math.log10 on complex, real part

Percentage Accurate: 52.9% → 99.1%

Time: 10.1s

Alternatives: 7

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: 52.9% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.1%

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

   1. hypot-def 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. Step-by-step derivation

   1. expm1-log1p-u 76.8%

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

   2. expm1-udef 76.7%

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

   3. log1p-udef 76.7%

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

   4. add-exp-log 99.1%

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

5. Applied egg-rr 99.1%

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

6. Final simplification 99.1%

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

Alternative 2: 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 48.1%

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

   1. hypot-def 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. Final simplification 99.1%

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

Alternative 3: 45.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.8 \cdot 10^{-84}:\\ \;\;\;\;-1 + \left(1 - \frac{\log \left(\frac{-1}{re}\right)}{\log 10}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.8e-84)
   (+ -1.0 (- 1.0 (/ (log (/ -1.0 re)) (log 10.0))))
   (/ 1.0 (/ (log 10.0) (log im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.8e-84) {
		tmp = -1.0 + (1.0 - (log((-1.0 / re)) / log(10.0)));
	} else {
		tmp = 1.0 / (log(10.0) / log(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-6.8d-84)) then
        tmp = (-1.0d0) + (1.0d0 - (log(((-1.0d0) / re)) / log(10.0d0)))
    else
        tmp = 1.0d0 / (log(10.0d0) / log(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -6.8e-84) {
		tmp = -1.0 + (1.0 - (Math.log((-1.0 / re)) / Math.log(10.0)));
	} else {
		tmp = 1.0 / (Math.log(10.0) / Math.log(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.8e-84:
		tmp = -1.0 + (1.0 - (math.log((-1.0 / re)) / math.log(10.0)))
	else:
		tmp = 1.0 / (math.log(10.0) / math.log(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.8e-84)
		tmp = Float64(-1.0 + Float64(1.0 - Float64(log(Float64(-1.0 / re)) / log(10.0))));
	else
		tmp = Float64(1.0 / Float64(log(10.0) / log(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6.8e-84)
		tmp = -1.0 + (1.0 - (log((-1.0 / re)) / log(10.0)));
	else
		tmp = 1.0 / (log(10.0) / log(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.8e-84], N[(-1.0 + N[(1.0 - N[(N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Log[10.0], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -6.80000000000000042e-84

   1. Initial program 48.4%

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

      1. hypot-def 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. Step-by-step derivation

      1. expm1-log1p-u 83.3%

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

      2. expm1-udef 83.3%

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

      3. log1p-udef 83.3%

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

      4. add-exp-log 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in re around -inf 78.9%

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

      1. mul-1-neg 78.9%

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

      2. unsub-neg 78.9%

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

   8. Simplified 78.9%

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

   if -6.80000000000000042e-84 < re

   1. Initial program 48.0%

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

      1. hypot-def 99.0%

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

   3. Simplified 99.0%

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

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in re around 0 31.9%

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

      1. unpow-1 31.9%

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

   8. Applied egg-rr 31.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.8 \cdot 10^{-84}:\\ \;\;\;\;-1 + \left(1 - \frac{\log \left(\frac{-1}{re}\right)}{\log 10}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \]

Alternative 4: 45.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.45 \cdot 10^{-80}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.45e-80)
   (/ (log (- re)) (log 10.0))
   (/ 1.0 (/ (log 10.0) (log im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.45e-80) {
		tmp = log(-re) / log(10.0);
	} else {
		tmp = 1.0 / (log(10.0) / log(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.45d-80)) then
        tmp = log(-re) / log(10.0d0)
    else
        tmp = 1.0d0 / (log(10.0d0) / log(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.45e-80) {
		tmp = Math.log(-re) / Math.log(10.0);
	} else {
		tmp = 1.0 / (Math.log(10.0) / Math.log(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.45e-80:
		tmp = math.log(-re) / math.log(10.0)
	else:
		tmp = 1.0 / (math.log(10.0) / math.log(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.45e-80)
		tmp = Float64(log(Float64(-re)) / log(10.0));
	else
		tmp = Float64(1.0 / Float64(log(10.0) / log(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.45e-80)
		tmp = log(-re) / log(10.0);
	else
		tmp = 1.0 / (log(10.0) / log(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.45e-80], N[(N[Log[(-re)], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Log[10.0], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.44999999999999999e-80

   1. Initial program 48.9%

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

      1. hypot-def 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. Taylor expanded in re around -inf 79.8%

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

      1. associate-*r/ 79.8%

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

      2. mul-1-neg 79.8%

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

   6. Simplified 79.8%

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

      1. add-sqr-sqrt 65.2%

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

      2. sqrt-unprod 66.6%

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

      3. sqr-neg 66.6%

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

      4. sqrt-unprod 1.2%

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

      5. *-un-lft-identity 1.2%

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

      6. times-frac 1.2%

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

   8. Applied egg-rr 65.2%

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

      1. /-rgt-identity 65.2%

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

      2. associate-*r/ 65.2%

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

      3. rem-square-sqrt 79.8%

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

   10. Simplified 79.8%

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

   if -1.44999999999999999e-80 < re

   1. Initial program 47.7%

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

      1. hypot-def 99.0%

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

   3. Simplified 99.0%

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

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in re around 0 31.7%

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

      1. unpow-1 31.7%

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

   8. Applied egg-rr 31.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.45 \cdot 10^{-80}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \]

Alternative 5: 45.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{-88}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.6e-88) (/ (log (- re)) (log 10.0)) (/ (log im) (log 10.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.6e-88) {
		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 (re <= (-3.6d-88)) 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 (re <= -3.6e-88) {
		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 re <= -3.6e-88:
		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 (re <= -3.6e-88)
		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 (re <= -3.6e-88)
		tmp = log(-re) / log(10.0);
	else
		tmp = log(im) / log(10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.6e-88], 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 re < -3.5999999999999999e-88

   1. Initial program 49.0%

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

      1. hypot-def 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. Taylor expanded in re around -inf 79.1%

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

      1. associate-*r/ 79.1%

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

      2. mul-1-neg 79.1%

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

   6. Simplified 79.1%

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

      1. add-sqr-sqrt 63.7%

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

      2. sqrt-unprod 65.3%

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

      3. sqr-neg 65.3%

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

      4. sqrt-unprod 1.4%

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

      5. *-un-lft-identity 1.4%

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

      6. times-frac 1.4%

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

   8. Applied egg-rr 63.6%

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

      1. /-rgt-identity 63.6%

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

      2. associate-*r/ 63.7%

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

      3. rem-square-sqrt 79.1%

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

   10. Simplified 79.1%

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

   if -3.5999999999999999e-88 < re

   1. Initial program 47.7%

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

      1. hypot-def 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in re around 0 32.1%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{-88}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

Alternative 6: 3.1% 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 48.1%

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

   1. hypot-def 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. Taylor expanded in re around 0 25.7%

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

   1. frac-2neg 25.7%

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

   2. div-inv 25.6%

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

   3. neg-log 25.7%

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

   4. metadata-eval 25.7%

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

6. Applied egg-rr 25.7%

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

   1. log-rec 25.7%

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

   2. associate-*r/ 25.7%

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

   3. *-rgt-identity 25.7%

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

   4. log-rec 25.7%

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

8. Simplified 25.7%

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

   1. div-inv 25.7%

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

   2. add-sqr-sqrt 6.6%

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

   3. sqrt-unprod 7.0%

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

   4. sqr-neg 7.0%

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

   5. sqrt-unprod 0.4%

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

   6. add-sqr-sqrt 3.2%

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

10. Applied egg-rr 3.2%

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

    1. associate-*r/ 3.2%

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

    2. *-rgt-identity 3.2%

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

12. Simplified 3.2%

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

13. Final simplification 3.2%

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

Alternative 7: 27.5% 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 48.1%

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

   1. hypot-def 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. Taylor expanded in re around 0 25.7%

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

5. Final simplification 25.7%

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

Reproduce

herbie shell --seed 2023196 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))