math.log10 on complex, real part

Percentage Accurate: 51.5% → 99.6%

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: 51.5% 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. +-commutative 51.7%

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

   2. +-commutative 51.7%

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

   3. sqr-neg 51.7%

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

   4. sqr-neg 51.7%

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

   5. sqr-neg 51.7%

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

   6. sqr-neg 51.7%

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

   7. hypot-define 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. Add Preprocessing
5. 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%

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

6. Applied egg-rr 99.0%

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

   1. unpow-1 99.0%

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

   2. clear-num 99.1%

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

   3. frac-2neg 99.1%

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

   4. neg-log 99.0%

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

   5. metadata-eval 99.0%

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

8. Applied egg-rr 99.0%

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

   1. neg-mul-1 99.0%

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

   2. *-commutative 99.0%

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

   3. metadata-eval 99.0%

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

   4. associate-*l* 99.1%

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

   5. *-commutative 99.1%

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

   6. associate-*r/ 99.4%

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

   7. *-commutative 99.4%

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

   8. associate-*r* 99.6%

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

10. Applied egg-rr 99.6%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. +-commutative 51.7%

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

   2. +-commutative 51.7%

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

   3. sqr-neg 51.7%

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

   4. sqr-neg 51.7%

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

   5. sqr-neg 51.7%

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

   6. sqr-neg 51.7%

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

   7. hypot-define 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. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 99.1%

      \[\leadsto \frac{\log \color{blue}{\left(\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}}{\log 10} \]

   2. pow3 99.1%

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}^{3}\right)}}{\log 10} \]

   3. log-pow 99.1%

      \[\leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}}{\log 10} \]

6. Applied egg-rr 99.1%

   \[\leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}}{\log 10} \]
7. Step-by-step derivation

   1. div-inv 98.6%

      \[\leadsto \color{blue}{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)\right) \cdot \frac{1}{\log 10}} \]

   2. *-commutative 98.6%

      \[\leadsto \color{blue}{\left(\log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot 3\right)} \cdot \frac{1}{\log 10} \]

   3. associate-*l* 98.9%

      \[\leadsto \color{blue}{\log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \left(3 \cdot \frac{1}{\log 10}\right)} \]

   4. frac-2neg 98.9%

      \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \left(3 \cdot \color{blue}{\frac{-1}{-\log 10}}\right) \]

   5. metadata-eval 98.9%

      \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \left(3 \cdot \frac{\color{blue}{-1}}{-\log 10}\right) \]

   6. neg-log 99.4%

      \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \left(3 \cdot \frac{-1}{\color{blue}{\log \left(\frac{1}{10}\right)}}\right) \]

   7. metadata-eval 99.4%

      \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \left(3 \cdot \frac{-1}{\log \color{blue}{0.1}}\right) \]

8. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \left(3 \cdot \frac{-1}{\log 0.1}\right)} \]
9. Step-by-step derivation

   1. associate-*r/ 99.4%

      \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \color{blue}{\frac{3 \cdot -1}{\log 0.1}} \]

   2. metadata-eval 99.4%

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

10. Simplified 99.4%

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

    1. pow1/3 99.6%

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

    2. log-pow 99.4%

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

12. Applied egg-rr 99.4%

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

13. Final simplification 99.4%

    \[\leadsto \frac{-3}{\log 0.1} \cdot \left(0.3333333333333333 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \]
14. Add Preprocessing

Alternative 3: 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 51.7%

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

   1. +-commutative 51.7%

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

   2. +-commutative 51.7%

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

   3. sqr-neg 51.7%

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

   4. sqr-neg 51.7%

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

   5. sqr-neg 51.7%

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

   6. sqr-neg 51.7%

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

   7. hypot-define 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. Add Preprocessing
5. Add Preprocessing

Alternative 4: 26.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((-3.0d0) / log(0.1d0)) * (0.3333333333333333d0 * log(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. +-commutative 51.7%

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

   2. +-commutative 51.7%

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

   3. sqr-neg 51.7%

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

   4. sqr-neg 51.7%

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

   5. sqr-neg 51.7%

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

   6. sqr-neg 51.7%

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

   7. hypot-define 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. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 99.1%

      \[\leadsto \frac{\log \color{blue}{\left(\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}}{\log 10} \]

   2. pow3 99.1%

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}^{3}\right)}}{\log 10} \]

   3. log-pow 99.1%

      \[\leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}}{\log 10} \]

6. Applied egg-rr 99.1%

   \[\leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}}{\log 10} \]
7. Step-by-step derivation

   1. div-inv 98.6%

      \[\leadsto \color{blue}{\left(3 \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)\right) \cdot \frac{1}{\log 10}} \]

   2. *-commutative 98.6%

      \[\leadsto \color{blue}{\left(\log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot 3\right)} \cdot \frac{1}{\log 10} \]

   3. associate-*l* 98.9%

      \[\leadsto \color{blue}{\log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \left(3 \cdot \frac{1}{\log 10}\right)} \]

   4. frac-2neg 98.9%

      \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \left(3 \cdot \color{blue}{\frac{-1}{-\log 10}}\right) \]

   5. metadata-eval 98.9%

      \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \left(3 \cdot \frac{\color{blue}{-1}}{-\log 10}\right) \]

   6. neg-log 99.4%

      \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \left(3 \cdot \frac{-1}{\color{blue}{\log \left(\frac{1}{10}\right)}}\right) \]

   7. metadata-eval 99.4%

      \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \left(3 \cdot \frac{-1}{\log \color{blue}{0.1}}\right) \]

8. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \left(3 \cdot \frac{-1}{\log 0.1}\right)} \]
9. Step-by-step derivation

   1. associate-*r/ 99.4%

      \[\leadsto \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \color{blue}{\frac{3 \cdot -1}{\log 0.1}} \]

   2. metadata-eval 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in re around 0 25.5%

    \[\leadsto \color{blue}{-3 \cdot \frac{\log \left({1}^{0.16666666666666666} \cdot \sqrt[3]{im}\right)}{\log 0.1}} \]
12. Step-by-step derivation

    1. metadata-eval 25.5%

       \[\leadsto -3 \cdot \frac{\log \left({1}^{0.16666666666666666} \cdot \sqrt[3]{im}\right)}{\log \color{blue}{\left(1 + -0.9\right)}} \]

    2. log1p-undefine 25.5%

       \[\leadsto -3 \cdot \frac{\log \left({1}^{0.16666666666666666} \cdot \sqrt[3]{im}\right)}{\color{blue}{\mathsf{log1p}\left(-0.9\right)}} \]

    3. pow-base-1 25.5%

       \[\leadsto -3 \cdot \frac{\log \left(\color{blue}{1} \cdot \sqrt[3]{im}\right)}{\mathsf{log1p}\left(-0.9\right)} \]

    4. *-lft-identity 25.5%

       \[\leadsto -3 \cdot \frac{\log \color{blue}{\left(\sqrt[3]{im}\right)}}{\mathsf{log1p}\left(-0.9\right)} \]

    5. associate-*r/ 25.6%

       \[\leadsto \color{blue}{\frac{-3 \cdot \log \left(\sqrt[3]{im}\right)}{\mathsf{log1p}\left(-0.9\right)}} \]

    6. *-rgt-identity 25.6%

       \[\leadsto \frac{-3 \cdot \log \left(\sqrt[3]{im}\right)}{\color{blue}{\mathsf{log1p}\left(-0.9\right) \cdot 1}} \]

    7. times-frac 25.5%

       \[\leadsto \color{blue}{\frac{-3}{\mathsf{log1p}\left(-0.9\right)} \cdot \frac{\log \left(\sqrt[3]{im}\right)}{1}} \]

    8. log1p-undefine 25.5%

       \[\leadsto \frac{-3}{\color{blue}{\log \left(1 + -0.9\right)}} \cdot \frac{\log \left(\sqrt[3]{im}\right)}{1} \]

    9. metadata-eval 25.6%

       \[\leadsto \frac{-3}{\log \color{blue}{0.1}} \cdot \frac{\log \left(\sqrt[3]{im}\right)}{1} \]

    10. /-rgt-identity 25.6%

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

13. Simplified 25.6%

    \[\leadsto \color{blue}{\frac{-3}{\log 0.1} \cdot \log \left(\sqrt[3]{im}\right)} \]
14. Step-by-step derivation

    1. pow1/3 25.7%

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

    2. log-pow 25.6%

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

15. Applied egg-rr 25.6%

    \[\leadsto \frac{-3}{\log 0.1} \cdot \color{blue}{\left(0.3333333333333333 \cdot \log im\right)} \]
16. Add Preprocessing

Alternative 5: 26.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \frac{\log im \cdot -0.3333333333333333}{\log 0.1} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 3.0d0 * ((log(im) * (-0.3333333333333333d0)) / log(0.1d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. +-commutative 51.7%

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

   2. +-commutative 51.7%

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

   3. sqr-neg 51.7%

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

   4. sqr-neg 51.7%

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

   5. sqr-neg 51.7%

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

   6. sqr-neg 51.7%

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

   7. hypot-define 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. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 99.1%

      \[\leadsto \frac{\log \color{blue}{\left(\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}}{\log 10} \]

   2. pow3 99.1%

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}^{3}\right)}}{\log 10} \]

   3. log-pow 99.1%

      \[\leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}}{\log 10} \]

6. Applied egg-rr 99.1%

   \[\leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}}{\log 10} \]

7. Taylor expanded in re around 0 25.5%

   \[\leadsto \color{blue}{3 \cdot \frac{\log \left({1}^{0.16666666666666666} \cdot \sqrt[3]{im}\right)}{\log 10}} \]
8. Step-by-step derivation

   1. pow-base-1 25.5%

      \[\leadsto 3 \cdot \frac{\log \left(\color{blue}{1} \cdot \sqrt[3]{im}\right)}{\log 10} \]

   2. *-lft-identity 25.5%

      \[\leadsto 3 \cdot \frac{\log \color{blue}{\left(\sqrt[3]{im}\right)}}{\log 10} \]

9. Simplified 25.5%

   \[\leadsto \color{blue}{3 \cdot \frac{\log \left(\sqrt[3]{im}\right)}{\log 10}} \]
10. Step-by-step derivation

    1. pow1/3 25.7%

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

    2. log-pow 25.6%

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

11. Applied egg-rr 25.4%

    \[\leadsto 3 \cdot \frac{\color{blue}{0.3333333333333333 \cdot \log im}}{\log 10} \]
12. Step-by-step derivation

    1. div-inv 25.3%

       \[\leadsto 3 \cdot \color{blue}{\left(\left(0.3333333333333333 \cdot \log im\right) \cdot \frac{1}{\log 10}\right)} \]

    2. *-commutative 25.3%

       \[\leadsto 3 \cdot \left(\color{blue}{\left(\log im \cdot 0.3333333333333333\right)} \cdot \frac{1}{\log 10}\right) \]

    3. frac-2neg 25.3%

       \[\leadsto 3 \cdot \left(\left(\log im \cdot 0.3333333333333333\right) \cdot \color{blue}{\frac{-1}{-\log 10}}\right) \]

    4. metadata-eval 25.3%

       \[\leadsto 3 \cdot \left(\left(\log im \cdot 0.3333333333333333\right) \cdot \frac{\color{blue}{-1}}{-\log 10}\right) \]

    5. neg-log 25.6%

       \[\leadsto 3 \cdot \left(\left(\log im \cdot 0.3333333333333333\right) \cdot \frac{-1}{\color{blue}{\log \left(\frac{1}{10}\right)}}\right) \]

    6. metadata-eval 25.6%

       \[\leadsto 3 \cdot \left(\left(\log im \cdot 0.3333333333333333\right) \cdot \frac{-1}{\log \color{blue}{0.1}}\right) \]

    7. associate-*l* 25.6%

       \[\leadsto 3 \cdot \color{blue}{\left(\log im \cdot \left(0.3333333333333333 \cdot \frac{-1}{\log 0.1}\right)\right)} \]

13. Applied egg-rr 25.6%

    \[\leadsto 3 \cdot \color{blue}{\left(\log im \cdot \left(0.3333333333333333 \cdot \frac{-1}{\log 0.1}\right)\right)} \]
14. Step-by-step derivation

    1. associate-*r/ 25.6%

       \[\leadsto 3 \cdot \left(\log im \cdot \color{blue}{\frac{0.3333333333333333 \cdot -1}{\log 0.1}}\right) \]

    2. metadata-eval 25.6%

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

    3. associate-*r/ 25.6%

       \[\leadsto 3 \cdot \color{blue}{\frac{\log im \cdot -0.3333333333333333}{\log 0.1}} \]

15. Simplified 25.6%

    \[\leadsto 3 \cdot \color{blue}{\frac{\log im \cdot -0.3333333333333333}{\log 0.1}} \]
16. Add Preprocessing

Alternative 6: 26.9% 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 51.7%

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

   1. +-commutative 51.7%

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

   2. +-commutative 51.7%

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

   3. sqr-neg 51.7%

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

   4. sqr-neg 51.7%

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

   5. sqr-neg 51.7%

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

   6. sqr-neg 51.7%

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

   7. hypot-define 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. Add Preprocessing

5. Taylor expanded in re around 0 25.6%

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

Reproduce

herbie shell --seed 2024145 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))