math.log10 on complex, real part

Percentage Accurate: 50.0% → 99.0%

Time: 9.2s

Alternatives: 5

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: 50.0% 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

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \log \left({im_m}^{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\log 10}\right)\right)\right)}\right) \end{array} \]

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 (log1p (expm1 (/ 1.0 (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(pow(im_m, log1p(expm1((1.0 / 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.pow(im_m, Math.log1p(Math.expm1((1.0 / 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.pow(im_m, math.log1p(math.expm1((1.0 / 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 log((im_m ^ log1p(expm1(Float64(1.0 / 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[Log[N[Power[im$95$m, N[Log[1 + N[(Exp[N[(1.0 / N[Log[10.0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 54.4%

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

   1. +-commutative 54.4%

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

   2. +-commutative 54.4%

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

   3. sqr-neg 54.4%

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

   4. sqr-neg 54.4%

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

   5. +-commutative 54.4%

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

   6. +-commutative 54.4%

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

   7. hypot-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in re around 0 35.9%

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

   1. add-log-exp 35.9%

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

   2. div-inv 35.7%

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

   3. exp-to-pow 35.8%

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

   4. frac-2neg 35.8%

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

   5. metadata-eval 35.8%

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

   6. neg-log 35.9%

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

   7. metadata-eval 35.9%

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

6. Applied egg-rr 35.9%

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

   1. metadata-eval 35.9%

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

   2. metadata-eval 35.9%

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

   3. neg-log 35.8%

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

   4. frac-2neg 35.8%

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

   5. log1p-expm1-u 36.2%

      \[\leadsto \log \left({im}^{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\log 10}\right)\right)\right)}}\right) \]

8. Applied egg-rr 36.2%

   \[\leadsto \log \left({im}^{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\log 10}\right)\right)\right)}}\right) \]

9. Final simplification 36.2%

   \[\leadsto \log \left({im}^{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\log 10}\right)\right)\right)}\right) \]

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

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

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

Derivation

1. Initial program 54.4%

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

   1. +-commutative 54.4%

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

   2. +-commutative 54.4%

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

   3. sqr-neg 54.4%

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

   4. sqr-neg 54.4%

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

   5. +-commutative 54.4%

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

   6. +-commutative 54.4%

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

   7. hypot-def 99.1%

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

3. Simplified 99.1%

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

   1. *-un-lft-identity 99.1%

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

   2. add-sqr-sqrt 99.1%

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

   3. times-frac 99.1%

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

   4. pow1/2 99.1%

      \[\leadsto \frac{1}{\color{blue}{{\log 10}^{0.5}}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}} \]

   5. pow-flip 99.1%

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

   6. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

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

   1. metadata-eval 99.1%

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

   2. pow-flip 99.1%

      \[\leadsto \color{blue}{\frac{1}{{\log 10}^{0.5}}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}} \]

   3. pow1/2 99.1%

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

   4. times-frac 99.1%

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

   5. add-sqr-sqrt 99.1%

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

   6. associate-/l* 99.1%

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

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

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

Alternative 3: 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 54.4%

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

   1. +-commutative 54.4%

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

   2. +-commutative 54.4%

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

   3. sqr-neg 54.4%

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

   4. sqr-neg 54.4%

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

   5. +-commutative 54.4%

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

   6. +-commutative 54.4%

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

   7. hypot-def 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 4: 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(Float64(-log(im_m)) / 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 54.4%

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

   1. +-commutative 54.4%

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

   2. +-commutative 54.4%

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

   3. sqr-neg 54.4%

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

   4. sqr-neg 54.4%

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

   5. +-commutative 54.4%

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

   6. +-commutative 54.4%

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

   7. hypot-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in re around 0 35.9%

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

   1. div-inv 35.7%

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

   2. frac-2neg 35.7%

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

   3. metadata-eval 35.7%

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

   4. neg-log 35.9%

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

   5. metadata-eval 35.9%

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

6. Applied egg-rr 35.9%

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

   1. *-commutative 35.9%

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

   2. associate-*l/ 35.9%

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

   3. neg-mul-1 35.9%

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

8. Simplified 35.9%

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

9. Final simplification 35.9%

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

Alternative 5: 98.3% accurate, 1.5× speedup

Math

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

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 54.4%

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

   1. +-commutative 54.4%

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

   2. +-commutative 54.4%

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

   3. sqr-neg 54.4%

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

   4. sqr-neg 54.4%

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

   5. +-commutative 54.4%

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

   6. +-commutative 54.4%

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

   7. hypot-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in re around 0 35.9%

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

5. Final simplification 35.9%

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

Reproduce

herbie shell --seed 2023333 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))