math.log/2 on complex, real part

Percentage Accurate: 51.0% → 99.4%

Time: 9.7s

Alternatives: 5

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (/
  (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))

C

double code(double re, double im, double base) {
	return ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0d0)) / ((log(base) * log(base)) + (0.0d0 * 0.0d0))
end function

Java

public static double code(double re, double im, double base) {
	return ((Math.log(Math.sqrt(((re * re) + (im * im)))) * Math.log(base)) + (Math.atan2(im, re) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}

Python

def code(re, im, base):
	return ((math.log(math.sqrt(((re * re) + (im * im)))) * math.log(base)) + (math.atan2(im, re) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

Julia

function code(re, im, base)
	return Float64(Float64(Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * log(base)) + Float64(atan(im, re) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(N[ArcTan[im / re], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (/
  (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))

C

double code(double re, double im, double base) {
	return ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0d0)) / ((log(base) * log(base)) + (0.0d0 * 0.0d0))
end function

Java

public static double code(double re, double im, double base) {
	return ((Math.log(Math.sqrt(((re * re) + (im * im)))) * Math.log(base)) + (Math.atan2(im, re) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}

Python

def code(re, im, base):
	return ((math.log(math.sqrt(((re * re) + (im * im)))) * math.log(base)) + (math.atan2(im, re) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

Julia

function code(re, im, base)
	return Float64(Float64(Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * log(base)) + Float64(atan(im, re) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(N[ArcTan[im / re], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Step-by-step derivation

   1. mul0-rgt 51.7%

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

   2. +-rgt-identity 51.7%

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

   3. metadata-eval 51.7%

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

   4. +-rgt-identity 51.7%

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

   5. times-frac 51.8%

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

   6. *-inverses 51.8%

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

   7. *-rgt-identity 51.8%

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

   8. hypot-def 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 2: 44.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log \left(-re\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\log base} \cdot \log im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (if (<= im 1.2e-104)
   (/ 1.0 (/ (log base) (log (- re))))
   (* (/ 1.0 (log base)) (log im))))

C

double code(double re, double im, double base) {
	double tmp;
	if (im <= 1.2e-104) {
		tmp = 1.0 / (log(base) / log(-re));
	} else {
		tmp = (1.0 / log(base)) * log(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    real(8) :: tmp
    if (im <= 1.2d-104) then
        tmp = 1.0d0 / (log(base) / log(-re))
    else
        tmp = (1.0d0 / log(base)) * log(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im, double base) {
	double tmp;
	if (im <= 1.2e-104) {
		tmp = 1.0 / (Math.log(base) / Math.log(-re));
	} else {
		tmp = (1.0 / Math.log(base)) * Math.log(im);
	}
	return tmp;
}

Python

def code(re, im, base):
	tmp = 0
	if im <= 1.2e-104:
		tmp = 1.0 / (math.log(base) / math.log(-re))
	else:
		tmp = (1.0 / math.log(base)) * math.log(im)
	return tmp

Julia

function code(re, im, base)
	tmp = 0.0
	if (im <= 1.2e-104)
		tmp = Float64(1.0 / Float64(log(base) / log(Float64(-re))));
	else
		tmp = Float64(Float64(1.0 / log(base)) * log(im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im, base)
	tmp = 0.0;
	if (im <= 1.2e-104)
		tmp = 1.0 / (log(base) / log(-re));
	else
		tmp = (1.0 / log(base)) * log(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_, base_] := If[LessEqual[im, 1.2e-104], N[(1.0 / N[(N[Log[base], $MachinePrecision] / N[Log[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Log[base], $MachinePrecision]), $MachinePrecision] * N[Log[im], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.2e-104

   1. Initial program 51.5%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   2. Step-by-step derivation

      1. mul0-rgt 51.5%

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

      2. +-rgt-identity 51.5%

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

      3. metadata-eval 51.5%

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

      4. +-rgt-identity 51.5%

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

      5. times-frac 51.5%

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

      6. *-inverses 51.5%

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

      7. *-rgt-identity 51.5%

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

      8. hypot-def 99.4%

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

   3. Simplified 99.4%

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

      1. add-cube-cbrt 98.0%

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

      2. pow3 98.0%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in re around -inf 34.0%

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

      1. associate-*r/ 34.5%

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

      2. mul-1-neg 34.5%

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

   8. Simplified 34.0%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{-\log \left(\frac{-1}{re}\right)}{\log base}}}\right)}^{3} \]
   9. Step-by-step derivation

      1. rem-cube-cbrt 34.5%

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

      2. clear-num 34.5%

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

      3. neg-log 34.5%

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

      4. frac-2neg 34.5%

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

      5. metadata-eval 34.5%

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

      6. remove-double-div 34.5%

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

   10. Applied egg-rr 34.5%

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

   if 1.2e-104 < im

   1. Initial program 52.2%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   2. Step-by-step derivation

      1. mul0-rgt 52.2%

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

      2. +-rgt-identity 52.2%

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

      3. metadata-eval 52.2%

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

      4. +-rgt-identity 52.2%

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

      5. times-frac 52.4%

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

      6. *-inverses 52.4%

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

      7. *-rgt-identity 52.4%

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

      8. hypot-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in re around 0 70.9%

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

      1. div-inv 70.9%

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

      2. frac-2neg 70.9%

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

      3. metadata-eval 70.9%

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

      4. add-sqr-sqrt 28.1%

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

      5. sqrt-unprod 29.4%

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

      6. sqr-neg 29.4%

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

      7. sqrt-prod 1.2%

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

      8. add-sqr-sqrt 4.2%

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

   6. Applied egg-rr 4.2%

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

      1. *-commutative 4.2%

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

      2. associate-*l/ 4.2%

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

      3. neg-mul-1 4.2%

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

   8. Simplified 4.2%

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

      1. add-sqr-sqrt 3.1%

         \[\leadsto \frac{\color{blue}{\sqrt{-\log im} \cdot \sqrt{-\log im}}}{\log base} \]

      2. sqrt-unprod 63.6%

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

      3. sqr-neg 63.6%

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

      4. sqrt-unprod 60.3%

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

      5. add-sqr-sqrt 70.9%

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

      6. clear-num 70.9%

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

      7. associate-/r/ 70.9%

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

   10. Applied egg-rr 70.9%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log \left(-re\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\log base} \cdot \log im\\ \end{array} \]

Alternative 3: 44.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\log base} \cdot \log im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (if (<= im 7.2e-105)
   (/ (log (- re)) (log base))
   (* (/ 1.0 (log base)) (log im))))

C

double code(double re, double im, double base) {
	double tmp;
	if (im <= 7.2e-105) {
		tmp = log(-re) / log(base);
	} else {
		tmp = (1.0 / log(base)) * log(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    real(8) :: tmp
    if (im <= 7.2d-105) then
        tmp = log(-re) / log(base)
    else
        tmp = (1.0d0 / log(base)) * log(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im, double base) {
	double tmp;
	if (im <= 7.2e-105) {
		tmp = Math.log(-re) / Math.log(base);
	} else {
		tmp = (1.0 / Math.log(base)) * Math.log(im);
	}
	return tmp;
}

Python

def code(re, im, base):
	tmp = 0
	if im <= 7.2e-105:
		tmp = math.log(-re) / math.log(base)
	else:
		tmp = (1.0 / math.log(base)) * math.log(im)
	return tmp

Julia

function code(re, im, base)
	tmp = 0.0
	if (im <= 7.2e-105)
		tmp = Float64(log(Float64(-re)) / log(base));
	else
		tmp = Float64(Float64(1.0 / log(base)) * log(im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im, base)
	tmp = 0.0;
	if (im <= 7.2e-105)
		tmp = log(-re) / log(base);
	else
		tmp = (1.0 / log(base)) * log(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_, base_] := If[LessEqual[im, 7.2e-105], N[(N[Log[(-re)], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Log[base], $MachinePrecision]), $MachinePrecision] * N[Log[im], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.19999999999999929e-105

   1. Initial program 51.5%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   2. Step-by-step derivation

      1. mul0-rgt 51.5%

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

      2. +-rgt-identity 51.5%

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

      3. metadata-eval 51.5%

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

      4. +-rgt-identity 51.5%

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

      5. times-frac 51.5%

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

      6. *-inverses 51.5%

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

      7. *-rgt-identity 51.5%

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

      8. hypot-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in re around -inf 34.5%

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

      1. associate-*r/ 34.5%

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

      2. mul-1-neg 34.5%

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

   6. Simplified 34.5%

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

      1. expm1-log1p-u 28.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\right)\right)} \]

      2. expm1-udef 28.4%

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

      3. neg-log 28.4%

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

      4. frac-2neg 28.4%

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

      5. metadata-eval 28.4%

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

      6. remove-double-div 28.4%

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

   8. Applied egg-rr 28.4%

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

      1. expm1-def 28.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(-re\right)}{\log base}\right)\right)} \]

      2. expm1-log1p 34.5%

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

   10. Simplified 34.5%

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

   if 7.19999999999999929e-105 < im

   1. Initial program 52.2%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   2. Step-by-step derivation

      1. mul0-rgt 52.2%

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

      2. +-rgt-identity 52.2%

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

      3. metadata-eval 52.2%

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

      4. +-rgt-identity 52.2%

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

      5. times-frac 52.4%

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

      6. *-inverses 52.4%

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

      7. *-rgt-identity 52.4%

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

      8. hypot-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in re around 0 70.9%

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

      1. div-inv 70.9%

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

      2. frac-2neg 70.9%

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

      3. metadata-eval 70.9%

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

      4. add-sqr-sqrt 28.1%

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

      5. sqrt-unprod 29.4%

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

      6. sqr-neg 29.4%

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

      7. sqrt-prod 1.2%

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

      8. add-sqr-sqrt 4.2%

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

   6. Applied egg-rr 4.2%

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

      1. *-commutative 4.2%

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

      2. associate-*l/ 4.2%

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

      3. neg-mul-1 4.2%

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

   8. Simplified 4.2%

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

      1. add-sqr-sqrt 3.1%

         \[\leadsto \frac{\color{blue}{\sqrt{-\log im} \cdot \sqrt{-\log im}}}{\log base} \]

      2. sqrt-unprod 63.6%

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

      3. sqr-neg 63.6%

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

      4. sqrt-unprod 60.3%

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

      5. add-sqr-sqrt 70.9%

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

      6. clear-num 70.9%

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

      7. associate-/r/ 70.9%

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

   10. Applied egg-rr 70.9%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\log base} \cdot \log im\\ \end{array} \]

Alternative 4: 44.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9 \cdot 10^{-105}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (if (<= im 9e-105) (/ (log (- re)) (log base)) (/ (log im) (log base))))

C

double code(double re, double im, double base) {
	double tmp;
	if (im <= 9e-105) {
		tmp = log(-re) / log(base);
	} else {
		tmp = log(im) / log(base);
	}
	return tmp;
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    real(8) :: tmp
    if (im <= 9d-105) then
        tmp = log(-re) / log(base)
    else
        tmp = log(im) / log(base)
    end if
    code = tmp
end function

Java

public static double code(double re, double im, double base) {
	double tmp;
	if (im <= 9e-105) {
		tmp = Math.log(-re) / Math.log(base);
	} else {
		tmp = Math.log(im) / Math.log(base);
	}
	return tmp;
}

Python

def code(re, im, base):
	tmp = 0
	if im <= 9e-105:
		tmp = math.log(-re) / math.log(base)
	else:
		tmp = math.log(im) / math.log(base)
	return tmp

Julia

function code(re, im, base)
	tmp = 0.0
	if (im <= 9e-105)
		tmp = Float64(log(Float64(-re)) / log(base));
	else
		tmp = Float64(log(im) / log(base));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im, base)
	tmp = 0.0;
	if (im <= 9e-105)
		tmp = log(-re) / log(base);
	else
		tmp = log(im) / log(base);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_, base_] := If[LessEqual[im, 9e-105], N[(N[Log[(-re)], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision], N[(N[Log[im], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 8.9999999999999995e-105

   1. Initial program 51.5%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   2. Step-by-step derivation

      1. mul0-rgt 51.5%

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

      2. +-rgt-identity 51.5%

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

      3. metadata-eval 51.5%

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

      4. +-rgt-identity 51.5%

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

      5. times-frac 51.5%

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

      6. *-inverses 51.5%

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

      7. *-rgt-identity 51.5%

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

      8. hypot-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in re around -inf 34.5%

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

      1. associate-*r/ 34.5%

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

      2. mul-1-neg 34.5%

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

   6. Simplified 34.5%

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

      1. expm1-log1p-u 28.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\right)\right)} \]

      2. expm1-udef 28.4%

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

      3. neg-log 28.4%

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

      4. frac-2neg 28.4%

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

      5. metadata-eval 28.4%

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

      6. remove-double-div 28.4%

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

   8. Applied egg-rr 28.4%

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

      1. expm1-def 28.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(-re\right)}{\log base}\right)\right)} \]

      2. expm1-log1p 34.5%

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

   10. Simplified 34.5%

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

   if 8.9999999999999995e-105 < im

   1. Initial program 52.2%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   2. Step-by-step derivation

      1. mul0-rgt 52.2%

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

      2. +-rgt-identity 52.2%

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

      3. metadata-eval 52.2%

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

      4. +-rgt-identity 52.2%

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

      5. times-frac 52.4%

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

      6. *-inverses 52.4%

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

      7. *-rgt-identity 52.4%

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

      8. hypot-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in re around 0 70.9%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9 \cdot 10^{-105}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array} \]

Alternative 5: 27.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_, base_] := N[(N[Log[im], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Step-by-step derivation

   1. mul0-rgt 51.7%

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

   2. +-rgt-identity 51.7%

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

   3. metadata-eval 51.7%

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

   4. +-rgt-identity 51.7%

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

   5. times-frac 51.8%

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

   6. *-inverses 51.8%

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

   7. *-rgt-identity 51.8%

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

   8. hypot-def 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in re around 0 28.4%

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

5. Final simplification 28.4%

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

Reproduce

herbie shell --seed 2023193 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))