math.log10 on complex, real part

Percentage Accurate: 50.7% → 99.1%

Time: 8.2s

Alternatives: 7

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: 50.7% 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 51.2%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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: 25.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(--2\right) \cdot \log im\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (* 0.5 (fma (/ re im) (/ re im) (* (- -2.0) (log im)))) (log 10.0)))

C

double code(double re, double im) {
	return (0.5 * fma((re / im), (re / im), (-(-2.0) * log(im)))) / log(10.0);
}

Julia

function code(re, im)
	return Float64(Float64(0.5 * fma(Float64(re / im), Float64(re / im), Float64(Float64(-(-2.0)) * log(im)))) / log(10.0))
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[(N[(re / im), $MachinePrecision] * N[(re / im), $MachinePrecision] + N[((--2.0) * N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.2%

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

   1. lift-hypot.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-hypot.f64 99.1

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

   4. remove-double-div N/A

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

   5. unpow-1 N/A

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

   6. lift-pow.f64 N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow-1 N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 99.0

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

6. Applied rewrites 99.0%

   \[\leadsto \frac{\color{blue}{\frac{-1}{\frac{-1}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}}}{\log 10} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. metadata-eval N/A

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

   5. *-lft-identity 99.1

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

   6. lift-log.f64 N/A

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

   7. lift-hypot.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. pow1/2 N/A

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

   11. log-pow N/A

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

   12. lift-log.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 51.2

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

   15. lift-fma.f64 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lower-fma.f64 N/A

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

   19. lower-*.f64 51.2

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

8. Applied rewrites 51.2%

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

9. Taylor expanded in im around inf

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

    1. +-commutative N/A

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

    2. unpow2 N/A

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

    3. unpow2 N/A

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

    4. times-frac N/A

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

    5. lower-fma.f64 N/A

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

    6. lower-/.f64 N/A

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

    7. lower-/.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. log-rec N/A

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

    11. lower-neg.f64 N/A

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

    12. lower-log.f64 21.4

        \[\leadsto \frac{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(-\color{blue}{\log im}\right) \cdot -2\right) \cdot 0.5}{\log 10} \]

11. Applied rewrites 21.4%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(-\log im\right) \cdot -2\right)} \cdot 0.5}{\log 10} \]

12. Final simplification 21.4%

    \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \left(--2\right) \cdot \log im\right)}{\log 10} \]
13. Add Preprocessing

Alternative 3: 25.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right) \cdot -0.5}{\log 0.1} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (* (fma (/ re im) (/ re im) (* 2.0 (log im))) -0.5) (log 0.1)))

C

double code(double re, double im) {
	return (fma((re / im), (re / im), (2.0 * log(im))) * -0.5) / log(0.1);
}

Julia

function code(re, im)
	return Float64(Float64(fma(Float64(re / im), Float64(re / im), Float64(2.0 * log(im))) * -0.5) / log(0.1))
end

Wolfram

code[re_, im_] := N[(N[(N[(N[(re / im), $MachinePrecision] * N[(re / im), $MachinePrecision] + N[(2.0 * N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / N[Log[0.1], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.2%

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

   1. lift-neg.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-hypot.f64 N/A

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

   4. pow1/2 N/A

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

   5. +-commutative N/A

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

   6. log-pow N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 51.3

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

6. Applied rewrites 51.3%

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

7. Taylor expanded in im around inf

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. log-rec N/A

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

   9. mul-1-neg N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lower-log.f64 21.4

       \[\leadsto \frac{-0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \color{blue}{\log im}\right)}{\log 0.1} \]

9. Applied rewrites 21.4%

   \[\leadsto \frac{-0.5 \cdot \color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right)}}{\log 0.1} \]

10. Final simplification 21.4%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right) \cdot -0.5}{\log 0.1} \]
11. Add Preprocessing

Alternative 4: 27.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\log 10}{\log im}} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ 1.0 (/ (log 10.0) (log im))))

C

double code(double re, double im) {
	return 1.0 / (log(10.0) / log(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 / (log(10.0d0) / log(im))
end function

Java

public static double code(double re, double im) {
	return 1.0 / (Math.log(10.0) / Math.log(im));
}

Python

def code(re, im):
	return 1.0 / (math.log(10.0) / math.log(im))

Julia

function code(re, im)
	return Float64(1.0 / Float64(log(10.0) / log(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 / (log(10.0) / log(im));
end

Wolfram

code[re_, im_] := N[(1.0 / N[(N[Log[10.0], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 23.4

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

5. Applied rewrites 23.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]

   4. lower-/.f64 23.4

      \[\leadsto \frac{1}{\color{blue}{\frac{\log 10}{\log im}}} \]

7. Applied rewrites 23.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]
8. Add Preprocessing

Alternative 5: 27.6% 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 51.2%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 23.4

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

5. Applied rewrites 23.4%

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

Alternative 6: 3.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (re * re) * (0.5d0 / log(10.0d0))
end function

Java

public static double code(double re, double im) {
	return (re * re) * (0.5 / Math.log(10.0));
}

Python

def code(re, im):
	return (re * re) * (0.5 / math.log(10.0))

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = (re * re) * (0.5 / log(10.0));
end

Wolfram

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

Derivation

1. Initial program 51.2%

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

   1. lift-neg.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-hypot.f64 N/A

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

   4. pow1/2 N/A

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

   5. +-commutative N/A

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

   6. log-pow N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 51.3

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

6. Applied rewrites 51.3%

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

7. Taylor expanded in im around inf

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. log-rec N/A

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

   9. mul-1-neg N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lower-log.f64 21.4

       \[\leadsto \frac{-0.5 \cdot \mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \color{blue}{\log im}\right)}{\log 0.1} \]

9. Applied rewrites 21.4%

   \[\leadsto \frac{-0.5 \cdot \color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right)}}{\log 0.1} \]

11. Applied rewrites 3.8%

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

12. Final simplification 3.8%

    \[\leadsto \left(re \cdot re\right) \cdot \frac{0.5}{\log 10} \]
13. Add Preprocessing

Alternative 7: 3.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{0}{\log 10} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ 0.0 (log 10.0)))

C

double code(double re, double im) {
	return 0.0 / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0 / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return 0.0 / Math.log(10.0);
}

Python

def code(re, im):
	return 0.0 / math.log(10.0)

Julia

function code(re, im)
	return Float64(0.0 / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0 / log(10.0);
end

Wolfram

code[re_, im_] := N[(0.0 / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 23.4

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

5. Applied rewrites 23.4%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. unpow-1 N/A

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

   4. sqr-pow N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower-pow.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-pow.f64 23.5

       \[\leadsto {\log 10}^{-0.5} \cdot \left(\color{blue}{{\log 10}^{-0.5}} \cdot \log im\right) \]

7. Applied rewrites 23.5%

   \[\leadsto \color{blue}{{\log 10}^{-0.5} \cdot \left({\log 10}^{-0.5} \cdot \log im\right)} \]

9. Applied rewrites 3.1%

   \[\leadsto \color{blue}{\frac{0}{\log 10}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024283 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))