math.log10 on complex, real part

Percentage Accurate: 51.1% → 99.5%

Time: 10.1s

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: 51.1% 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 53.0%

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

   1. +-commutative 53.0%

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

   2. +-commutative 53.0%

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

   3. sqr-neg 53.0%

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

   4. sqr-neg 53.0%

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

   5. +-commutative 53.0%

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

   6. +-commutative 53.0%

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

   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. Add Preprocessing
5. 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%

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

   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%

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

6. Applied egg-rr 99.1%

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

   1. clear-num 99.0%

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

   2. un-div-inv 99.1%

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

8. Applied egg-rr 99.1%

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

   1. associate-/r/ 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 2: 27.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \log \left({im}^{\left({\left({\log 10}^{-0.5}\right)}^{2}\right)}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (log (pow im (pow (pow (log 10.0) -0.5) 2.0))))

C

double code(double re, double im) {
	return log(pow(im, pow(pow(log(10.0), -0.5), 2.0)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log((im ** ((log(10.0d0) ** (-0.5d0)) ** 2.0d0)))
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.pow(im, Math.pow(Math.pow(Math.log(10.0), -0.5), 2.0)));
}

Python

def code(re, im):
	return math.log(math.pow(im, math.pow(math.pow(math.log(10.0), -0.5), 2.0)))

Julia

function code(re, im)
	return log((im ^ ((log(10.0) ^ -0.5) ^ 2.0)))
end

MATLAB

function tmp = code(re, im)
	tmp = log((im ^ ((log(10.0) ^ -0.5) ^ 2.0)));
end

Wolfram

code[re_, im_] := N[Log[N[Power[im, N[Power[N[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.0%

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

   1. +-commutative 53.0%

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

   2. +-commutative 53.0%

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

   3. sqr-neg 53.0%

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

   4. sqr-neg 53.0%

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

   5. +-commutative 53.0%

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

   6. +-commutative 53.0%

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

   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. Add Preprocessing

5. Taylor expanded in re around 0 24.5%

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

   1. add-log-exp 24.5%

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

   2. div-inv 24.4%

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

   3. exp-to-pow 24.4%

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

   4. frac-2neg 24.4%

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

   5. metadata-eval 24.4%

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

   6. neg-log 24.5%

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

   7. metadata-eval 24.5%

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

7. Applied egg-rr 24.5%

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

   1. metadata-eval 24.5%

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

   2. metadata-eval 24.5%

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

   3. neg-log 24.4%

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

   4. frac-2neg 24.4%

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

   5. inv-pow 24.4%

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

   6. metadata-eval 24.4%

      \[\leadsto \log \left({im}^{\left({\log 10}^{\color{blue}{\left(-0.5 + -0.5\right)}}\right)}\right) \]

   7. pow-prod-up 24.7%

      \[\leadsto \log \left({im}^{\color{blue}{\left({\log 10}^{-0.5} \cdot {\log 10}^{-0.5}\right)}}\right) \]

   8. pow2 24.7%

      \[\leadsto \log \left({im}^{\color{blue}{\left({\left({\log 10}^{-0.5}\right)}^{2}\right)}}\right) \]

9. Applied egg-rr 24.7%

   \[\leadsto \log \left({im}^{\color{blue}{\left({\left({\log 10}^{-0.5}\right)}^{2}\right)}}\right) \]

10. Final simplification 24.7%

    \[\leadsto \log \left({im}^{\left({\left({\log 10}^{-0.5}\right)}^{2}\right)}\right) \]
11. Add Preprocessing

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 53.0%

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

   1. +-commutative 53.0%

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

   2. +-commutative 53.0%

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

   3. sqr-neg 53.0%

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

   4. sqr-neg 53.0%

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

   5. +-commutative 53.0%

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

   6. +-commutative 53.0%

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

   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. Add Preprocessing
5. Step-by-step derivation

   1. frac-2neg 99.0%

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

   2. div-inv 98.5%

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

   3. neg-log 99.1%

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

   4. metadata-eval 99.1%

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

6. Applied egg-rr 99.1%

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

   1. associate-*r/ 99.1%

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

   2. *-rgt-identity 99.1%

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

8. Simplified 99.1%

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

9. Final simplification 99.1%

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

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 53.0%

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

   1. +-commutative 53.0%

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

   2. +-commutative 53.0%

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

   3. sqr-neg 53.0%

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

   4. sqr-neg 53.0%

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

   5. +-commutative 53.0%

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

   6. +-commutative 53.0%

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

   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. Add Preprocessing

5. Final simplification 99.0%

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

Alternative 5: 27.6% 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 53.0%

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

   1. +-commutative 53.0%

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

   2. +-commutative 53.0%

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

   3. sqr-neg 53.0%

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

   4. sqr-neg 53.0%

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

   5. +-commutative 53.0%

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

   6. +-commutative 53.0%

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

   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. Add Preprocessing

5. Taylor expanded in re around 0 24.5%

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

   1. div-inv 24.4%

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

   2. frac-2neg 24.4%

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

   3. metadata-eval 24.4%

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

   4. neg-log 24.5%

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

   5. metadata-eval 24.5%

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

7. Applied egg-rr 24.5%

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

   1. *-commutative 24.5%

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

   2. associate-*l/ 24.5%

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

   3. neg-mul-1 24.5%

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

9. Simplified 24.5%

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

10. Final simplification 24.5%

    \[\leadsto \frac{-\log im}{\log 0.1} \]
11. Add Preprocessing

Alternative 6: 27.6% 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 53.0%

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

   1. +-commutative 53.0%

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

   2. +-commutative 53.0%

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

   3. sqr-neg 53.0%

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

   4. sqr-neg 53.0%

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

   5. +-commutative 53.0%

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

   6. +-commutative 53.0%

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

   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. Add Preprocessing

5. Taylor expanded in re around 0 24.5%

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

6. Final simplification 24.5%

   \[\leadsto \frac{\log im}{\log 10} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))