math.log10 on complex, real part

Percentage Accurate: 50.5% → 99.5%

Time: 11.2s

Alternatives: 7

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: 50.5% 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 52.2%

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

   1. sqr-neg 52.2%

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

   2. sqr-neg 52.2%

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

   3. sqr-neg 52.2%

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

   4. sqr-neg 52.2%

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

   5. +-commutative 52.2%

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

   6. +-commutative 52.2%

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

   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. add-sqr-sqrt 99.1%

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

   2. associate-/r* 98.6%

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

   3. div-inv 99.2%

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

   4. metadata-eval 99.2%

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

   5. sqrt-div 99.2%

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

   6. pow1 99.2%

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

   7. pow-flip 99.2%

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

   8. sqrt-pow1 99.2%

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

   9. metadata-eval 99.2%

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

   10. metadata-eval 99.2%

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

5. Applied egg-rr 99.2%

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

   1. *-commutative 99.2%

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

   2. associate-*r/ 99.1%

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

   3. associate-/l* 99.2%

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

   4. associate-/r/ 99.6%

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

7. Simplified 99.6%

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

8. 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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

   1. sqr-neg 52.2%

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

   2. sqr-neg 52.2%

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

   3. sqr-neg 52.2%

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

   4. sqr-neg 52.2%

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

   5. +-commutative 52.2%

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

   6. +-commutative 52.2%

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

   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. add-sqr-sqrt 99.1%

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

   2. associate-/r* 98.6%

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

   3. div-inv 99.2%

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

   4. metadata-eval 99.2%

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

   5. sqrt-div 99.2%

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

   6. pow1 99.2%

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

   7. pow-flip 99.2%

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

   8. sqrt-pow1 99.2%

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

   9. metadata-eval 99.2%

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

   10. metadata-eval 99.2%

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

5. Applied egg-rr 99.2%

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

   1. *-commutative 99.2%

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

   2. associate-*r/ 99.1%

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

   3. associate-/l* 99.2%

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

   4. associate-/r/ 99.6%

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

7. Simplified 99.6%

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

   1. associate-*l/ 99.1%

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

   2. clear-num 99.1%

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

   3. *-commutative 99.1%

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

   4. associate-/l/ 99.1%

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

   5. pow1/2 99.1%

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

   6. pow-div 99.1%

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

   7. metadata-eval 99.1%

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

   8. pow1 99.1%

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

9. Applied egg-rr 99.1%

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

10. Final simplification 99.1%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 0.1} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (- (/ (log (hypot re im)) (log 0.1))))

C

double code(double re, double im) {
	return -(log(hypot(re, im)) / log(0.1));
}

Java

public static double code(double re, double im) {
	return -(Math.log(Math.hypot(re, im)) / Math.log(0.1));
}

Python

def code(re, im):
	return -(math.log(math.hypot(re, im)) / math.log(0.1))

Julia

function code(re, im)
	return Float64(-Float64(log(hypot(re, im)) / log(0.1)))
end

MATLAB

function tmp = code(re, im)
	tmp = -(log(hypot(re, im)) / log(0.1));
end

Wolfram

code[re_, im_] := (-N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[0.1], $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 52.2%

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

   1. sqr-neg 52.2%

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

   2. sqr-neg 52.2%

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

   3. sqr-neg 52.2%

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

   4. sqr-neg 52.2%

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

   5. +-commutative 52.2%

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

   6. +-commutative 52.2%

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

   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. add-sqr-sqrt 99.1%

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

   2. associate-/r* 98.6%

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

   3. div-inv 99.2%

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

   4. metadata-eval 99.2%

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

   5. sqrt-div 99.2%

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

   6. pow1 99.2%

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

   7. pow-flip 99.2%

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

   8. sqrt-pow1 99.2%

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

   9. metadata-eval 99.2%

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

   10. metadata-eval 99.2%

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

5. Applied egg-rr 99.2%

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

   1. *-commutative 99.2%

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

   2. associate-*r/ 99.1%

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

   3. associate-/l* 99.2%

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

   4. associate-/r/ 99.6%

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

7. Simplified 99.6%

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

   1. *-commutative 99.6%

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

   2. clear-num 98.6%

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

   3. un-div-inv 99.1%

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

   4. pow1/2 99.1%

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

   5. pow-div 99.1%

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

   6. metadata-eval 99.1%

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

   7. pow1 99.1%

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

   8. frac-2neg 99.1%

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

   9. add-log-exp 99.1%

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

   10. neg-log 99.1%

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

   11. rem-exp-log 99.1%

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

   12. metadata-eval 99.1%

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

9. Applied egg-rr 99.1%

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

10. Final simplification 99.1%

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

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 52.2%

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

   1. sqr-neg 52.2%

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

   2. sqr-neg 52.2%

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

   3. sqr-neg 52.2%

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

   4. sqr-neg 52.2%

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

   5. +-commutative 52.2%

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

   6. +-commutative 52.2%

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

   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 5: 27.1% 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 52.2%

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

   1. sqr-neg 52.2%

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

   2. sqr-neg 52.2%

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

   3. sqr-neg 52.2%

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

   4. sqr-neg 52.2%

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

   5. +-commutative 52.2%

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

   6. +-commutative 52.2%

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

   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 28.6%

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

   1. clear-num 28.5%

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

   2. inv-pow 28.5%

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

6. Applied egg-rr 28.5%

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

   1. unpow-1 28.5%

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

8. Simplified 28.5%

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

9. Final simplification 28.5%

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

Alternative 6: 27.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ -\frac{\log im}{\log 0.1} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (- (/ (log im) (log 0.1))))

C

double code(double re, double im) {
	return -(log(im) / log(0.1));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -(log(im) / log(0.1d0))
end function

Java

public static double code(double re, double im) {
	return -(Math.log(im) / Math.log(0.1));
}

Python

def code(re, im):
	return -(math.log(im) / math.log(0.1))

Julia

function code(re, im)
	return Float64(-Float64(log(im) / log(0.1)))
end

MATLAB

function tmp = code(re, im)
	tmp = -(log(im) / log(0.1));
end

Wolfram

code[re_, im_] := (-N[(N[Log[im], $MachinePrecision] / N[Log[0.1], $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 52.2%

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

   1. sqr-neg 52.2%

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

   2. sqr-neg 52.2%

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

   3. sqr-neg 52.2%

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

   4. sqr-neg 52.2%

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

   5. +-commutative 52.2%

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

   6. +-commutative 52.2%

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

   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 28.6%

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

   1. frac-2neg 28.6%

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

   2. distribute-frac-neg 28.6%

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

   3. add-log-exp 28.6%

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

   4. neg-log 28.6%

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

   5. rem-exp-log 28.7%

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

   6. metadata-eval 28.7%

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

6. Applied egg-rr 28.7%

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

   1. distribute-neg-frac 28.7%

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

8. Simplified 28.7%

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

9. Final simplification 28.7%

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

Alternative 7: 27.1% 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 52.2%

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

   1. sqr-neg 52.2%

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

   2. sqr-neg 52.2%

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

   3. sqr-neg 52.2%

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

   4. sqr-neg 52.2%

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

   5. +-commutative 52.2%

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

   6. +-commutative 52.2%

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

   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 28.6%

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

5. Final simplification 28.6%

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

Reproduce

herbie shell --seed 2023305 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))