math.log10 on complex, imaginary part

Percentage Accurate: 98.7% → 99.8%

Time: 9.2s

Alternatives: 3

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (atan2 im re) (log 10.0)))

C

double code(double re, double im) {
	return atan2(im, re) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = atan2(im, re) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.atan2(im, re) / Math.log(10.0);
}

Python

def code(re, im):
	return math.atan2(im, re) / math.log(10.0)

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = atan2(im, re) / log(10.0);
end

Wolfram

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

Initial Program: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (atan2 im re) (log 10.0)))

C

double code(double re, double im) {
	return atan2(im, re) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = atan2(im, re) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.atan2(im, re) / Math.log(10.0);
}

Python

def code(re, im):
	return math.atan2(im, re) / math.log(10.0)

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = atan2(im, re) / log(10.0);
end

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\tan^{-1}_* \frac{im}{re}}{-\log 0.1} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (atan2 im re) (- (log 0.1))))

C

double code(double re, double im) {
	return atan2(im, re) / -log(0.1);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = atan2(im, re) / -log(0.1d0)
end function

Java

public static double code(double re, double im) {
	return Math.atan2(im, re) / -Math.log(0.1);
}

Python

def code(re, im):
	return math.atan2(im, re) / -math.log(0.1)

Julia

function code(re, im)
	return Float64(atan(im, re) / Float64(-log(0.1)))
end

MATLAB

function tmp = code(re, im)
	tmp = atan2(im, re) / -log(0.1);
end

Wolfram

code[re_, im_] := N[(N[ArcTan[im / re], $MachinePrecision] / (-N[Log[0.1], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 98.6%

   \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \]
2. Step-by-step derivation

   1. clear-num 98.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}} \]

   2. frac-2neg 98.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{-\log 10}{-\tan^{-1}_* \frac{im}{re}}}} \]

   3. div-inv 98.4%

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

   4. associate-/r* 97.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{-\log 10}}{\frac{1}{-\tan^{-1}_* \frac{im}{re}}}} \]

   5. inv-pow 97.6%

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

   6. sqr-pow 0.0%

      \[\leadsto \frac{\color{blue}{{\left(-\log 10\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(-\log 10\right)}^{\left(\frac{-1}{2}\right)}}}{\frac{1}{-\tan^{-1}_* \frac{im}{re}}} \]

   7. pow-prod-down 7.4%

      \[\leadsto \frac{\color{blue}{{\left(\left(-\log 10\right) \cdot \left(-\log 10\right)\right)}^{\left(\frac{-1}{2}\right)}}}{\frac{1}{-\tan^{-1}_* \frac{im}{re}}} \]

   8. sqr-neg 7.4%

      \[\leadsto \frac{{\color{blue}{\left(\log 10 \cdot \log 10\right)}}^{\left(\frac{-1}{2}\right)}}{\frac{1}{-\tan^{-1}_* \frac{im}{re}}} \]

   9. pow-prod-down 7.4%

      \[\leadsto \frac{\color{blue}{{\log 10}^{\left(\frac{-1}{2}\right)} \cdot {\log 10}^{\left(\frac{-1}{2}\right)}}}{\frac{1}{-\tan^{-1}_* \frac{im}{re}}} \]

   10. div-inv 7.4%

       \[\leadsto \frac{{\log 10}^{\left(\frac{-1}{2}\right)} \cdot {\log 10}^{\left(\frac{-1}{2}\right)}}{\color{blue}{1 \cdot \frac{1}{-\tan^{-1}_* \frac{im}{re}}}} \]

   11. times-frac 7.4%

       \[\leadsto \color{blue}{\frac{{\log 10}^{\left(\frac{-1}{2}\right)}}{1} \cdot \frac{{\log 10}^{\left(\frac{-1}{2}\right)}}{\frac{1}{-\tan^{-1}_* \frac{im}{re}}}} \]

3. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{{\log 0.1}^{-0.5} \cdot \frac{{\log 0.1}^{-0.5}}{\frac{-1}{\tan^{-1}_* \frac{im}{re}}}} \]
4. Step-by-step derivation

   1. *-commutative 0.0%

      \[\leadsto \color{blue}{\frac{{\log 0.1}^{-0.5}}{\frac{-1}{\tan^{-1}_* \frac{im}{re}}} \cdot {\log 0.1}^{-0.5}} \]

   2. associate-/r/ 0.0%

      \[\leadsto \color{blue}{\left(\frac{{\log 0.1}^{-0.5}}{-1} \cdot \tan^{-1}_* \frac{im}{re}\right)} \cdot {\log 0.1}^{-0.5} \]

   3. associate-*r* 0.0%

      \[\leadsto \color{blue}{\frac{{\log 0.1}^{-0.5}}{-1} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot {\log 0.1}^{-0.5}\right)} \]

   4. associate-*l/ 0.0%

      \[\leadsto \color{blue}{\frac{{\log 0.1}^{-0.5} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot {\log 0.1}^{-0.5}\right)}{-1}} \]

   5. *-commutative 0.0%

      \[\leadsto \frac{\color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot {\log 0.1}^{-0.5}\right) \cdot {\log 0.1}^{-0.5}}}{-1} \]

   6. associate-*l* 0.0%

      \[\leadsto \frac{\color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \left({\log 0.1}^{-0.5} \cdot {\log 0.1}^{-0.5}\right)}}{-1} \]

   7. pow-sqr 99.8%

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \color{blue}{{\log 0.1}^{\left(2 \cdot -0.5\right)}}}{-1} \]

   8. metadata-eval 99.8%

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot {\log 0.1}^{\color{blue}{-1}}}{-1} \]

   9. unpow-1 99.8%

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{1}{\log 0.1}}}{-1} \]

   10. associate-*r/ 99.8%

       \[\leadsto \frac{\color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot 1}{\log 0.1}}}{-1} \]

   11. associate-/l* 99.8%

       \[\leadsto \frac{\color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\frac{\log 0.1}{1}}}}{-1} \]

   12. /-rgt-identity 99.8%

       \[\leadsto \frac{\frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\log 0.1}}}{-1} \]

   13. associate-/l/ 99.8%

       \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{-1 \cdot \log 0.1}} \]

   14. mul-1-neg 99.8%

       \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{-\log 0.1}} \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{-\log 0.1}} \]

6. Final simplification 99.8%

   \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{-\log 0.1} \]

Alternative 2: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (atan2 im re) (log 10.0)))

C

double code(double re, double im) {
	return atan2(im, re) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = atan2(im, re) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.atan2(im, re) / Math.log(10.0);
}

Python

def code(re, im):
	return math.atan2(im, re) / math.log(10.0)

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = atan2(im, re) / log(10.0);
end

Wolfram

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

Derivation

1. Initial program 98.6%

   \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \]

2. Final simplification 98.6%

   \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \]

Alternative 3: 27.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\tan^{-1}_* \frac{im}{re}}{2} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (atan2 im re) 2.0))

C

double code(double re, double im) {
	return atan2(im, re) / 2.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = atan2(im, re) / 2.0d0
end function

Java

public static double code(double re, double im) {
	return Math.atan2(im, re) / 2.0;
}

Python

def code(re, im):
	return math.atan2(im, re) / 2.0

Julia

function code(re, im)
	return Float64(atan(im, re) / 2.0)
end

MATLAB

function tmp = code(re, im)
	tmp = atan2(im, re) / 2.0;
end

Wolfram

code[re_, im_] := N[(N[ArcTan[im / re], $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 98.6%

   \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \]
2. Step-by-step derivation

   1. clear-num 98.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}} \]

   2. frac-2neg 98.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{-\log 10}{-\tan^{-1}_* \frac{im}{re}}}} \]

   3. div-inv 98.4%

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

   4. associate-/r* 97.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{-\log 10}}{\frac{1}{-\tan^{-1}_* \frac{im}{re}}}} \]

   5. inv-pow 97.6%

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

   6. sqr-pow 0.0%

      \[\leadsto \frac{\color{blue}{{\left(-\log 10\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(-\log 10\right)}^{\left(\frac{-1}{2}\right)}}}{\frac{1}{-\tan^{-1}_* \frac{im}{re}}} \]

   7. pow-prod-down 7.4%

      \[\leadsto \frac{\color{blue}{{\left(\left(-\log 10\right) \cdot \left(-\log 10\right)\right)}^{\left(\frac{-1}{2}\right)}}}{\frac{1}{-\tan^{-1}_* \frac{im}{re}}} \]

   8. sqr-neg 7.4%

      \[\leadsto \frac{{\color{blue}{\left(\log 10 \cdot \log 10\right)}}^{\left(\frac{-1}{2}\right)}}{\frac{1}{-\tan^{-1}_* \frac{im}{re}}} \]

   9. pow-prod-down 7.4%

      \[\leadsto \frac{\color{blue}{{\log 10}^{\left(\frac{-1}{2}\right)} \cdot {\log 10}^{\left(\frac{-1}{2}\right)}}}{\frac{1}{-\tan^{-1}_* \frac{im}{re}}} \]

   10. div-inv 7.4%

       \[\leadsto \frac{{\log 10}^{\left(\frac{-1}{2}\right)} \cdot {\log 10}^{\left(\frac{-1}{2}\right)}}{\color{blue}{1 \cdot \frac{1}{-\tan^{-1}_* \frac{im}{re}}}} \]

   11. times-frac 7.4%

       \[\leadsto \color{blue}{\frac{{\log 10}^{\left(\frac{-1}{2}\right)}}{1} \cdot \frac{{\log 10}^{\left(\frac{-1}{2}\right)}}{\frac{1}{-\tan^{-1}_* \frac{im}{re}}}} \]

3. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{{\log 0.1}^{-0.5} \cdot \frac{{\log 0.1}^{-0.5}}{\frac{-1}{\tan^{-1}_* \frac{im}{re}}}} \]
4. Step-by-step derivation

   1. *-commutative 0.0%

      \[\leadsto \color{blue}{\frac{{\log 0.1}^{-0.5}}{\frac{-1}{\tan^{-1}_* \frac{im}{re}}} \cdot {\log 0.1}^{-0.5}} \]

   2. associate-/r/ 0.0%

      \[\leadsto \color{blue}{\left(\frac{{\log 0.1}^{-0.5}}{-1} \cdot \tan^{-1}_* \frac{im}{re}\right)} \cdot {\log 0.1}^{-0.5} \]

   3. associate-*r* 0.0%

      \[\leadsto \color{blue}{\frac{{\log 0.1}^{-0.5}}{-1} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot {\log 0.1}^{-0.5}\right)} \]

   4. associate-*l/ 0.0%

      \[\leadsto \color{blue}{\frac{{\log 0.1}^{-0.5} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot {\log 0.1}^{-0.5}\right)}{-1}} \]

   5. *-commutative 0.0%

      \[\leadsto \frac{\color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot {\log 0.1}^{-0.5}\right) \cdot {\log 0.1}^{-0.5}}}{-1} \]

   6. associate-*l* 0.0%

      \[\leadsto \frac{\color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \left({\log 0.1}^{-0.5} \cdot {\log 0.1}^{-0.5}\right)}}{-1} \]

   7. pow-sqr 99.8%

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \color{blue}{{\log 0.1}^{\left(2 \cdot -0.5\right)}}}{-1} \]

   8. metadata-eval 99.8%

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot {\log 0.1}^{\color{blue}{-1}}}{-1} \]

   9. unpow-1 99.8%

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{1}{\log 0.1}}}{-1} \]

   10. associate-*r/ 99.8%

       \[\leadsto \frac{\color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot 1}{\log 0.1}}}{-1} \]

   11. associate-/l* 99.8%

       \[\leadsto \frac{\color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\frac{\log 0.1}{1}}}}{-1} \]

   12. /-rgt-identity 99.8%

       \[\leadsto \frac{\frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\log 0.1}}}{-1} \]

   13. associate-/l/ 99.8%

       \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{-1 \cdot \log 0.1}} \]

   14. mul-1-neg 99.8%

       \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{-\log 0.1}} \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{-\log 0.1}} \]

7. Applied egg-rr 26.6%

   \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{-\color{blue}{-2}} \]

8. Final simplification 26.6%

   \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{2} \]

Reproduce

herbie shell --seed 2023297 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))