math.log/2 on complex, real part

Percentage Accurate: 51.8% → 99.4%

Time: 11.1s

Alternatives: 4

Speedup: 1.8×

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: 51.8% 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, 1.8× 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 52.5%

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

3. Taylor expanded in base around 0

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

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. lower-hypot.f64 N/A

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

   7. lower-log.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 2: 25.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

double code(double re, double im, double base) {
	return fma(((0.5 * re) / im), (re / im), log(im)) / log(base);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 52.5%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. mul0-rgt N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. lift-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. metadata-eval N/A

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

   10. +-rgt-identity N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/r* N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in re around 0

   \[\leadsto \frac{\color{blue}{\log im + \frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}}}{\log base} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}} + \log im}}{\log base} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot {re}^{2}}{{im}^{2}}} + \log im}{\log base} \]

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. times-frac N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot re}{im} \cdot \frac{re}{im}} + \log im}{\log base} \]

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-log.f64 26.6

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

7. Applied rewrites 26.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot re}{im}, \frac{re}{im}, \log im\right)}}{\log base} \]
8. Add Preprocessing

Alternative 3: 26.9% accurate, 2.7× 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 52.5%

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

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{\log im}{\log base}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. lower-log.f64 28.1

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

5. Applied rewrites 28.1%

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

Alternative 4: 3.0% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.5%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. mul0-rgt N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. lift-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. metadata-eval N/A

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

   10. +-rgt-identity N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/r* N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in re around 0

   \[\leadsto \frac{\color{blue}{\log im + \frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}}}{\log base} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}} + \log im}}{\log base} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot {re}^{2}}{{im}^{2}}} + \log im}{\log base} \]

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. times-frac N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot re}{im} \cdot \frac{re}{im}} + \log im}{\log base} \]

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-log.f64 26.6

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

7. Applied rewrites 26.6%

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

8. Taylor expanded in re around inf

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

10. Applied rewrites 3.1%

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

Reproduce

herbie shell --seed 2024255 
(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))))