math.log10 on complex, real part

Percentage Accurate: 51.3% → 99.0%

Time: 8.2s

Alternatives: 7

Speedup: 1.5×

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: 51.3% 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: 99.0% accurate, 0.6× speedup

Math

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

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (log (pow im_m (sqrt (pow (log 10.0) -2.0)))))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return log(pow(im_m, sqrt(pow(log(10.0), -2.0))));
}

Fortran

re_m = abs(re)
im_m = abs(im)
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 ** sqrt((log(10.0d0) ** (-2.0d0)))))
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return Math.log(Math.pow(im_m, Math.sqrt(Math.pow(Math.log(10.0), -2.0))));
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return math.log(math.pow(im_m, math.sqrt(math.pow(math.log(10.0), -2.0))))

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return log((im_m ^ sqrt((log(10.0) ^ -2.0))))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = log((im_m ^ sqrt((log(10.0) ^ -2.0))));
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $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[Log[N[Power[im$95$m, N[Sqrt[N[Power[N[Log[10.0], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 51.9%

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

   1. +-commutative 51.9%

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

   2. +-commutative 51.9%

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

   3. sqr-neg 51.9%

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

   4. sqr-neg 51.9%

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

   5. sqr-neg 51.9%

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

   6. sqr-neg 51.9%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in re around 0 30.6%

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

   1. add-log-exp 30.6%

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

   2. div-inv 30.4%

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

   3. exp-to-pow 30.4%

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

   4. frac-2neg 30.4%

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

   5. metadata-eval 30.4%

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

   6. neg-log 30.6%

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

   7. metadata-eval 30.6%

      \[\leadsto \log \left({im}^{\left(\frac{-1}{\log \color{blue}{0.1}}\right)}\right) \]

7. Applied egg-rr 30.6%

   \[\leadsto \color{blue}{\log \left({im}^{\left(\frac{-1}{\log 0.1}\right)}\right)} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 30.8%

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

   2. pow2 30.8%

      \[\leadsto \log \left({im}^{\color{blue}{\left({\left(\sqrt{\frac{-1}{\log 0.1}}\right)}^{2}\right)}}\right) \]

9. Applied egg-rr 30.8%

   \[\leadsto \log \left({im}^{\color{blue}{\left({\left(\sqrt{\frac{-1}{\log 0.1}}\right)}^{2}\right)}}\right) \]
10. Step-by-step derivation

    1. unpow2 30.8%

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

    2. add-sqr-sqrt 30.6%

       \[\leadsto \log \left({im}^{\color{blue}{\left(\frac{-1}{\log 0.1}\right)}}\right) \]

    3. frac-2neg 30.6%

       \[\leadsto \log \left({im}^{\color{blue}{\left(\frac{--1}{-\log 0.1}\right)}}\right) \]

    4. metadata-eval 30.6%

       \[\leadsto \log \left({im}^{\left(\frac{\color{blue}{1}}{-\log 0.1}\right)}\right) \]

    5. neg-log 30.4%

       \[\leadsto \log \left({im}^{\left(\frac{1}{\color{blue}{\log \left(\frac{1}{0.1}\right)}}\right)}\right) \]

    6. metadata-eval 30.4%

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

    7. add-sqr-sqrt 30.8%

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

    8. sqrt-unprod 30.4%

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

    9. inv-pow 30.4%

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

    10. inv-pow 30.4%

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

    11. pow-prod-up 30.8%

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

    12. metadata-eval 30.8%

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

11. Applied egg-rr 30.8%

    \[\leadsto \log \left({im}^{\color{blue}{\left(\sqrt{{\log 10}^{-2}}\right)}}\right) \]
12. Add Preprocessing

Alternative 2: 99.1% accurate, 0.8× speedup

Math

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

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
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 10.0) (log (hypot re_m im_m))) -1.0))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $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[Power[N[(N[Log[10.0], $MachinePrecision] / N[Log[N[Sqrt[re$95$m ^ 2 + im$95$m ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 51.9%

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

   1. +-commutative 51.9%

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

   2. +-commutative 51.9%

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

   3. sqr-neg 51.9%

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

   4. sqr-neg 51.9%

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

   5. sqr-neg 51.9%

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

   6. sqr-neg 51.9%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

   1. clear-num 99.1%

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

   2. inv-pow 99.1%

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

6. Applied egg-rr 99.1%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
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 0.1))))

C

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

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
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(0.1);
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[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(0.1)

Julia

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

MATLAB

re_m = abs(re);
im_m = abs(im);
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(0.1);
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $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[0.1], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 51.9%

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

   1. +-commutative 51.9%

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

   2. +-commutative 51.9%

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

   3. sqr-neg 51.9%

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

   4. sqr-neg 51.9%

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

   5. sqr-neg 51.9%

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

   6. sqr-neg 51.9%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

   1. div-inv 98.5%

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

   2. frac-2neg 98.5%

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

   3. metadata-eval 98.5%

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

   4. neg-log 99.0%

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

   5. metadata-eval 99.0%

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

6. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{-1}{\log 0.1}} \]
7. Step-by-step derivation

   1. *-commutative 99.0%

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

   2. associate-*l/ 99.0%

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

   3. neg-mul-1 99.0%

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

   4. distribute-neg-frac 99.0%

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

   5. distribute-neg-frac2 99.0%

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

8. Simplified 99.0%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

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

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
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

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

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
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

re_m = math.fabs(re)
im_m = math.fabs(im)
[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

re_m = abs(re)
im_m = abs(im)
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

re_m = abs(re);
im_m = abs(im);
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

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $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 51.9%

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

   1. +-commutative 51.9%

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

   2. +-commutative 51.9%

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

   3. sqr-neg 51.9%

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

   4. sqr-neg 51.9%

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

   5. sqr-neg 51.9%

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

   6. sqr-neg 51.9%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

Alternative 5: 98.3% accurate, 1.5× speedup

Math

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

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
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

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

Fortran

re_m = abs(re)
im_m = abs(im)
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

re_m = Math.abs(re);
im_m = Math.abs(im);
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

re_m = math.fabs(re)
im_m = math.fabs(im)
[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

re_m = abs(re)
im_m = abs(im)
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

re_m = abs(re);
im_m = abs(im);
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

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $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 51.9%

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

   1. +-commutative 51.9%

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

   2. +-commutative 51.9%

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

   3. sqr-neg 51.9%

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

   4. sqr-neg 51.9%

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

   5. sqr-neg 51.9%

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

   6. sqr-neg 51.9%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in re around 0 30.6%

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

   1. clear-num 30.6%

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

   2. inv-pow 30.6%

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

7. Applied egg-rr 30.6%

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

   1. unpow-1 30.6%

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

9. Simplified 30.6%

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

Alternative 6: 98.3% accurate, 1.5× speedup

Math

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

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
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 0.1))))

C

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

Fortran

re_m = abs(re)
im_m = abs(im)
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(0.1d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $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[0.1], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 51.9%

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

   1. +-commutative 51.9%

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

   2. +-commutative 51.9%

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

   3. sqr-neg 51.9%

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

   4. sqr-neg 51.9%

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

   5. sqr-neg 51.9%

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

   6. sqr-neg 51.9%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in re around 0 30.6%

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

   1. div-inv 30.4%

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

   2. frac-2neg 30.4%

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

   3. metadata-eval 30.4%

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

   4. neg-log 30.5%

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

   5. metadata-eval 30.5%

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

7. Applied egg-rr 30.5%

   \[\leadsto \color{blue}{\log im \cdot \frac{-1}{\log 0.1}} \]
8. Step-by-step derivation

   1. *-commutative 30.5%

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

   2. associate-*l/ 30.6%

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

   3. neg-mul-1 30.6%

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

   4. distribute-neg-frac 30.6%

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

   5. distribute-neg-frac2 30.6%

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

9. Simplified 30.6%

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

Alternative 7: 98.4% accurate, 1.5× speedup

Math

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

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
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

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

Fortran

re_m = abs(re)
im_m = abs(im)
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

re_m = Math.abs(re);
im_m = Math.abs(im);
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

re_m = math.fabs(re)
im_m = math.fabs(im)
[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

re_m = abs(re)
im_m = abs(im)
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

re_m = abs(re);
im_m = abs(im);
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

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $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 51.9%

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

   1. +-commutative 51.9%

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

   2. +-commutative 51.9%

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

   3. sqr-neg 51.9%

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

   4. sqr-neg 51.9%

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

   5. sqr-neg 51.9%

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

   6. sqr-neg 51.9%

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

   7. hypot-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in re around 0 30.6%

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

Reproduce

herbie shell --seed 2024111 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))