math.log10 on complex, imaginary part

Percentage Accurate: 98.7% → 99.5%

Time: 4.6s

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (/ (/ -1.0 (log 0.1)) (/ 1.0 (atan2 im re))))

C

double code(double re, double im) {
	return (-1.0 / log(0.1)) / (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) / log(0.1d0)) / (1.0d0 / atan2(im, re))
end function

Java

public static double code(double re, double im) {
	return (-1.0 / Math.log(0.1)) / (1.0 / Math.atan2(im, re));
}

Python

def code(re, im):
	return (-1.0 / math.log(0.1)) / (1.0 / math.atan2(im, re))

Julia

function code(re, im)
	return Float64(Float64(-1.0 / log(0.1)) / Float64(1.0 / atan(im, re)))
end

MATLAB

function tmp = code(re, im)
	tmp = (-1.0 / log(0.1)) / (1.0 / atan2(im, re));
end

Wolfram

code[re_, im_] := N[(N[(-1.0 / N[Log[0.1], $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[ArcTan[im / re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.7%

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

   1. add-cbrt-cube 87.7%

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

   2. pow3 87.7%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)}^{3}}} \]

3. Applied egg-rr 87.7%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)}^{3}}} \]
4. Step-by-step derivation

   1. rem-cbrt-cube 98.7%

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

   2. clear-num 98.7%

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

   3. div-inv 98.7%

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

   4. associate-/r* 97.9%

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

   5. frac-2neg 97.9%

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

   6. metadata-eval 97.9%

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

   7. neg-log 99.7%

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

   8. metadata-eval 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

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 2023172 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))