math.log10 on complex, real part

Percentage Accurate: 51.8% → 98.6%

Time: 8.7s

Alternatives: 8

Speedup: 1.1×

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: 51.8% 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: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{-1}{\frac{\log 0.1}{\log \left(\mathsf{fma}\left(0.5 \cdot re\_m, \frac{re\_m}{im\_m}, im\_m\right)\right)}} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (/ -1.0 (/ (log 0.1) (log (fma (* 0.5 re_m) (/ re_m im_m) im_m)))))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return -1.0 / (log(0.1) / log(fma((0.5 * re_m), (re_m / im_m), im_m)));
}

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(-1.0 / Float64(log(0.1) / log(fma(Float64(0.5 * re_m), Float64(re_m / im_m), im_m))))
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(-1.0 / N[(N[Log[0.1], $MachinePrecision] / N[Log[N[(N[(0.5 * re$95$m), $MachinePrecision] * N[(re$95$m / im$95$m), $MachinePrecision] + im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.4%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. lower-fma.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 29.2

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

5. Applied rewrites 29.2%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{fma}\left(\frac{0.5}{im}, re \cdot re, im\right)\right)}}{\log 10} \]
6. Step-by-step derivation

7. Applied rewrites 30.4%

   \[\leadsto \frac{\log \left(\mathsf{fma}\left(\frac{re}{im}, \color{blue}{0.5 \cdot re}, im\right)\right)}{\log 10} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. metadata-eval N/A

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

   4. neg-log N/A

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

   5. lift-log.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. associate-/r* N/A

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

   8. clear-num N/A

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

9. Applied rewrites 30.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{\log 0.1}{\log \left(\mathsf{fma}\left(0.5 \cdot re, \frac{re}{im}, im\right)\right) \cdot -1}}} \]

10. Final simplification 30.3%

    \[\leadsto \frac{-1}{\frac{\log 0.1}{\log \left(\mathsf{fma}\left(0.5 \cdot re, \frac{re}{im}, im\right)\right)}} \]
11. Add Preprocessing

Alternative 2: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{\log \left(\mathsf{fma}\left(\frac{re\_m}{im\_m}, 0.5 \cdot re\_m, im\_m\right)\right)}{\log 10} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (/ (log (fma (/ re_m im_m) (* 0.5 re_m) im_m)) (log 10.0)))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return log(fma((re_m / im_m), (0.5 * re_m), im_m)) / log(10.0);
}

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(log(fma(Float64(re_m / im_m), Float64(0.5 * re_m), im_m)) / log(10.0))
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[Log[N[(N[(re$95$m / im$95$m), $MachinePrecision] * N[(0.5 * re$95$m), $MachinePrecision] + im$95$m), $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.4%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. lower-fma.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 29.2

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

5. Applied rewrites 29.2%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{fma}\left(\frac{0.5}{im}, re \cdot re, im\right)\right)}}{\log 10} \]
6. Step-by-step derivation

7. Applied rewrites 30.4%

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

Alternative 3: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{1}{\frac{\log 10}{\log im\_m}} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m) :precision binary64 (/ 1.0 (/ (log 10.0) (log im_m))))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return 1.0 / (log(10.0) / log(im_m));
}

Fortran

im_m = abs(im)
re_m = abs(re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function code(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = 1.0d0 / (log(10.0d0) / log(im_m))
end function

Java

im_m = Math.abs(im);
re_m = Math.abs(re);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return 1.0 / (Math.log(10.0) / Math.log(im_m));
}

Python

im_m = math.fabs(im)
re_m = math.fabs(re)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return 1.0 / (math.log(10.0) / math.log(im_m))

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(1.0 / Float64(log(10.0) / log(im_m)))
end

MATLAB

im_m = abs(im);
re_m = abs(re);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = 1.0 / (log(10.0) / log(im_m));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(1.0 / N[(N[Log[10.0], $MachinePrecision] / N[Log[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.4%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 30.2

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

5. Applied rewrites 30.2%

   \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]
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 30.2

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

7. Applied rewrites 30.2%

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

Alternative 4: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{\log im\_m}{\log 10} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m) :precision binary64 (/ (log im_m) (log 10.0)))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return log(im_m) / log(10.0);
}

Fortran

im_m = abs(im)
re_m = abs(re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function code(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = log(im_m) / log(10.0d0)
end function

Java

im_m = Math.abs(im);
re_m = Math.abs(re);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return Math.log(im_m) / Math.log(10.0);
}

Python

im_m = math.fabs(im)
re_m = math.fabs(re)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return math.log(im_m) / math.log(10.0)

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(log(im_m) / log(10.0))
end

MATLAB

im_m = abs(im);
re_m = abs(re);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = log(im_m) / log(10.0);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[Log[im$95$m], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.4%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 30.2

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

5. Applied rewrites 30.2%

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

Alternative 5: 3.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{\frac{re\_m}{im\_m} \cdot re\_m}{-2 \cdot \left(im\_m \cdot \log 0.1\right)} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (/ (* (/ re_m im_m) re_m) (* -2.0 (* im_m (log 0.1)))))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return ((re_m / im_m) * re_m) / (-2.0 * (im_m * log(0.1)));
}

Fortran

im_m = abs(im)
re_m = abs(re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function code(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = ((re_m / im_m) * re_m) / ((-2.0d0) * (im_m * log(0.1d0)))
end function

Java

im_m = Math.abs(im);
re_m = Math.abs(re);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return ((re_m / im_m) * re_m) / (-2.0 * (im_m * Math.log(0.1)));
}

Python

im_m = math.fabs(im)
re_m = math.fabs(re)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return ((re_m / im_m) * re_m) / (-2.0 * (im_m * math.log(0.1)))

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(Float64(Float64(re_m / im_m) * re_m) / Float64(-2.0 * Float64(im_m * log(0.1))))
end

MATLAB

im_m = abs(im);
re_m = abs(re);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = ((re_m / im_m) * re_m) / (-2.0 * (im_m * log(0.1)));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[(N[(re$95$m / im$95$m), $MachinePrecision] * re$95$m), $MachinePrecision] / N[(-2.0 * N[(im$95$m * N[Log[0.1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.4%

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

3. Taylor expanded in re around 0

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-/r* N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-log.f64 N/A

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

   17. lower-log.f64 28.9

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

5. Applied rewrites 28.9%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.3%

   \[\leadsto \frac{0.5}{\log 10 \cdot im} \cdot \color{blue}{\frac{re \cdot re}{im}} \]
9. Step-by-step derivation

10. Applied rewrites 3.3%

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

11. Final simplification 3.3%

    \[\leadsto \frac{\frac{re}{im} \cdot re}{-2 \cdot \left(im \cdot \log 0.1\right)} \]
12. Add Preprocessing

Alternative 6: 3.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{re\_m \cdot re\_m}{im\_m} \cdot \frac{-0.5}{im\_m \cdot \log 0.1} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (* (/ (* re_m re_m) im_m) (/ -0.5 (* im_m (log 0.1)))))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return ((re_m * re_m) / im_m) * (-0.5 / (im_m * log(0.1)));
}

Fortran

im_m = abs(im)
re_m = abs(re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function code(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = ((re_m * re_m) / im_m) * ((-0.5d0) / (im_m * log(0.1d0)))
end function

Java

im_m = Math.abs(im);
re_m = Math.abs(re);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return ((re_m * re_m) / im_m) * (-0.5 / (im_m * Math.log(0.1)));
}

Python

im_m = math.fabs(im)
re_m = math.fabs(re)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return ((re_m * re_m) / im_m) * (-0.5 / (im_m * math.log(0.1)))

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(Float64(Float64(re_m * re_m) / im_m) * Float64(-0.5 / Float64(im_m * log(0.1))))
end

MATLAB

im_m = abs(im);
re_m = abs(re);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = ((re_m * re_m) / im_m) * (-0.5 / (im_m * log(0.1)));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[(N[(re$95$m * re$95$m), $MachinePrecision] / im$95$m), $MachinePrecision] * N[(-0.5 / N[(im$95$m * N[Log[0.1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.4%

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

3. Taylor expanded in re around 0

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-/r* N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-log.f64 N/A

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

   17. lower-log.f64 28.9

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

5. Applied rewrites 28.9%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.3%

   \[\leadsto \frac{0.5}{\log 10 \cdot im} \cdot \color{blue}{\frac{re \cdot re}{im}} \]
9. Step-by-step derivation

10. Applied rewrites 3.3%

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

11. Final simplification 3.3%

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

Alternative 7: 3.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \left(\frac{0.5}{\log 10 \cdot im\_m} \cdot \frac{re\_m}{im\_m}\right) \cdot re\_m \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (* (* (/ 0.5 (* (log 10.0) im_m)) (/ re_m im_m)) re_m))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return ((0.5 / (log(10.0) * im_m)) * (re_m / im_m)) * re_m;
}

Fortran

im_m = abs(im)
re_m = abs(re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function code(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = ((0.5d0 / (log(10.0d0) * im_m)) * (re_m / im_m)) * re_m
end function

Java

im_m = Math.abs(im);
re_m = Math.abs(re);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return ((0.5 / (Math.log(10.0) * im_m)) * (re_m / im_m)) * re_m;
}

Python

im_m = math.fabs(im)
re_m = math.fabs(re)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return ((0.5 / (math.log(10.0) * im_m)) * (re_m / im_m)) * re_m

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(Float64(Float64(0.5 / Float64(log(10.0) * im_m)) * Float64(re_m / im_m)) * re_m)
end

MATLAB

im_m = abs(im);
re_m = abs(re);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = ((0.5 / (log(10.0) * im_m)) * (re_m / im_m)) * re_m;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[(N[(0.5 / N[(N[Log[10.0], $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(re$95$m / im$95$m), $MachinePrecision]), $MachinePrecision] * re$95$m), $MachinePrecision]

Derivation

1. Initial program 57.4%

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

3. Taylor expanded in re around 0

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-/r* N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-log.f64 N/A

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

   17. lower-log.f64 28.9

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

5. Applied rewrites 28.9%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.3%

   \[\leadsto \frac{0.5}{\log 10 \cdot im} \cdot \color{blue}{\frac{re \cdot re}{im}} \]
9. Step-by-step derivation

10. Applied rewrites 3.3%

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

11. Final simplification 3.3%

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

Alternative 8: 3.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{re\_m}{\left(\log 10 \cdot im\_m\right) \cdot im\_m} \cdot \left(0.5 \cdot re\_m\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (* (/ re_m (* (* (log 10.0) im_m) im_m)) (* 0.5 re_m)))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return (re_m / ((log(10.0) * im_m) * im_m)) * (0.5 * re_m);
}

Fortran

im_m = abs(im)
re_m = abs(re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function code(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = (re_m / ((log(10.0d0) * im_m) * im_m)) * (0.5d0 * re_m)
end function

Java

im_m = Math.abs(im);
re_m = Math.abs(re);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return (re_m / ((Math.log(10.0) * im_m) * im_m)) * (0.5 * re_m);
}

Python

im_m = math.fabs(im)
re_m = math.fabs(re)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return (re_m / ((math.log(10.0) * im_m) * im_m)) * (0.5 * re_m)

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(Float64(re_m / Float64(Float64(log(10.0) * im_m) * im_m)) * Float64(0.5 * re_m))
end

MATLAB

im_m = abs(im);
re_m = abs(re);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = (re_m / ((log(10.0) * im_m) * im_m)) * (0.5 * re_m);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[(re$95$m / N[(N[(N[Log[10.0], $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.5 * re$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.4%

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

3. Taylor expanded in re around 0

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-/r* N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-log.f64 N/A

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

   17. lower-log.f64 28.9

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

5. Applied rewrites 28.9%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.3%

   \[\leadsto \frac{0.5}{\log 10 \cdot im} \cdot \color{blue}{\frac{re \cdot re}{im}} \]
9. Step-by-step derivation

10. Applied rewrites 3.3%

    \[\leadsto \frac{\frac{-0.5}{\log 0.1}}{\frac{im}{re \cdot re} \cdot \color{blue}{im}} \]
11. Step-by-step derivation

12. Applied rewrites 3.1%

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

Reproduce

herbie shell --seed 2024249 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))