math.log10 on complex, real part

Percentage Accurate: 52.1% → 99.4%

Time: 9.8s

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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\log 10}^{-0.5} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (pow (log 10.0) -0.5) (* (/ 1.0 (sqrt (log 10.0))) (log (hypot re im)))))

C

double code(double re, double im) {
	return pow(log(10.0), -0.5) * ((1.0 / 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) * ((1.0 / 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) * ((1.0 / math.sqrt(math.log(10.0))) * math.log(math.hypot(re, im)))

Julia

function code(re, im)
	return Float64((log(10.0) ^ -0.5) * Float64(Float64(1.0 / sqrt(log(10.0))) * log(hypot(re, im))))
end

MATLAB

function tmp = code(re, im)
	tmp = (log(10.0) ^ -0.5) * ((1.0 / sqrt(log(10.0))) * log(hypot(re, im)));
end

Wolfram

code[re_, im_] := N[(N[Power[N[Log[10.0], $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(1.0 / N[Sqrt[N[Log[10.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

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

   1. +-commutative 54.7%

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

   2. +-commutative 54.7%

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

   3. sqr-neg 54.7%

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

   4. sqr-neg 54.7%

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

   5. sqr-neg 54.7%

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

   6. sqr-neg 54.7%

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

   7. hypot-define 99.2%

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

3. Simplified 99.2%

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

   1. clear-num 99.1%

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

   2. inv-pow 99.1%

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

   3. add-sqr-sqrt 99.1%

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

   4. associate-/l* 98.8%

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

   5. unpow-prod-down 99.3%

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

   6. inv-pow 99.3%

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

   7. pow1/2 99.3%

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

   8. pow-flip 99.3%

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

   9. metadata-eval 99.3%

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

6. Applied egg-rr 99.3%

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

   1. unpow-1 99.3%

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

   2. associate-/r/ 99.5%

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

8. Simplified 99.5%

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

Alternative 2: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

   1. +-commutative 54.7%

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

   2. +-commutative 54.7%

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

   3. sqr-neg 54.7%

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

   4. sqr-neg 54.7%

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

   5. sqr-neg 54.7%

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

   6. sqr-neg 54.7%

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

   7. hypot-define 99.2%

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

3. Simplified 99.2%

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

   1. *-un-lft-identity 99.2%

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

   2. add-sqr-sqrt 99.2%

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

   3. times-frac 99.3%

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

   4. pow1/2 99.3%

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

   5. pow-flip 99.3%

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

   6. metadata-eval 99.3%

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

6. Applied egg-rr 99.3%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot 3}{\log 0.1} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* -0.3333333333333333 (/ (* (log (hypot re im)) 3.0) (log 0.1))))

C

double code(double re, double im) {
	return -0.3333333333333333 * ((log(hypot(re, im)) * 3.0) / log(0.1));
}

Java

public static double code(double re, double im) {
	return -0.3333333333333333 * ((Math.log(Math.hypot(re, im)) * 3.0) / Math.log(0.1));
}

Python

def code(re, im):
	return -0.3333333333333333 * ((math.log(math.hypot(re, im)) * 3.0) / math.log(0.1))

Julia

function code(re, im)
	return Float64(-0.3333333333333333 * Float64(Float64(log(hypot(re, im)) * 3.0) / log(0.1)))
end

MATLAB

function tmp = code(re, im)
	tmp = -0.3333333333333333 * ((log(hypot(re, im)) * 3.0) / log(0.1));
end

Wolfram

code[re_, im_] := N[(-0.3333333333333333 * N[(N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision] / N[Log[0.1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

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

   1. +-commutative 54.7%

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

   2. +-commutative 54.7%

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

   3. sqr-neg 54.7%

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

   4. sqr-neg 54.7%

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

   5. sqr-neg 54.7%

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

   6. sqr-neg 54.7%

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

   7. hypot-define 99.2%

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

3. Simplified 99.2%

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

   1. clear-num 99.1%

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

   2. inv-pow 99.1%

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

   3. add-sqr-sqrt 99.1%

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

   4. associate-/l* 98.8%

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

   5. unpow-prod-down 99.3%

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

   6. inv-pow 99.3%

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

   7. pow1/2 99.3%

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

   8. pow-flip 99.3%

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

   9. metadata-eval 99.3%

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

6. Applied egg-rr 99.3%

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

   1. unpow-1 99.3%

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

   2. associate-/r/ 99.5%

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

8. Simplified 99.5%

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

   1. metadata-eval 99.5%

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

   2. sqrt-pow2 99.5%

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

   3. inv-pow 99.5%

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

   4. associate-*l/ 99.3%

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

   5. *-un-lft-identity 99.3%

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

   6. times-frac 99.2%

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

   7. *-un-lft-identity 99.2%

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

   8. add-sqr-sqrt 99.2%

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

   9. frac-2neg 99.2%

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

   10. neg-log 98.9%

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

   11. metadata-eval 98.9%

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

10. Applied egg-rr 98.9%

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

    1. add-cbrt-cube 36.8%

       \[\leadsto \frac{-\log \color{blue}{\left(\sqrt[3]{\left(\mathsf{hypot}\left(re, im\right) \cdot \mathsf{hypot}\left(re, im\right)\right) \cdot \mathsf{hypot}\left(re, im\right)}\right)}}{\log 0.1} \]

    2. pow1/3 37.0%

       \[\leadsto \frac{-\log \color{blue}{\left({\left(\left(\mathsf{hypot}\left(re, im\right) \cdot \mathsf{hypot}\left(re, im\right)\right) \cdot \mathsf{hypot}\left(re, im\right)\right)}^{0.3333333333333333}\right)}}{\log 0.1} \]

    3. log-pow 36.9%

       \[\leadsto \frac{-\color{blue}{0.3333333333333333 \cdot \log \left(\left(\mathsf{hypot}\left(re, im\right) \cdot \mathsf{hypot}\left(re, im\right)\right) \cdot \mathsf{hypot}\left(re, im\right)\right)}}{\log 0.1} \]

    4. pow3 36.9%

       \[\leadsto \frac{-0.3333333333333333 \cdot \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{3}\right)}}{\log 0.1} \]

    5. log-pow 99.1%

       \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)}}{\log 0.1} \]

12. Applied egg-rr 99.1%

    \[\leadsto \frac{-\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)}}{\log 0.1} \]

14. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{3 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 0.1}} \]

15. Final simplification 99.3%

    \[\leadsto -0.3333333333333333 \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot 3}{\log 0.1} \]
16. 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 54.7%

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

   1. +-commutative 54.7%

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

   2. +-commutative 54.7%

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

   3. sqr-neg 54.7%

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

   4. sqr-neg 54.7%

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

   5. sqr-neg 54.7%

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

   6. sqr-neg 54.7%

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

   7. hypot-define 99.2%

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

3. Simplified 99.2%

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

Alternative 5: 27.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(\frac{1}{im}\right)}{\log 10} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

   1. +-commutative 54.7%

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

   2. +-commutative 54.7%

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

   3. sqr-neg 54.7%

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

   4. sqr-neg 54.7%

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

   5. sqr-neg 54.7%

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

   6. sqr-neg 54.7%

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

   7. hypot-define 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in im around inf 24.7%

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

6. Final simplification 24.7%

   \[\leadsto \frac{-\log \left(\frac{1}{im}\right)}{\log 10} \]
7. Add Preprocessing

Alternative 6: 27.4% 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 54.7%

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

   1. +-commutative 54.7%

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

   2. +-commutative 54.7%

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

   3. sqr-neg 54.7%

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

   4. sqr-neg 54.7%

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

   5. sqr-neg 54.7%

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

   6. sqr-neg 54.7%

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

   7. hypot-define 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in re around 0 24.7%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))