math.log10 on complex, imaginary part

Percentage Accurate: 98.7% → 99.8%

Time: 5.2s

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\log 10}^{0.25}\\ \frac{\frac{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}{t_0}}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (pow (log 10.0) 0.25)))
   (/ (/ (/ (atan2 im re) (sqrt (log 10.0))) t_0) t_0)))

C

double code(double re, double im) {
	double t_0 = pow(log(10.0), 0.25);
	return ((atan2(im, re) / sqrt(log(10.0))) / t_0) / t_0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    t_0 = log(10.0d0) ** 0.25d0
    code = ((atan2(im, re) / sqrt(log(10.0d0))) / t_0) / t_0
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(Math.log(10.0), 0.25);
	return ((Math.atan2(im, re) / Math.sqrt(Math.log(10.0))) / t_0) / t_0;
}

Python

def code(re, im):
	t_0 = math.pow(math.log(10.0), 0.25)
	return ((math.atan2(im, re) / math.sqrt(math.log(10.0))) / t_0) / t_0

Julia

function code(re, im)
	t_0 = log(10.0) ^ 0.25
	return Float64(Float64(Float64(atan(im, re) / sqrt(log(10.0))) / t_0) / t_0)
end

MATLAB

function tmp = code(re, im)
	t_0 = log(10.0) ^ 0.25;
	tmp = ((atan2(im, re) / sqrt(log(10.0))) / t_0) / t_0;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Power[N[Log[10.0], $MachinePrecision], 0.25], $MachinePrecision]}, N[(N[(N[(N[ArcTan[im / re], $MachinePrecision] / N[Sqrt[N[Log[10.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 98.7%

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

   1. expm1-log1p-u 86.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)\right)} \]

   2. expm1-udef 67.2%

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

   3. log1p-udef 67.2%

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

   4. add-exp-log 79.1%

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

3. Applied egg-rr 79.1%

   \[\leadsto \color{blue}{\left(1 + \frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right) - 1} \]
4. Step-by-step derivation

   1. associate--l+ 79.1%

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

5. Simplified 79.1%

   \[\leadsto \color{blue}{1 + \left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} - 1\right)} \]
6. Step-by-step derivation

   1. associate-+r- 79.1%

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

   2. add-exp-log 67.2%

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

   3. log1p-udef 67.2%

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

   4. expm1-udef 86.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)\right)} \]

   5. expm1-log1p-u 98.7%

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

   6. add-sqr-sqrt 98.7%

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

   7. associate-/r* 98.7%

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

   8. add-sqr-sqrt 97.8%

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

   9. associate-/r* 97.8%

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

   10. pow1/2 97.8%

       \[\leadsto \frac{\frac{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}{\sqrt{\color{blue}{{\log 10}^{0.5}}}}}{\sqrt{\sqrt{\log 10}}} \]

   11. sqrt-pow1 98.7%

       \[\leadsto \frac{\frac{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}{\color{blue}{{\log 10}^{\left(\frac{0.5}{2}\right)}}}}{\sqrt{\sqrt{\log 10}}} \]

   12. metadata-eval 98.7%

       \[\leadsto \frac{\frac{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}{{\log 10}^{\color{blue}{0.25}}}}{\sqrt{\sqrt{\log 10}}} \]

   13. pow1/2 98.7%

       \[\leadsto \frac{\frac{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}{{\log 10}^{0.25}}}{\sqrt{\color{blue}{{\log 10}^{0.5}}}} \]

   14. sqrt-pow1 99.8%

       \[\leadsto \frac{\frac{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}{{\log 10}^{0.25}}}{\color{blue}{{\log 10}^{\left(\frac{0.5}{2}\right)}}} \]

   15. metadata-eval 99.8%

       \[\leadsto \frac{\frac{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}{{\log 10}^{0.25}}}{{\log 10}^{\color{blue}{0.25}}} \]

7. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}{{\log 10}^{0.25}}}{{\log 10}^{0.25}}} \]

8. Final simplification 99.8%

   \[\leadsto \frac{\frac{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}{{\log 10}^{0.25}}}{{\log 10}^{0.25}} \]

Alternative 2: 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(-Float64(atan(im, re) / 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 98.7%

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

   1. frac-2neg 98.7%

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

   2. div-inv 98.6%

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

   3. neg-log 99.7%

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

   4. metadata-eval 99.7%

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

3. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 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 98.7%

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

2. Final simplification 98.7%

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

Reproduce

herbie shell --seed 2023178 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))