math.log/2 on complex, real part

Percentage Accurate: 51.9% → 99.4%

Time: 16.5s

Alternatives: 6

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: 51.9% 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 46.6%

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

   1. fma-define 46.6%

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

   2. +-commutative 46.6%

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

   3. mul0-rgt 46.6%

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

   4. mul0-rgt 46.6%

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

   5. fma-define 46.6%

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

3. Simplified 99.5%

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

Alternative 2: 24.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{\frac{re}{im} \cdot \frac{re}{im} + \log im \cdot 2}{\log base} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (* 0.5 (/ (+ (* (/ re im) (/ re im)) (* (log im) 2.0)) (log base))))

C

double code(double re, double im, double base) {
	return 0.5 * ((((re / im) * (re / im)) + (log(im) * 2.0)) / 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 = 0.5d0 * ((((re / im) * (re / im)) + (log(im) * 2.0d0)) / log(base))
end function

Java

public static double code(double re, double im, double base) {
	return 0.5 * ((((re / im) * (re / im)) + (Math.log(im) * 2.0)) / Math.log(base));
}

Python

def code(re, im, base):
	return 0.5 * ((((re / im) * (re / im)) + (math.log(im) * 2.0)) / math.log(base))

Julia

function code(re, im, base)
	return Float64(0.5 * Float64(Float64(Float64(Float64(re / im) * Float64(re / im)) + Float64(log(im) * 2.0)) / log(base)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = 0.5 * ((((re / im) * (re / im)) + (log(im) * 2.0)) / log(base));
end

Wolfram

code[re_, im_, base_] := N[(0.5 * N[(N[(N[(N[(re / im), $MachinePrecision] * N[(re / im), $MachinePrecision]), $MachinePrecision] + N[(N[Log[im], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.6%

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

   1. fma-define 46.6%

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

   2. +-commutative 46.6%

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

   3. mul0-rgt 46.6%

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

   4. mul0-rgt 46.6%

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

   5. fma-define 46.6%

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

3. Simplified 99.5%

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

   1. expm1-log1p-u 74.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log base}\right)\right)} \]

   2. expm1-undefine 74.1%

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

   3. hypot-define 38.6%

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

   4. pow1/2 38.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{0.5}\right)}}{\log base}\right)} - 1 \]

   5. log-pow 38.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log base}\right)} - 1 \]

6. Applied egg-rr 38.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}{\log base}\right)} - 1} \]
7. Step-by-step derivation

   1. sub-neg 38.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}{\log base}\right)} + \left(-1\right)} \]

   2. metadata-eval 38.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}{\log base}\right)} + \color{blue}{-1} \]

   3. +-commutative 38.6%

      \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\frac{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}{\log base}\right)}} \]

   4. log1p-undefine 38.6%

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

   5. rem-exp-log 46.3%

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

   6. *-commutative 46.3%

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

   7. associate-/l* 46.3%

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

8. Simplified 46.3%

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

9. Taylor expanded in im around inf 26.2%

   \[\leadsto -1 + \left(1 + \color{blue}{\left(-2 \cdot \log \left(\frac{1}{im}\right) + \frac{{re}^{2}}{{im}^{2}}\right)} \cdot \frac{0.5}{\log base}\right) \]
10. Step-by-step derivation

    1. +-commutative 26.2%

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

    2. unpow2 26.2%

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

    3. unpow2 26.2%

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

    4. times-frac 28.7%

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

    5. *-commutative 28.7%

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

    6. log-rec 28.7%

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

    7. sub0-neg 28.7%

       \[\leadsto -1 + \left(1 + \left(\frac{re}{im} \cdot \frac{re}{im} + \color{blue}{\left(0 - \log im\right)} \cdot -2\right) \cdot \frac{0.5}{\log base}\right) \]

11. Simplified 28.7%

    \[\leadsto -1 + \left(1 + \color{blue}{\left(\frac{re}{im} \cdot \frac{re}{im} + \left(0 - \log im\right) \cdot -2\right)} \cdot \frac{0.5}{\log base}\right) \]

12. Taylor expanded in base around 0 26.4%

    \[\leadsto \color{blue}{0.5 \cdot \frac{2 \cdot \log im + \frac{{re}^{2}}{{im}^{2}}}{\log base}} \]
13. Step-by-step derivation

    1. +-commutative 26.4%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{\frac{{re}^{2}}{{im}^{2}} + 2 \cdot \log im}}{\log base} \]

    2. unpow2 26.4%

       \[\leadsto 0.5 \cdot \frac{\frac{\color{blue}{re \cdot re}}{{im}^{2}} + 2 \cdot \log im}{\log base} \]

    3. unpow2 26.4%

       \[\leadsto 0.5 \cdot \frac{\frac{re \cdot re}{\color{blue}{im \cdot im}} + 2 \cdot \log im}{\log base} \]

    4. times-frac 28.8%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{\frac{re}{im} \cdot \frac{re}{im}} + 2 \cdot \log im}{\log base} \]

    5. *-commutative 28.8%

       \[\leadsto 0.5 \cdot \frac{\frac{re}{im} \cdot \frac{re}{im} + \color{blue}{\log im \cdot 2}}{\log base} \]

14. Simplified 28.8%

    \[\leadsto \color{blue}{0.5 \cdot \frac{\frac{re}{im} \cdot \frac{re}{im} + \log im \cdot 2}{\log base}} \]
15. Add Preprocessing

Alternative 3: 26.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{1}{im}\right)}{-\log base} \end{array} \]

FPCore

(FPCore (re im base) :precision binary64 (/ (log (/ 1.0 im)) (- (log base))))

C

double code(double re, double im, double base) {
	return log((1.0 / 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((1.0d0 / im)) / -log(base)
end function

Java

public static double code(double re, double im, double base) {
	return Math.log((1.0 / im)) / -Math.log(base);
}

Python

def code(re, im, base):
	return math.log((1.0 / im)) / -math.log(base)

Julia

function code(re, im, base)
	return Float64(log(Float64(1.0 / im)) / Float64(-log(base)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = log((1.0 / im)) / -log(base);
end

Wolfram

code[re_, im_, base_] := N[(N[Log[N[(1.0 / im), $MachinePrecision]], $MachinePrecision] / (-N[Log[base], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 46.6%

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

   1. fma-define 46.6%

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

   2. +-commutative 46.6%

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

   3. mul0-rgt 46.6%

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

   4. mul0-rgt 46.6%

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

   5. fma-define 46.6%

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

3. Simplified 99.5%

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

5. Taylor expanded in im around inf 30.9%

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

6. Final simplification 30.9%

   \[\leadsto \frac{\log \left(\frac{1}{im}\right)}{-\log base} \]
7. Add Preprocessing

Alternative 4: 26.5% 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 46.6%

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

   1. fma-define 46.6%

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

   2. +-commutative 46.6%

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

   3. mul0-rgt 46.6%

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

   4. mul0-rgt 46.6%

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

   5. fma-define 46.6%

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

3. Simplified 99.5%

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

5. Taylor expanded in re around 0 30.9%

   \[\leadsto \color{blue}{\frac{\log im}{\log base}} \]
6. Add Preprocessing

Alternative 5: 3.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \left(\frac{re}{im} \cdot \frac{re}{im}\right) \cdot \frac{0.5}{\log base} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (* (* (/ re im) (/ re im)) (/ 0.5 (log base))))

C

double code(double re, double im, double base) {
	return ((re / im) * (re / im)) * (0.5 / 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 = ((re / im) * (re / im)) * (0.5d0 / log(base))
end function

Java

public static double code(double re, double im, double base) {
	return ((re / im) * (re / im)) * (0.5 / Math.log(base));
}

Python

def code(re, im, base):
	return ((re / im) * (re / im)) * (0.5 / math.log(base))

Julia

function code(re, im, base)
	return Float64(Float64(Float64(re / im) * Float64(re / im)) * Float64(0.5 / log(base)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((re / im) * (re / im)) * (0.5 / log(base));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(re / im), $MachinePrecision] * N[(re / im), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[Log[base], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.6%

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

   1. fma-define 46.6%

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

   2. +-commutative 46.6%

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

   3. mul0-rgt 46.6%

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

   4. mul0-rgt 46.6%

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

   5. fma-define 46.6%

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

3. Simplified 99.5%

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

   1. expm1-log1p-u 74.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log base}\right)\right)} \]

   2. expm1-undefine 74.1%

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

   3. hypot-define 38.6%

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

   4. pow1/2 38.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{0.5}\right)}}{\log base}\right)} - 1 \]

   5. log-pow 38.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log base}\right)} - 1 \]

6. Applied egg-rr 38.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}{\log base}\right)} - 1} \]
7. Step-by-step derivation

   1. sub-neg 38.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}{\log base}\right)} + \left(-1\right)} \]

   2. metadata-eval 38.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}{\log base}\right)} + \color{blue}{-1} \]

   3. +-commutative 38.6%

      \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\frac{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}{\log base}\right)}} \]

   4. log1p-undefine 38.6%

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

   5. rem-exp-log 46.3%

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

   6. *-commutative 46.3%

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

   7. associate-/l* 46.3%

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

8. Simplified 46.3%

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

9. Taylor expanded in im around inf 26.2%

   \[\leadsto -1 + \left(1 + \color{blue}{\left(-2 \cdot \log \left(\frac{1}{im}\right) + \frac{{re}^{2}}{{im}^{2}}\right)} \cdot \frac{0.5}{\log base}\right) \]
10. Step-by-step derivation

    1. +-commutative 26.2%

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

    2. unpow2 26.2%

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

    3. unpow2 26.2%

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

    4. times-frac 28.7%

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

    5. *-commutative 28.7%

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

    6. log-rec 28.7%

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

    7. sub0-neg 28.7%

       \[\leadsto -1 + \left(1 + \left(\frac{re}{im} \cdot \frac{re}{im} + \color{blue}{\left(0 - \log im\right)} \cdot -2\right) \cdot \frac{0.5}{\log base}\right) \]

11. Simplified 28.7%

    \[\leadsto -1 + \left(1 + \color{blue}{\left(\frac{re}{im} \cdot \frac{re}{im} + \left(0 - \log im\right) \cdot -2\right)} \cdot \frac{0.5}{\log base}\right) \]

12. Taylor expanded in re around inf 2.8%

    \[\leadsto \color{blue}{0.5 \cdot \frac{{re}^{2}}{{im}^{2} \cdot \log base}} \]
13. Step-by-step derivation

    1. associate-*r/ 2.8%

       \[\leadsto \color{blue}{\frac{0.5 \cdot {re}^{2}}{{im}^{2} \cdot \log base}} \]

    2. unpow2 2.8%

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

    3. *-commutative 2.8%

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

    4. unpow2 2.8%

       \[\leadsto \frac{0.5 \cdot \left(re \cdot re\right)}{\log base \cdot \color{blue}{\left(im \cdot im\right)}} \]

14. Simplified 2.8%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \left(re \cdot re\right)}{\log base \cdot \left(im \cdot im\right)}} \]
15. Step-by-step derivation

    1. times-frac 2.8%

       \[\leadsto \color{blue}{\frac{0.5}{\log base} \cdot \frac{re \cdot re}{im \cdot im}} \]

    2. frac-times 3.3%

       \[\leadsto \frac{0.5}{\log base} \cdot \color{blue}{\left(\frac{re}{im} \cdot \frac{re}{im}\right)} \]

16. Applied egg-rr 3.3%

    \[\leadsto \color{blue}{\frac{0.5}{\log base} \cdot \left(\frac{re}{im} \cdot \frac{re}{im}\right)} \]

17. Final simplification 3.3%

    \[\leadsto \left(\frac{re}{im} \cdot \frac{re}{im}\right) \cdot \frac{0.5}{\log base} \]
18. Add Preprocessing

Alternative 6: 3.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ re \cdot \frac{\frac{re}{im}}{im \cdot \left(\log base \cdot 2\right)} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (* re (/ (/ re im) (* im (* (log base) 2.0)))))

C

double code(double re, double im, double base) {
	return re * ((re / im) / (im * (log(base) * 2.0)));
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = re * ((re / im) / (im * (log(base) * 2.0d0)))
end function

Java

public static double code(double re, double im, double base) {
	return re * ((re / im) / (im * (Math.log(base) * 2.0)));
}

Python

def code(re, im, base):
	return re * ((re / im) / (im * (math.log(base) * 2.0)))

Julia

function code(re, im, base)
	return Float64(re * Float64(Float64(re / im) / Float64(im * Float64(log(base) * 2.0))))
end

MATLAB

function tmp = code(re, im, base)
	tmp = re * ((re / im) / (im * (log(base) * 2.0)));
end

Wolfram

code[re_, im_, base_] := N[(re * N[(N[(re / im), $MachinePrecision] / N[(im * N[(N[Log[base], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.6%

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

   1. fma-define 46.6%

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

   2. +-commutative 46.6%

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

   3. mul0-rgt 46.6%

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

   4. mul0-rgt 46.6%

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

   5. fma-define 46.6%

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

3. Simplified 99.5%

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

   1. expm1-log1p-u 74.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log base}\right)\right)} \]

   2. expm1-undefine 74.1%

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

   3. hypot-define 38.6%

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

   4. pow1/2 38.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{0.5}\right)}}{\log base}\right)} - 1 \]

   5. log-pow 38.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log base}\right)} - 1 \]

6. Applied egg-rr 38.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}{\log base}\right)} - 1} \]
7. Step-by-step derivation

   1. sub-neg 38.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}{\log base}\right)} + \left(-1\right)} \]

   2. metadata-eval 38.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}{\log base}\right)} + \color{blue}{-1} \]

   3. +-commutative 38.6%

      \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\frac{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}{\log base}\right)}} \]

   4. log1p-undefine 38.6%

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

   5. rem-exp-log 46.3%

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

   6. *-commutative 46.3%

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

   7. associate-/l* 46.3%

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

8. Simplified 46.3%

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

9. Taylor expanded in im around inf 26.2%

   \[\leadsto -1 + \left(1 + \color{blue}{\left(-2 \cdot \log \left(\frac{1}{im}\right) + \frac{{re}^{2}}{{im}^{2}}\right)} \cdot \frac{0.5}{\log base}\right) \]
10. Step-by-step derivation

    1. +-commutative 26.2%

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

    2. unpow2 26.2%

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

    3. unpow2 26.2%

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

    4. times-frac 28.7%

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

    5. *-commutative 28.7%

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

    6. log-rec 28.7%

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

    7. sub0-neg 28.7%

       \[\leadsto -1 + \left(1 + \left(\frac{re}{im} \cdot \frac{re}{im} + \color{blue}{\left(0 - \log im\right)} \cdot -2\right) \cdot \frac{0.5}{\log base}\right) \]

11. Simplified 28.7%

    \[\leadsto -1 + \left(1 + \color{blue}{\left(\frac{re}{im} \cdot \frac{re}{im} + \left(0 - \log im\right) \cdot -2\right)} \cdot \frac{0.5}{\log base}\right) \]

12. Taylor expanded in re around inf 2.8%

    \[\leadsto \color{blue}{0.5 \cdot \frac{{re}^{2}}{{im}^{2} \cdot \log base}} \]
13. Step-by-step derivation

    1. associate-*r/ 2.8%

       \[\leadsto \color{blue}{\frac{0.5 \cdot {re}^{2}}{{im}^{2} \cdot \log base}} \]

    2. unpow2 2.8%

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

    3. *-commutative 2.8%

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

    4. unpow2 2.8%

       \[\leadsto \frac{0.5 \cdot \left(re \cdot re\right)}{\log base \cdot \color{blue}{\left(im \cdot im\right)}} \]

14. Simplified 2.8%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \left(re \cdot re\right)}{\log base \cdot \left(im \cdot im\right)}} \]
15. Step-by-step derivation

    1. times-frac 2.8%

       \[\leadsto \color{blue}{\frac{0.5}{\log base} \cdot \frac{re \cdot re}{im \cdot im}} \]

    2. clear-num 2.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{\log base}{0.5}}} \cdot \frac{re \cdot re}{im \cdot im} \]

    3. frac-times 2.8%

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

    4. *-un-lft-identity 2.8%

       \[\leadsto \frac{\color{blue}{re \cdot re}}{\frac{\log base}{0.5} \cdot \left(im \cdot im\right)} \]

    5. div-inv 2.8%

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

    6. metadata-eval 2.8%

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

16. Applied egg-rr 2.8%

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

    1. unpow2 2.8%

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

    2. times-frac 3.0%

       \[\leadsto \color{blue}{\frac{re}{\log base \cdot 2} \cdot \frac{re}{{im}^{2}}} \]

    3. unpow2 3.0%

       \[\leadsto \frac{re}{\log base \cdot 2} \cdot \frac{re}{\color{blue}{im \cdot im}} \]

    4. associate-/r* 3.2%

       \[\leadsto \frac{re}{\log base \cdot 2} \cdot \color{blue}{\frac{\frac{re}{im}}{im}} \]

    5. times-frac 3.3%

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

    6. associate-/l* 3.2%

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

    7. *-commutative 3.2%

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

    8. *-commutative 3.2%

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

18. Simplified 3.2%

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

19. Final simplification 3.2%

    \[\leadsto re \cdot \frac{\frac{re}{im}}{im \cdot \left(\log base \cdot 2\right)} \]
20. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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))))