math.log10 on complex, real part

Percentage Accurate: 51.1% → 99.1%

Time: 6.7s

Alternatives: 5

Speedup: 0.8×

Specification

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Initial Program: 51.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (log (hypot re im)) (log 10.0)))

C

double code(double re, double im) {
	return log(hypot(re, im)) / log(10.0);
}

Java

public static double code(double re, double im) {
	return Math.log(Math.hypot(re, im)) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.hypot(re, im)) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(hypot(re, im)) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(hypot(re, im)) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}{\log 10} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re} + im \cdot im}\right)}{\log 10} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   5. lower-hypot.f64 99.1

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

4. Applied rewrites 99.1%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]
5. Add Preprocessing

Alternative 2: 25.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{0.5 \cdot re}{im}, \frac{re}{im}, \log im\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (fma (/ (* 0.5 re) im) (/ re im) (log im)) (log 10.0)))

C

double code(double re, double im) {
	return fma(((0.5 * re) / im), (re / im), log(im)) / log(10.0);
}

Julia

function code(re, im)
	return Float64(fma(Float64(Float64(0.5 * re) / im), Float64(re / im), log(im)) / log(10.0))
end

Wolfram

code[re_, im_] := N[(N[(N[(N[(0.5 * re), $MachinePrecision] / im), $MachinePrecision] * N[(re / im), $MachinePrecision] + N[Log[im], $MachinePrecision]), $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}{\log 10} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re} + im \cdot im}\right)}{\log 10} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   5. lower-hypot.f64 99.1

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

4. Applied rewrites 99.1%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

5. Taylor expanded in re around 0

   \[\leadsto \frac{\color{blue}{\log im + \frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}}}{\log 10} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}} + \log im}}{\log 10} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot {re}^{2}}{{im}^{2}}} + \log im}{\log 10} \]

   3. unpow2 N/A

      \[\leadsto \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(re \cdot re\right)}}{{im}^{2}} + \log im}{\log 10} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{2} \cdot re\right) \cdot re}}{{im}^{2}} + \log im}{\log 10} \]

   5. unpow2 N/A

      \[\leadsto \frac{\frac{\left(\frac{1}{2} \cdot re\right) \cdot re}{\color{blue}{im \cdot im}} + \log im}{\log 10} \]

   6. times-frac N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot re}{im} \cdot \frac{re}{im}} + \log im}{\log 10} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{2} \cdot re}{im}, \frac{re}{im}, \log im\right)}}{\log 10} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2} \cdot re}{im}}, \frac{re}{im}, \log im\right)}{\log 10} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{\frac{1}{2} \cdot re}}{im}, \frac{re}{im}, \log im\right)}{\log 10} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{1}{2} \cdot re}{im}, \color{blue}{\frac{re}{im}}, \log im\right)}{\log 10} \]

   11. lower-log.f64 25.4

       \[\leadsto \frac{\mathsf{fma}\left(\frac{0.5 \cdot re}{im}, \frac{re}{im}, \color{blue}{\log im}\right)}{\log 10} \]

7. Applied rewrites 25.4%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot re}{im}, \frac{re}{im}, \log im\right)}}{\log 10} \]
8. Add Preprocessing

Alternative 3: 27.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\log im}{\log 10} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (log im) (log 10.0)))

C

double code(double re, double im) {
	return log(im) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(im) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(im) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(im) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(im) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(im) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]
4. Step-by-step derivation

   1. lower-log.f64 26.6

      \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]

5. Applied rewrites 26.6%

   \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]
6. Add Preprocessing

Alternative 4: 3.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.5}{im} \cdot \frac{re \cdot re}{im}}{\log 0.1} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (* (/ -0.5 im) (/ (* re re) im)) (log 0.1)))

C

double code(double re, double im) {
	return ((-0.5 / im) * ((re * re) / im)) / log(0.1);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (((-0.5d0) / im) * ((re * re) / im)) / log(0.1d0)
end function

Java

public static double code(double re, double im) {
	return ((-0.5 / im) * ((re * re) / im)) / Math.log(0.1);
}

Python

def code(re, im):
	return ((-0.5 / im) * ((re * re) / im)) / math.log(0.1)

Julia

function code(re, im)
	return Float64(Float64(Float64(-0.5 / im) * Float64(Float64(re * re) / im)) / log(0.1))
end

MATLAB

function tmp = code(re, im)
	tmp = ((-0.5 / im) * ((re * re) / im)) / log(0.1);
end

Wolfram

code[re_, im_] := N[(N[(N[(-0.5 / im), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision] / N[Log[0.1], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}{\mathsf{neg}\left(\log 10\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}{\mathsf{neg}\left(\log 10\right)}} \]

   4. lower-neg.f64 N/A

      \[\leadsto \frac{\color{blue}{-\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\mathsf{neg}\left(\log 10\right)} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \frac{-\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\mathsf{neg}\left(\log 10\right)} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{-\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}{\mathsf{neg}\left(\log 10\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{-\log \left(\sqrt{\color{blue}{im \cdot im + re \cdot re}}\right)}{\mathsf{neg}\left(\log 10\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{-\log \left(\sqrt{\color{blue}{im \cdot im} + re \cdot re}\right)}{\mathsf{neg}\left(\log 10\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{-\log \left(\sqrt{im \cdot im + \color{blue}{re \cdot re}}\right)}{\mathsf{neg}\left(\log 10\right)} \]

   10. lower-hypot.f64 N/A

       \[\leadsto \frac{-\log \color{blue}{\left(\mathsf{hypot}\left(im, re\right)\right)}}{\mathsf{neg}\left(\log 10\right)} \]

   11. lift-log.f64 N/A

       \[\leadsto \frac{-\log \left(\mathsf{hypot}\left(im, re\right)\right)}{\mathsf{neg}\left(\color{blue}{\log 10}\right)} \]

   12. neg-log N/A

       \[\leadsto \frac{-\log \left(\mathsf{hypot}\left(im, re\right)\right)}{\color{blue}{\log \left(\frac{1}{10}\right)}} \]

   13. lower-log.f64 N/A

       \[\leadsto \frac{-\log \left(\mathsf{hypot}\left(im, re\right)\right)}{\color{blue}{\log \left(\frac{1}{10}\right)}} \]

   14. metadata-eval 99.0

       \[\leadsto \frac{-\log \left(\mathsf{hypot}\left(im, re\right)\right)}{\log \color{blue}{0.1}} \]

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\frac{-\log \left(\mathsf{hypot}\left(im, re\right)\right)}{\log 0.1}} \]

5. Taylor expanded in re around 0

   \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \frac{{re}^{2}}{{im}^{2}} - \log im}}{\log \frac{1}{10}} \]
6. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \frac{{re}^{2}}{{im}^{2}} - \log im}}{\log \frac{1}{10}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{2} \cdot {re}^{2}}{{im}^{2}}} - \log im}{\log \frac{1}{10}} \]

   3. unpow2 N/A

      \[\leadsto \frac{\frac{\frac{-1}{2} \cdot \color{blue}{\left(re \cdot re\right)}}{{im}^{2}} - \log im}{\log \frac{1}{10}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{-1}{2} \cdot re\right) \cdot re}}{{im}^{2}} - \log im}{\log \frac{1}{10}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\frac{\left(\frac{-1}{2} \cdot re\right) \cdot re}{\color{blue}{im \cdot im}} - \log im}{\log \frac{1}{10}} \]

   6. times-frac N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{2} \cdot re}{im} \cdot \frac{re}{im}} - \log im}{\log \frac{1}{10}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{2} \cdot re}{im} \cdot \frac{re}{im}} - \log im}{\log \frac{1}{10}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{2} \cdot re}{im}} \cdot \frac{re}{im} - \log im}{\log \frac{1}{10}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{-1}{2} \cdot re}}{im} \cdot \frac{re}{im} - \log im}{\log \frac{1}{10}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{-1}{2} \cdot re}{im} \cdot \color{blue}{\frac{re}{im}} - \log im}{\log \frac{1}{10}} \]

   11. lower-log.f64 25.4

       \[\leadsto \frac{\frac{-0.5 \cdot re}{im} \cdot \frac{re}{im} - \color{blue}{\log im}}{\log 0.1} \]

7. Applied rewrites 25.4%

   \[\leadsto \frac{\color{blue}{\frac{-0.5 \cdot re}{im} \cdot \frac{re}{im} - \log im}}{\log 0.1} \]

8. Taylor expanded in re around inf

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

10. Applied rewrites 12.8%

    \[\leadsto \frac{\left(\frac{\log im}{\left(-re\right) \cdot re} - \frac{0.5}{im \cdot im}\right) \cdot \color{blue}{\left(re \cdot re\right)}}{\log 0.1} \]

11. Taylor expanded in re around inf

    \[\leadsto \frac{\frac{-1}{2} \cdot \color{blue}{\frac{{re}^{2}}{{im}^{2}}}}{\log \frac{1}{10}} \]
12. Step-by-step derivation

13. Applied rewrites 3.1%

    \[\leadsto \frac{\frac{-0.5}{im} \cdot \color{blue}{\frac{re \cdot re}{im}}}{\log 0.1} \]
14. Add Preprocessing

Alternative 5: 2.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.5}{im \cdot im} \cdot \left(re \cdot re\right)}{\log 0.1} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (* (/ -0.5 (* im im)) (* re re)) (log 0.1)))

C

double code(double re, double im) {
	return ((-0.5 / (im * im)) * (re * re)) / log(0.1);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (((-0.5d0) / (im * im)) * (re * re)) / log(0.1d0)
end function

Java

public static double code(double re, double im) {
	return ((-0.5 / (im * im)) * (re * re)) / Math.log(0.1);
}

Python

def code(re, im):
	return ((-0.5 / (im * im)) * (re * re)) / math.log(0.1)

Julia

function code(re, im)
	return Float64(Float64(Float64(-0.5 / Float64(im * im)) * Float64(re * re)) / log(0.1))
end

MATLAB

function tmp = code(re, im)
	tmp = ((-0.5 / (im * im)) * (re * re)) / log(0.1);
end

Wolfram

code[re_, im_] := N[(N[(N[(-0.5 / N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] / N[Log[0.1], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}{\mathsf{neg}\left(\log 10\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}{\mathsf{neg}\left(\log 10\right)}} \]

   4. lower-neg.f64 N/A

      \[\leadsto \frac{\color{blue}{-\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\mathsf{neg}\left(\log 10\right)} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \frac{-\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\mathsf{neg}\left(\log 10\right)} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{-\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}{\mathsf{neg}\left(\log 10\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{-\log \left(\sqrt{\color{blue}{im \cdot im + re \cdot re}}\right)}{\mathsf{neg}\left(\log 10\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{-\log \left(\sqrt{\color{blue}{im \cdot im} + re \cdot re}\right)}{\mathsf{neg}\left(\log 10\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{-\log \left(\sqrt{im \cdot im + \color{blue}{re \cdot re}}\right)}{\mathsf{neg}\left(\log 10\right)} \]

   10. lower-hypot.f64 N/A

       \[\leadsto \frac{-\log \color{blue}{\left(\mathsf{hypot}\left(im, re\right)\right)}}{\mathsf{neg}\left(\log 10\right)} \]

   11. lift-log.f64 N/A

       \[\leadsto \frac{-\log \left(\mathsf{hypot}\left(im, re\right)\right)}{\mathsf{neg}\left(\color{blue}{\log 10}\right)} \]

   12. neg-log N/A

       \[\leadsto \frac{-\log \left(\mathsf{hypot}\left(im, re\right)\right)}{\color{blue}{\log \left(\frac{1}{10}\right)}} \]

   13. lower-log.f64 N/A

       \[\leadsto \frac{-\log \left(\mathsf{hypot}\left(im, re\right)\right)}{\color{blue}{\log \left(\frac{1}{10}\right)}} \]

   14. metadata-eval 99.0

       \[\leadsto \frac{-\log \left(\mathsf{hypot}\left(im, re\right)\right)}{\log \color{blue}{0.1}} \]

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\frac{-\log \left(\mathsf{hypot}\left(im, re\right)\right)}{\log 0.1}} \]

5. Taylor expanded in re around 0

   \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \frac{{re}^{2}}{{im}^{2}} - \log im}}{\log \frac{1}{10}} \]
6. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \frac{{re}^{2}}{{im}^{2}} - \log im}}{\log \frac{1}{10}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{2} \cdot {re}^{2}}{{im}^{2}}} - \log im}{\log \frac{1}{10}} \]

   3. unpow2 N/A

      \[\leadsto \frac{\frac{\frac{-1}{2} \cdot \color{blue}{\left(re \cdot re\right)}}{{im}^{2}} - \log im}{\log \frac{1}{10}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{-1}{2} \cdot re\right) \cdot re}}{{im}^{2}} - \log im}{\log \frac{1}{10}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\frac{\left(\frac{-1}{2} \cdot re\right) \cdot re}{\color{blue}{im \cdot im}} - \log im}{\log \frac{1}{10}} \]

   6. times-frac N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{2} \cdot re}{im} \cdot \frac{re}{im}} - \log im}{\log \frac{1}{10}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{2} \cdot re}{im} \cdot \frac{re}{im}} - \log im}{\log \frac{1}{10}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{2} \cdot re}{im}} \cdot \frac{re}{im} - \log im}{\log \frac{1}{10}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{-1}{2} \cdot re}}{im} \cdot \frac{re}{im} - \log im}{\log \frac{1}{10}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{-1}{2} \cdot re}{im} \cdot \color{blue}{\frac{re}{im}} - \log im}{\log \frac{1}{10}} \]

   11. lower-log.f64 25.4

       \[\leadsto \frac{\frac{-0.5 \cdot re}{im} \cdot \frac{re}{im} - \color{blue}{\log im}}{\log 0.1} \]

7. Applied rewrites 25.4%

   \[\leadsto \frac{\color{blue}{\frac{-0.5 \cdot re}{im} \cdot \frac{re}{im} - \log im}}{\log 0.1} \]

8. Taylor expanded in re around inf

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

10. Applied rewrites 12.8%

    \[\leadsto \frac{\left(\frac{\log im}{\left(-re\right) \cdot re} - \frac{0.5}{im \cdot im}\right) \cdot \color{blue}{\left(re \cdot re\right)}}{\log 0.1} \]

11. Taylor expanded in re around inf

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

13. Applied rewrites 2.8%

    \[\leadsto \frac{\frac{-0.5}{im \cdot im} \cdot \left(re \cdot re\right)}{\log 0.1} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))