* \frac{im}{re} \cdot 0\right)\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+55}:\\ \;\;\;\;\frac{\log \left(\sqrt{{re}^{2} + {im}^{2}}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.6
Cost	33096

\[\begin{array}{l} t_0 := \log \left(\sqrt{{re}^{2} + {im}^{2}}\right)\\ \mathbf{if}\;im \leq -1.1 \cdot 10^{+91}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log base}\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{0.5}{\left(\log base \cdot 0.5\right) \cdot \frac{1}{t_0}}\\ \mathbf{elif}\;im \leq 7.6 \cdot 10^{-221}:\\ \;\;\;\;\frac{1}{\log base} \cdot \log \left(-re\right)\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+54}:\\ \;\;\;\;\frac{t_0}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 2
Error	17.5
Cost	32976

\[\begin{array}{l} t_0 := \frac{\log \left(\sqrt{{re}^{2} + {im}^{2}}\right)}{\log base}\\ \mathbf{if}\;im \leq -2.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log base}\\ \mathbf{elif}\;im \leq 7 \cdot 10^{-304}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{-224}:\\ \;\;\;\;\frac{1}{\log base} \cdot \log \left(-re\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 3
Error	17.6
Cost	32976

\[\begin{array}{l} t_0 := \frac{1}{\log base}\\ t_1 := \log \left(\sqrt{{re}^{2} + {im}^{2}}\right)\\ \mathbf{if}\;im \leq -1 \cdot 10^{+80}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log base}\\ \mathbf{elif}\;im \leq 2 \cdot 10^{-304}:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{elif}\;im \leq 7.6 \cdot 10^{-221}:\\ \;\;\;\;t_0 \cdot \log \left(-re\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{t_1}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 4
Error	26.1
Cost	13580

\[\begin{array}{l} t_0 := \frac{1}{\log base}\\ \mathbf{if}\;im \leq -6.6 \cdot 10^{-44}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log base}\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{-302}:\\ \;\;\;\;t_0 \cdot \log re\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{-219}:\\ \;\;\;\;t_0 \cdot \log \left(-re\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 5
Error	26.1
Cost	13520

\[\begin{array}{l} t_0 := \frac{\log re}{\log base}\\ \mathbf{if}\;im \leq -8.6 \cdot 10^{-44}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log base}\\ \mathbf{elif}\;im \leq 2 \cdot 10^{-303}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.36 \cdot 10^{-219}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 6
Error	26.0
Cost	13520

\[\begin{array}{l} \mathbf{if}\;im \leq -8.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log base}\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{-301}:\\ \;\;\;\;\frac{1}{\log base} \cdot \log re\\ \mathbf{elif}\;im \leq 2.05 \cdot 10^{-219}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 7
Error	26.0
Cost	13256

\[\begin{array}{l} \mathbf{if}\;im \leq -1.9 \cdot 10^{-43}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log base}\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 8
Error	36.2
Cost	13124

\[\begin{array}{l} \mathbf{if}\;re \leq 2.45 \cdot 10^{-141}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array} \]

Alternative 9
Error	46.7
Cost	12992

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

Reproduce

herbie shell --seed 2023066 
(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))))