math.log10 on complex, real part

Percentage Accurate: 51.3% → 99.7%

Time: 9.2s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

MATLAB

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

Wolfram

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

Initial Program: 51.3% 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: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (log (pow (hypot re im) (cbrt (pow (log 10.0) -3.0)))))

C

double code(double re, double im) {
	return log(pow(hypot(re, im), cbrt(pow(log(10.0), -3.0))));
}

Java

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

Julia

function code(re, im)
	return log((hypot(re, im) ^ cbrt((log(10.0) ^ -3.0))))
end

Wolfram

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

Derivation

1. Initial program 54.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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. add-log-exp 99.1%

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

   2. div-inv 98.6%

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

   3. exp-to-pow 98.5%

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

   4. frac-2neg 98.5%

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

   5. metadata-eval 98.5%

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

   6. neg-log 99.0%

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

   7. metadata-eval 99.0%

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

5. Applied egg-rr 99.0%

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

   1. add-cbrt-cube 98.0%

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

   2. pow1/3 99.0%

      \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\left({\left(\left(\frac{-1}{\log 0.1} \cdot \frac{-1}{\log 0.1}\right) \cdot \frac{-1}{\log 0.1}\right)}^{0.3333333333333333}\right)}}\right) \]

7. Applied egg-rr 99.8%

   \[\leadsto \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\left({\left({\log 10}^{-3}\right)}^{0.3333333333333333}\right)}}\right) \]
8. Step-by-step derivation

   1. unpow1/3 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 27.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{{\log 10}^{-2}} \cdot \log im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (sqrt (pow (log 10.0) -2.0)) (log im)))

C

double code(double re, double im) {
	return sqrt(pow(log(10.0), -2.0)) * log(im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt((log(10.0d0) ** (-2.0d0))) * log(im)
end function

Java

public static double code(double re, double im) {
	return Math.sqrt(Math.pow(Math.log(10.0), -2.0)) * Math.log(im);
}

Python

def code(re, im):
	return math.sqrt(math.pow(math.log(10.0), -2.0)) * math.log(im)

Julia

function code(re, im)
	return Float64(sqrt((log(10.0) ^ -2.0)) * log(im))
end

MATLAB

function tmp = code(re, im)
	tmp = sqrt((log(10.0) ^ -2.0)) * log(im);
end

Wolfram

code[re_, im_] := N[(N[Sqrt[N[Power[N[Log[10.0], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision] * N[Log[im], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 26.6%

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

   1. clear-num 26.6%

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

   2. associate-/r/ 26.5%

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

6. Applied egg-rr 26.5%

   \[\leadsto \color{blue}{\frac{1}{\log 10} \cdot \log im} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 26.7%

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

   2. pow1/2 26.7%

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

   3. pow1/2 26.7%

      \[\leadsto \left({\left(\frac{1}{\log 10}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{1}{\log 10}\right)}^{0.5}}\right) \cdot \log im \]

   4. pow-prod-down 26.5%

      \[\leadsto \color{blue}{{\left(\frac{1}{\log 10} \cdot \frac{1}{\log 10}\right)}^{0.5}} \cdot \log im \]

   5. inv-pow 26.5%

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

   6. add-cbrt-cube 26.5%

      \[\leadsto {\left({\log 10}^{\color{blue}{\left(\sqrt[3]{\left(-1 \cdot -1\right) \cdot -1}\right)}} \cdot \frac{1}{\log 10}\right)}^{0.5} \cdot \log im \]

   7. metadata-eval 26.5%

      \[\leadsto {\left({\log 10}^{\left(\sqrt[3]{\color{blue}{1} \cdot -1}\right)} \cdot \frac{1}{\log 10}\right)}^{0.5} \cdot \log im \]

   8. metadata-eval 26.5%

      \[\leadsto {\left({\log 10}^{\left(\sqrt[3]{\color{blue}{-1}}\right)} \cdot \frac{1}{\log 10}\right)}^{0.5} \cdot \log im \]

   9. inv-pow 26.5%

      \[\leadsto {\left({\log 10}^{\left(\sqrt[3]{-1}\right)} \cdot \color{blue}{{\log 10}^{-1}}\right)}^{0.5} \cdot \log im \]

   10. add-cbrt-cube 26.5%

       \[\leadsto {\left({\log 10}^{\left(\sqrt[3]{-1}\right)} \cdot {\log 10}^{\color{blue}{\left(\sqrt[3]{\left(-1 \cdot -1\right) \cdot -1}\right)}}\right)}^{0.5} \cdot \log im \]

   11. metadata-eval 26.5%

       \[\leadsto {\left({\log 10}^{\left(\sqrt[3]{-1}\right)} \cdot {\log 10}^{\left(\sqrt[3]{\color{blue}{1} \cdot -1}\right)}\right)}^{0.5} \cdot \log im \]

   12. metadata-eval 26.5%

       \[\leadsto {\left({\log 10}^{\left(\sqrt[3]{-1}\right)} \cdot {\log 10}^{\left(\sqrt[3]{\color{blue}{-1}}\right)}\right)}^{0.5} \cdot \log im \]

   13. pow-sqr 26.7%

       \[\leadsto {\color{blue}{\left({\log 10}^{\left(2 \cdot \sqrt[3]{-1}\right)}\right)}}^{0.5} \cdot \log im \]

   14. metadata-eval 26.7%

       \[\leadsto {\left({\log 10}^{\left(2 \cdot \sqrt[3]{\color{blue}{1 \cdot -1}}\right)}\right)}^{0.5} \cdot \log im \]

   15. metadata-eval 26.7%

       \[\leadsto {\left({\log 10}^{\left(2 \cdot \sqrt[3]{\color{blue}{\left(-1 \cdot -1\right)} \cdot -1}\right)}\right)}^{0.5} \cdot \log im \]

   16. add-cbrt-cube 26.7%

       \[\leadsto {\left({\log 10}^{\left(2 \cdot \color{blue}{-1}\right)}\right)}^{0.5} \cdot \log im \]

   17. metadata-eval 26.7%

       \[\leadsto {\left({\log 10}^{\color{blue}{-2}}\right)}^{0.5} \cdot \log im \]

8. Applied egg-rr 26.7%

   \[\leadsto \color{blue}{{\left({\log 10}^{-2}\right)}^{0.5}} \cdot \log im \]
9. Step-by-step derivation

   1. unpow1/2 26.7%

      \[\leadsto \color{blue}{\sqrt{{\log 10}^{-2}}} \cdot \log im \]

10. Simplified 26.7%

    \[\leadsto \color{blue}{\sqrt{{\log 10}^{-2}}} \cdot \log im \]

11. Final simplification 26.7%

    \[\leadsto \sqrt{{\log 10}^{-2}} \cdot \log im \]

Alternative 3: 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 54.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 71.3%

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

   2. expm1-udef 71.4%

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

   3. log1p-udef 71.3%

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

   4. add-exp-log 99.1%

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

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 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 54.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 5: 27.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\log 10}{\log im}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 / (log(10.0d0) / log(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(1.0 / N[(N[Log[10.0], $MachinePrecision] / N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 26.6%

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

   1. clear-num 26.6%

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

   2. inv-pow 26.6%

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

6. Applied egg-rr 26.6%

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

   1. unpow-1 26.6%

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

8. Simplified 26.6%

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

9. Final simplification 26.6%

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

Alternative 6: 27.8% 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 54.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 26.6%

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

5. Final simplification 26.6%

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

Reproduce

herbie shell --seed 2023201 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))