math.log10 on complex, real part

Percentage Accurate: 52.1% → 99.5%

Time: 16.2s

Alternatives: 6

Speedup: 1.0×

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: 52.1% 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: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (* (/ (pow (log 10.0) -0.5) (sqrt (log 10.0))) (log (hypot re im))))

C

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

Java

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

Python

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

Julia

function code(re, im)
	return Float64(Float64((log(10.0) ^ -0.5) / sqrt(log(10.0))) * log(hypot(re, im)))
end

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(N[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision] / N[Sqrt[N[Log[10.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.3%

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

   1. +-commutative 50.3%

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

   2. +-commutative 50.3%

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

   3. sqr-neg 50.3%

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

   4. sqr-neg 50.3%

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

   5. sqr-neg 50.3%

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

   6. sqr-neg 50.3%

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

   7. hypot-def 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. Step-by-step derivation

   1. *-un-lft-identity 99.0%

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

   2. add-sqr-sqrt 99.0%

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

   3. times-frac 99.1%

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

5. Applied egg-rr 99.1%

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

   1. clear-num 99.0%

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

   2. un-div-inv 99.1%

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

   3. pow1/2 99.1%

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

   4. pow-flip 99.1%

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

   5. metadata-eval 99.1%

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

7. Applied egg-rr 99.1%

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

   1. associate-/r/ 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 2: 27.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \cdot \log im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

   1. +-commutative 50.3%

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

   2. +-commutative 50.3%

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

   3. sqr-neg 50.3%

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

   4. sqr-neg 50.3%

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

   5. sqr-neg 50.3%

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

   6. sqr-neg 50.3%

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

   7. hypot-def 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. Step-by-step derivation

   1. *-un-lft-identity 99.0%

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

   2. add-sqr-sqrt 99.0%

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

   3. times-frac 99.1%

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

5. Applied egg-rr 99.1%

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

6. Taylor expanded in re around 0 25.9%

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

   1. associate-*l/ 25.9%

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

   2. *-un-lft-identity 25.9%

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

   3. associate-/l* 25.8%

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

   4. metadata-eval 25.8%

      \[\leadsto \frac{\sqrt{\frac{1}{\log \color{blue}{\left(1 + 9\right)}}}}{\frac{\sqrt{\log 10}}{\log im}} \]

   5. log1p-udef 25.8%

      \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{log1p}\left(9\right)}}}}{\frac{\sqrt{\log 10}}{\log im}} \]

   6. inv-pow 25.8%

      \[\leadsto \frac{\sqrt{\color{blue}{{\left(\mathsf{log1p}\left(9\right)\right)}^{-1}}}}{\frac{\sqrt{\log 10}}{\log im}} \]

   7. sqrt-pow1 25.8%

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{log1p}\left(9\right)\right)}^{\left(\frac{-1}{2}\right)}}}{\frac{\sqrt{\log 10}}{\log im}} \]

   8. log1p-udef 25.8%

      \[\leadsto \frac{{\color{blue}{\log \left(1 + 9\right)}}^{\left(\frac{-1}{2}\right)}}{\frac{\sqrt{\log 10}}{\log im}} \]

   9. metadata-eval 25.8%

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

   10. metadata-eval 25.8%

       \[\leadsto \frac{{\log 10}^{\color{blue}{-0.5}}}{\frac{\sqrt{\log 10}}{\log im}} \]

8. Applied egg-rr 25.8%

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

   1. associate-/r/ 25.9%

      \[\leadsto \color{blue}{\frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \cdot \log im} \]

10. Simplified 25.9%

    \[\leadsto \color{blue}{\frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \cdot \log im} \]

11. Final simplification 25.9%

    \[\leadsto \frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \cdot \log im \]

Alternative 3: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[Power[N[(N[Log[10.0], $MachinePrecision] / N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 50.3%

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

   1. +-commutative 50.3%

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

   2. +-commutative 50.3%

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

   3. sqr-neg 50.3%

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

   4. sqr-neg 50.3%

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

   5. sqr-neg 50.3%

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

   6. sqr-neg 50.3%

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

   7. hypot-def 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. Step-by-step derivation

   1. clear-num 99.0%

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

   2. inv-pow 99.0%

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

5. Applied egg-rr 99.0%

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

6. Final simplification 99.0%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.3%

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

   1. +-commutative 50.3%

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

   2. +-commutative 50.3%

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

   3. sqr-neg 50.3%

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

   4. sqr-neg 50.3%

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

   5. sqr-neg 50.3%

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

   6. sqr-neg 50.3%

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

   7. hypot-def 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. Final simplification 99.0%

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

Alternative 5: 27.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

   1. +-commutative 50.3%

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

   2. +-commutative 50.3%

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

   3. sqr-neg 50.3%

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

   4. sqr-neg 50.3%

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

   5. sqr-neg 50.3%

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

   6. sqr-neg 50.3%

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

   7. hypot-def 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. Taylor expanded in re around 0 25.8%

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

   1. clear-num 25.8%

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

   2. inv-pow 25.8%

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

6. Applied egg-rr 25.8%

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

   1. unpow-1 25.8%

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

8. Simplified 25.8%

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

9. Final simplification 25.8%

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

Alternative 6: 27.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

   1. +-commutative 50.3%

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

   2. +-commutative 50.3%

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

   3. sqr-neg 50.3%

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

   4. sqr-neg 50.3%

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

   5. sqr-neg 50.3%

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

   6. sqr-neg 50.3%

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

   7. hypot-def 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. Taylor expanded in re around 0 25.8%

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

5. Final simplification 25.8%

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

Reproduce

herbie shell --seed 2023283 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))