math.log10 on complex, real part

Percentage Accurate: 51.8% → 98.7%

Time: 7.7s

Alternatives: 9

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: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \left({\log 0.1}^{2} \cdot \left(-{\log 0.1}^{-3}\right)\right) \cdot \log im\_m \end{array} \]

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
 (* (* (pow (log 0.1) 2.0) (- (pow (log 0.1) -3.0))) (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 (pow(log(0.1), 2.0) * -pow(log(0.1), -3.0)) * 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 = ((log(0.1d0) ** 2.0d0) * -(log(0.1d0) ** (-3.0d0))) * 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 (Math.pow(Math.log(0.1), 2.0) * -Math.pow(Math.log(0.1), -3.0)) * 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 (math.pow(math.log(0.1), 2.0) * -math.pow(math.log(0.1), -3.0)) * 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(Float64((log(0.1) ^ 2.0) * Float64(-(log(0.1) ^ -3.0))) * 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 = ((log(0.1) ^ 2.0) * -(log(0.1) ^ -3.0)) * 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[(N[(N[Power[N[Log[0.1], $MachinePrecision], 2.0], $MachinePrecision] * (-N[Power[N[Log[0.1], $MachinePrecision], -3.0], $MachinePrecision])), $MachinePrecision] * N[Log[im$95$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

   \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]
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. lift-log.f64 N/A

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

   3. metadata-eval N/A

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

   4. neg-log N/A

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

   5. lift-log.f64 N/A

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

   6. neg-sub0 N/A

      \[\leadsto \frac{\log im}{\color{blue}{0 - \log \frac{1}{10}}} \]

   7. flip3-- N/A

      \[\leadsto \frac{\log im}{\color{blue}{\frac{{0}^{3} - {\log \frac{1}{10}}^{3}}{0 \cdot 0 + \left(\log \frac{1}{10} \cdot \log \frac{1}{10} + 0 \cdot \log \frac{1}{10}\right)}}} \]

   8. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{\log im}{{0}^{3} - {\log \frac{1}{10}}^{3}} \cdot \left(0 \cdot 0 + \left(\log \frac{1}{10} \cdot \log \frac{1}{10} + 0 \cdot \log \frac{1}{10}\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\log im}{{0}^{3} - {\log \frac{1}{10}}^{3}} \cdot \left(0 \cdot 0 + \left(\log \frac{1}{10} \cdot \log \frac{1}{10} + 0 \cdot \log \frac{1}{10}\right)\right)} \]

7. Applied rewrites 30.3%

   \[\leadsto \color{blue}{\frac{\log im}{-{\log 0.1}^{3}} \cdot {\log 0.1}^{2}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\log im}{-{\log \frac{1}{10}}^{3}} \cdot {\log \frac{1}{10}}^{2}} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\log im}{-{\log \frac{1}{10}}^{3}}} \cdot {\log \frac{1}{10}}^{2} \]

   3. div-inv N/A

      \[\leadsto \color{blue}{\left(\log im \cdot \frac{1}{-{\log \frac{1}{10}}^{3}}\right)} \cdot {\log \frac{1}{10}}^{2} \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\log im \cdot \left(\frac{1}{-{\log \frac{1}{10}}^{3}} \cdot {\log \frac{1}{10}}^{2}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\log im \cdot \left(\frac{1}{-{\log \frac{1}{10}}^{3}} \cdot {\log \frac{1}{10}}^{2}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \log im \cdot \color{blue}{\left(\frac{1}{-{\log \frac{1}{10}}^{3}} \cdot {\log \frac{1}{10}}^{2}\right)} \]

9. Applied rewrites 30.3%

   \[\leadsto \color{blue}{\log im \cdot \left(\left(-{\log 0.1}^{-3}\right) \cdot {\log 0.1}^{2}\right)} \]

10. Final simplification 30.3%

    \[\leadsto \left({\log 0.1}^{2} \cdot \left(-{\log 0.1}^{-3}\right)\right) \cdot \log im \]
11. Add Preprocessing

Alternative 2: 99.0% accurate, 0.7× speedup

Math

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

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 10.0) (log (hypot 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 1.0 / (log(10.0) / log(hypot(im_m, re_m)));
}

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(Math.hypot(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 1.0 / (math.log(10.0) / math.log(math.hypot(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(1.0 / Float64(log(10.0) / log(hypot(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 = 1.0 / (log(10.0) / log(hypot(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[(1.0 / N[(N[Log[10.0], $MachinePrecision] / N[Log[N[Sqrt[im$95$m ^ 2 + re$95$m ^ 2], $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. 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. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 57.4

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

   5. lift-sqrt.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-hypot.f64 99.0

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

4. Applied rewrites 99.0%

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

Alternative 3: 99.0% accurate, 0.7× speedup

Math

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

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 (hypot 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 -1.0 / (log(0.1) / log(hypot(im_m, re_m)));
}

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(0.1) / Math.log(Math.hypot(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 -1.0 / (math.log(0.1) / math.log(math.hypot(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(-1.0 / Float64(log(0.1) / log(hypot(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 = -1.0 / (log(0.1) / log(hypot(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[(-1.0 / N[(N[Log[0.1], $MachinePrecision] / N[Log[N[Sqrt[im$95$m ^ 2 + re$95$m ^ 2], $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. 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. clear-num N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 N/A

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

   6. distribute-neg-frac N/A

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

   7. lower-/.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. neg-log N/A

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

   10. lower-log.f64 N/A

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

   11. metadata-eval 57.4

       \[\leadsto \frac{-1}{\frac{\log \color{blue}{0.1}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}} \]

   12. lift-sqrt.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lower-hypot.f64 99.0

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

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\frac{-1}{\frac{\log 0.1}{\log \left(\mathsf{hypot}\left(im, re\right)\right)}}} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.8× speedup

Math

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

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 (hypot 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(hypot(re_m, im_m)) / log(10.0);
}

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(Math.hypot(re_m, 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(math.hypot(re_m, 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(hypot(re_m, 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(hypot(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[Sqrt[re$95$m ^ 2 + im$95$m ^ 2], $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. 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 99.0

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

4. Applied rewrites 99.0%

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

Alternative 5: 98.7% accurate, 0.9× speedup

Math

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

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
 (/
  (* 0.5 (fma (/ re_m im_m) (/ re_m im_m) (* (- -2.0) (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 (0.5 * fma((re_m / im_m), (re_m / im_m), (-(-2.0) * log(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(Float64(0.5 * fma(Float64(re_m / im_m), Float64(re_m / im_m), Float64(Float64(-(-2.0)) * 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[(0.5 * N[(N[(re$95$m / im$95$m), $MachinePrecision] * N[(re$95$m / im$95$m), $MachinePrecision] + N[((--2.0) * N[Log[im$95$m], $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. pow1/2 N/A

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

   4. pow-to-exp N/A

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

   5. rem-log-exp N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 57.4

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 57.4

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

4. Applied rewrites 57.4%

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

5. Taylor expanded in im around inf

   \[\leadsto \frac{\color{blue}{\left(-2 \cdot \log \left(\frac{1}{im}\right) + \frac{{re}^{2}}{{im}^{2}}\right)} \cdot \frac{1}{2}}{\log 10} \]
6. 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 28.9

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

7. Applied rewrites 28.9%

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

8. Final simplification 28.9%

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

Alternative 6: 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. frac-2neg N/A

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

   5. lift-log.f64 N/A

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

   6. neg-log N/A

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

   7. metadata-eval N/A

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

   8. lift-log.f64 N/A

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

   9. neg-mul-1 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. neg-mul-1 N/A

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

   16. lift-log.f64 N/A

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

   17. neg-log N/A

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

   18. metadata-eval N/A

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

   19. lift-log.f64 30.2

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

7. Applied rewrites 30.2%

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

Alternative 7: 98.2% accurate, 1.1× speedup

Math

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

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 0.1) (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(0.1) / 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(0.1d0) / 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(0.1) / 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(0.1) / 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(0.1) / 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(0.1) / 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[0.1], $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. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lift-log.f64 N/A

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

   6. metadata-eval N/A

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

   7. neg-log N/A

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

   8. lift-log.f64 N/A

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

   9. distribute-frac-neg N/A

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

   10. remove-double-neg N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 30.1

       \[\leadsto \frac{-1}{\color{blue}{\frac{\log 0.1}{\log im}}} \]

7. Applied rewrites 30.1%

   \[\leadsto \color{blue}{\frac{-1}{\frac{\log 0.1}{\log im}}} \]
8. Add Preprocessing

Alternative 8: 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 9: 3.0% accurate, 1.8× speedup

Math

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

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) -0.5) (* (* im_m 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) * -0.5) / ((im_m * 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) * (-0.5d0)) / ((im_m * 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) * -0.5) / ((im_m * 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) * -0.5) / ((im_m * 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) * -0.5) / Float64(Float64(im_m * 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) * -0.5) / ((im_m * 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] * -0.5), $MachinePrecision] / N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[Log[0.1], $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. frac-2neg N/A

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

   3. lift-log.f64 N/A

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

   4. neg-log N/A

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

   5. metadata-eval N/A

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

   6. lift-log.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-neg.f64 30.1

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

7. Applied rewrites 30.1%

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

8. Taylor expanded in re around 0

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. log-rec N/A

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

   4. +-commutative N/A

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

   5. associate-*r/ N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. unpow2 N/A

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

   12. unpow2 N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

10. Applied rewrites 28.8%

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

11. Taylor expanded in re around inf

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

13. Applied rewrites 2.9%

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

14. Final simplification 2.9%

    \[\leadsto \frac{\left(re \cdot re\right) \cdot -0.5}{\left(im \cdot im\right) \cdot \log 0.1} \]
15. 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)))