math.log10 on complex, real part

Percentage Accurate: 51.9% → 99.1%

Time: 9.1s

Alternatives: 8

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 51.9% 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: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\log 10}\\ \frac{1}{t_0} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (log 10.0)))) (* (/ 1.0 t_0) (/ (log (hypot re im)) t_0))))

C

double code(double re, double im) {
	double t_0 = sqrt(log(10.0));
	return (1.0 / t_0) * (log(hypot(re, im)) / t_0);
}

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(Math.log(10.0));
	return (1.0 / t_0) * (Math.log(Math.hypot(re, im)) / t_0);
}

Python

def code(re, im):
	t_0 = math.sqrt(math.log(10.0))
	return (1.0 / t_0) * (math.log(math.hypot(re, im)) / t_0)

Julia

function code(re, im)
	t_0 = sqrt(log(10.0))
	return Float64(Float64(1.0 / t_0) * Float64(log(hypot(re, im)) / t_0))
end

MATLAB

function tmp = code(re, im)
	t_0 = sqrt(log(10.0));
	tmp = (1.0 / t_0) * (log(hypot(re, im)) / t_0);
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[Log[10.0], $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 50.8%

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

   1. *-un-lft-identity 50.8%

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

   2. add-sqr-sqrt 50.8%

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

   3. times-frac 50.9%

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

   4. hypot-def 99.2%

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

3. Applied egg-rr 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(e^{\log \left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot 3}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (exp (* (log (cbrt (hypot re im))) 3.0))) (log 10.0)))

C

double code(double re, double im) {
	return log(exp((log(cbrt(hypot(re, im))) * 3.0))) / log(10.0);
}

Java

public static double code(double re, double im) {
	return Math.log(Math.exp((Math.log(Math.cbrt(Math.hypot(re, im))) * 3.0))) / Math.log(10.0);
}

Julia

function code(re, im)
	return Float64(log(exp(Float64(log(cbrt(hypot(re, im))) * 3.0))) / log(10.0))
end

Wolfram

code[re_, im_] := N[(N[Log[N[Exp[N[(N[Log[N[Power[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.8%

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

   1. add-cube-cbrt 50.8%

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

   2. pow3 50.8%

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

   3. pow-to-exp 50.8%

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

   4. hypot-def 99.1%

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

3. Applied egg-rr 99.1%

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

4. Final simplification 99.1%

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

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 50.8%

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

   1. frac-2neg 50.8%

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

   2. div-inv 50.5%

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

   3. hypot-def 98.6%

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

   4. neg-log 99.0%

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

   5. metadata-eval 99.0%

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

3. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\left(-\log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot \frac{1}{\log 0.1}} \]
4. 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%

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

5. Simplified 99.1%

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

6. Final simplification 99.1%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.8%

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

   1. add-log-exp 5.9%

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

   2. *-un-lft-identity 5.9%

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

   3. log-prod 5.9%

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

   4. metadata-eval 5.9%

      \[\leadsto \frac{\log \left(\color{blue}{0} + \log \left(e^{\sqrt{re \cdot re + im \cdot im}}\right)\right)}{\log 10} \]

   5. add-log-exp 50.8%

      \[\leadsto \frac{\log \left(0 + \color{blue}{\sqrt{re \cdot re + im \cdot im}}\right)}{\log 10} \]

   6. hypot-def 99.1%

      \[\leadsto \frac{\log \left(0 + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)}{\log 10} \]

3. Applied egg-rr 99.1%

   \[\leadsto \frac{\log \color{blue}{\left(0 + \mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]
4. Step-by-step derivation

   1. +-lft-identity 99.1%

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

5. Simplified 99.1%

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

6. Final simplification 99.1%

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

Alternative 5: 44.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.5e-165) (/ (log (/ -1.0 re)) (log 0.1)) (/ (log im) (log 10.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.5e-165) {
		tmp = log((-1.0 / re)) / log(0.1);
	} else {
		tmp = log(im) / log(10.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.5d-165) then
        tmp = log(((-1.0d0) / re)) / log(0.1d0)
    else
        tmp = log(im) / log(10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.5e-165) {
		tmp = Math.log((-1.0 / re)) / Math.log(0.1);
	} else {
		tmp = Math.log(im) / Math.log(10.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.5e-165:
		tmp = math.log((-1.0 / re)) / math.log(0.1)
	else:
		tmp = math.log(im) / math.log(10.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.5e-165)
		tmp = Float64(log(Float64(-1.0 / re)) / log(0.1));
	else
		tmp = Float64(log(im) / log(10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.5e-165)
		tmp = log((-1.0 / re)) / log(0.1);
	else
		tmp = log(im) / log(10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.5e-165], N[(N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision] / N[Log[0.1], $MachinePrecision]), $MachinePrecision], N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.4999999999999999e-165

   1. Initial program 49.4%

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

      1. frac-2neg 49.4%

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

      2. div-inv 49.1%

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

      3. hypot-def 98.6%

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

      4. neg-log 99.1%

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

      5. metadata-eval 99.1%

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

   3. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\left(-\log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \cdot \frac{1}{\log 0.1}} \]
   4. 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%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in re around -inf 29.7%

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

   if 2.4999999999999999e-165 < im

   1. Initial program 53.5%

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

   2. Taylor expanded in re around 0 64.0%

      \[\leadsto \frac{\log \color{blue}{im}}{\log 10} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\log 0.1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 6: 44.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.5e-165) (/ (log (- re)) (log 10.0)) (/ (log im) (log 10.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.5e-165) {
		tmp = log(-re) / log(10.0);
	} else {
		tmp = log(im) / log(10.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.5d-165) then
        tmp = log(-re) / log(10.0d0)
    else
        tmp = log(im) / log(10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.5e-165) {
		tmp = Math.log(-re) / Math.log(10.0);
	} else {
		tmp = Math.log(im) / Math.log(10.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.5e-165:
		tmp = math.log(-re) / math.log(10.0)
	else:
		tmp = math.log(im) / math.log(10.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.5e-165)
		tmp = Float64(log(Float64(-re)) / log(10.0));
	else
		tmp = Float64(log(im) / log(10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.5e-165)
		tmp = log(-re) / log(10.0);
	else
		tmp = log(im) / log(10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.5e-165], N[(N[Log[(-re)], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision], N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.4999999999999999e-165

   1. Initial program 49.4%

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

   2. Taylor expanded in re around -inf 29.7%

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

      1. mul-1-neg 29.7%

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

   4. Simplified 29.7%

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

   if 2.4999999999999999e-165 < im

   1. Initial program 53.5%

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

   2. Taylor expanded in re around 0 64.0%

      \[\leadsto \frac{\log \color{blue}{im}}{\log 10} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 7: 3.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return Float64(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 50.8%

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

2. Taylor expanded in re around 0 24.1%

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

   1. frac-2neg 24.1%

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

   2. div-inv 24.0%

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

   3. neg-log 24.1%

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

   4. metadata-eval 24.1%

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

4. Applied egg-rr 24.1%

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

   1. log-rec 24.1%

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

   2. associate-*r/ 24.1%

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

   3. *-rgt-identity 24.1%

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

   4. log-rec 24.1%

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

6. Simplified 24.1%

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

8. Applied egg-rr 2.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\log im}{\log 0.1}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 2.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log im}{\log 0.1}\right)\right)} \]

   2. expm1-log1p 2.5%

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

10. Simplified 2.5%

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

11. Final simplification 2.5%

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

Alternative 8: 28.2% 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 50.8%

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

2. Taylor expanded in re around 0 24.1%

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

3. Final simplification 24.1%

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

Reproduce

herbie shell --seed 2023187 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))