math.log10 on complex, real part

Percentage Accurate: 51.8% → 99.5%

Time: 8.8s

Alternatives: 5

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: 51.8% 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 59.0%

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

   1. +-commutative 59.0%

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

   2. +-commutative 59.0%

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

   3. sqr-neg 59.0%

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

   4. sqr-neg 59.0%

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

   5. sqr-neg 59.0%

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

   6. sqr-neg 59.0%

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

   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. Step-by-step derivation

   1. clear-num 99.2%

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

   2. un-div-inv 99.2%

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

   3. log1p-expm1-u 99.2%

      \[\leadsto \frac{{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\log 10\right)\right)\right)}}^{-0.5}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

   4. expm1-undefine 99.2%

      \[\leadsto \frac{{\left(\mathsf{log1p}\left(\color{blue}{e^{\log 10} - 1}\right)\right)}^{-0.5}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

   5. rem-exp-log 99.2%

      \[\leadsto \frac{{\left(\mathsf{log1p}\left(\color{blue}{10} - 1\right)\right)}^{-0.5}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

   6. metadata-eval 99.2%

      \[\leadsto \frac{{\left(\mathsf{log1p}\left(\color{blue}{9}\right)\right)}^{-0.5}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

   7. log1p-expm1-u 99.2%

      \[\leadsto \frac{{\left(\mathsf{log1p}\left(9\right)\right)}^{-0.5}}{\frac{\sqrt{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log 10\right)\right)}}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

   8. expm1-undefine 99.2%

      \[\leadsto \frac{{\left(\mathsf{log1p}\left(9\right)\right)}^{-0.5}}{\frac{\sqrt{\mathsf{log1p}\left(\color{blue}{e^{\log 10} - 1}\right)}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

   9. rem-exp-log 99.2%

      \[\leadsto \frac{{\left(\mathsf{log1p}\left(9\right)\right)}^{-0.5}}{\frac{\sqrt{\mathsf{log1p}\left(\color{blue}{10} - 1\right)}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

   10. metadata-eval 99.2%

       \[\leadsto \frac{{\left(\mathsf{log1p}\left(9\right)\right)}^{-0.5}}{\frac{\sqrt{\mathsf{log1p}\left(\color{blue}{9}\right)}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

8. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{{\left(\mathsf{log1p}\left(9\right)\right)}^{-0.5}}{\frac{\sqrt{\mathsf{log1p}\left(9\right)}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}} \]
9. Step-by-step derivation

   1. associate-/r/ 99.6%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{log1p}\left(9\right)\right)}^{-0.5}}{\sqrt{\mathsf{log1p}\left(9\right)}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)} \]

   2. log1p-undefine 99.6%

      \[\leadsto \frac{{\color{blue}{\log \left(1 + 9\right)}}^{-0.5}}{\sqrt{\mathsf{log1p}\left(9\right)}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right) \]

   3. metadata-eval 99.6%

      \[\leadsto \frac{{\log \color{blue}{10}}^{-0.5}}{\sqrt{\mathsf{log1p}\left(9\right)}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right) \]

   4. log1p-undefine 99.6%

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

   5. metadata-eval 99.6%

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

10. Simplified 99.6%

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

Alternative 2: 27.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.0%

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

   1. +-commutative 59.0%

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

   2. +-commutative 59.0%

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

   3. sqr-neg 59.0%

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

   4. sqr-neg 59.0%

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

   5. sqr-neg 59.0%

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

   6. sqr-neg 59.0%

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

   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. Step-by-step derivation

   1. clear-num 99.2%

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

   2. un-div-inv 99.2%

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

   3. log1p-expm1-u 99.2%

      \[\leadsto \frac{{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\log 10\right)\right)\right)}}^{-0.5}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

   4. expm1-undefine 99.2%

      \[\leadsto \frac{{\left(\mathsf{log1p}\left(\color{blue}{e^{\log 10} - 1}\right)\right)}^{-0.5}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

   5. rem-exp-log 99.2%

      \[\leadsto \frac{{\left(\mathsf{log1p}\left(\color{blue}{10} - 1\right)\right)}^{-0.5}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

   6. metadata-eval 99.2%

      \[\leadsto \frac{{\left(\mathsf{log1p}\left(\color{blue}{9}\right)\right)}^{-0.5}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

   7. log1p-expm1-u 99.2%

      \[\leadsto \frac{{\left(\mathsf{log1p}\left(9\right)\right)}^{-0.5}}{\frac{\sqrt{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log 10\right)\right)}}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

   8. expm1-undefine 99.2%

      \[\leadsto \frac{{\left(\mathsf{log1p}\left(9\right)\right)}^{-0.5}}{\frac{\sqrt{\mathsf{log1p}\left(\color{blue}{e^{\log 10} - 1}\right)}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

   9. rem-exp-log 99.2%

      \[\leadsto \frac{{\left(\mathsf{log1p}\left(9\right)\right)}^{-0.5}}{\frac{\sqrt{\mathsf{log1p}\left(\color{blue}{10} - 1\right)}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

   10. metadata-eval 99.2%

       \[\leadsto \frac{{\left(\mathsf{log1p}\left(9\right)\right)}^{-0.5}}{\frac{\sqrt{\mathsf{log1p}\left(\color{blue}{9}\right)}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

8. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{{\left(\mathsf{log1p}\left(9\right)\right)}^{-0.5}}{\frac{\sqrt{\mathsf{log1p}\left(9\right)}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}} \]
9. Step-by-step derivation

   1. associate-/r/ 99.6%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{log1p}\left(9\right)\right)}^{-0.5}}{\sqrt{\mathsf{log1p}\left(9\right)}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)} \]

   2. log1p-undefine 99.6%

      \[\leadsto \frac{{\color{blue}{\log \left(1 + 9\right)}}^{-0.5}}{\sqrt{\mathsf{log1p}\left(9\right)}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right) \]

   3. metadata-eval 99.6%

      \[\leadsto \frac{{\log \color{blue}{10}}^{-0.5}}{\sqrt{\mathsf{log1p}\left(9\right)}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right) \]

   4. log1p-undefine 99.6%

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

   5. metadata-eval 99.6%

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

10. Simplified 99.6%

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

11. Taylor expanded in re around 0 30.5%

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

Alternative 3: 27.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[re_, im_] := N[(N[Log[N[Power[im, 1/3], $MachinePrecision]], $MachinePrecision] * N[(-3.0 / N[Log[0.1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.0%

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

   1. +-commutative 59.0%

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

   2. +-commutative 59.0%

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

   3. sqr-neg 59.0%

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

   4. sqr-neg 59.0%

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

   5. sqr-neg 59.0%

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

   6. sqr-neg 59.0%

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

   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 30.3%

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

   1. frac-2neg 30.3%

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

   2. div-inv 30.2%

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

   3. neg-log 30.3%

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

   4. metadata-eval 30.3%

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

7. Applied egg-rr 30.3%

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

   1. log-rec 30.3%

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

   2. associate-*r/ 30.3%

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

   3. *-rgt-identity 30.3%

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

   4. log-rec 30.3%

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

   5. distribute-neg-frac 30.3%

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

   6. distribute-neg-frac2 30.3%

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

9. Simplified 30.3%

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

11. Applied egg-rr 30.3%

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

    1. associate-*r/ 30.3%

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

    2. clear-num 30.3%

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

13. Applied egg-rr 30.3%

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

    1. associate-/r/ 30.3%

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

    2. *-commutative 30.3%

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

    3. associate-*r* 30.3%

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

    4. /-rgt-identity 30.3%

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

    5. times-frac 30.3%

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

    6. *-rgt-identity 30.3%

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

    7. associate-*r/ 30.3%

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

    8. *-lft-identity 30.3%

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

    9. associate-*l/ 30.3%

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

    10. associate-/l* 30.4%

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

15. Simplified 30.4%

    \[\leadsto \color{blue}{\log \left(\sqrt[3]{im}\right) \cdot \frac{-3}{\log 0.1}} \]
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 59.0%

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

   1. +-commutative 59.0%

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

   2. +-commutative 59.0%

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

   3. sqr-neg 59.0%

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

   4. sqr-neg 59.0%

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

   5. sqr-neg 59.0%

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

   6. sqr-neg 59.0%

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

   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

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

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 59.0%

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

   1. +-commutative 59.0%

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

   2. +-commutative 59.0%

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

   3. sqr-neg 59.0%

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

   4. sqr-neg 59.0%

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

   5. sqr-neg 59.0%

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

   6. sqr-neg 59.0%

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

   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 30.3%

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

Reproduce

herbie shell --seed 2024141 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))