math.log10 on complex, imaginary part

Percentage Accurate: 98.7% → 99.8%

Time: 6.1s

Alternatives: 4

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. Add Preprocessing
3. Step-by-step derivation

   1. remove-double-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. neg-log N/A

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

   5. lower-log.f64 N/A

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

   6. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

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. Add Preprocessing
3. Add Preprocessing

Alternative 3: 22.8% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (/ 1.0 (/ 1.0 (atan2 im re))))

C

double code(double re, double im) {
	return 1.0 / (1.0 / atan2(im, re));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 / (1.0d0 / atan2(im, re))
end function

Java

public static double code(double re, double im) {
	return 1.0 / (1.0 / Math.atan2(im, re));
}

Python

def code(re, im):
	return 1.0 / (1.0 / math.atan2(im, re))

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = 1.0 / (1.0 / atan2(im, re));
end

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 98.4

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

4. Applied rewrites 98.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

6. Applied rewrites 98.4%

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

8. Applied rewrites 20.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{\tan^{-1}_* \frac{im}{re}}}} \]
9. Add Preprocessing

Alternative 4: 22.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return atan(im, re)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 98.4

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

4. Applied rewrites 98.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

6. Applied rewrites 98.4%

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

8. Applied rewrites 20.7%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))