math.log10 on complex, imaginary part

Percentage Accurate: 98.7% → 99.5%

Time: 7.0s

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

   1. expm1-log1p-u 86.4%

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

   2. expm1-undefine 66.7%

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

   3. log1p-undefine 66.7%

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

   4. rem-exp-log 79.0%

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

4. Applied egg-rr 79.0%

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

   1. add-exp-log 66.7%

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

   2. log1p-define 66.7%

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

   3. expm1-define 86.4%

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

   4. expm1-log1p-u 98.7%

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

   5. frac-2neg 98.7%

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

   6. neg-log 99.7%

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

   7. metadata-eval 99.7%

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

   8. neg-mul-1 99.7%

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

   9. associate-/l* 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

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. Add Preprocessing
3. Step-by-step derivation

   1. div-inv 98.6%

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

   2. frac-2neg 98.6%

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

   3. metadata-eval 98.6%

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

   4. log1p-expm1-u 98.6%

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

   5. expm1-undefine 98.6%

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

   6. exp-neg 98.6%

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

   7. rem-exp-log 98.6%

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

   8. metadata-eval 98.6%

      \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \frac{-1}{\mathsf{log1p}\left(\color{blue}{0.1} - 1\right)} \]

   9. metadata-eval 98.6%

      \[\leadsto \tan^{-1}_* \frac{im}{re} \cdot \frac{-1}{\mathsf{log1p}\left(\color{blue}{-0.9}\right)} \]

4. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \frac{-1}{\mathsf{log1p}\left(-0.9\right)}} \]
5. Step-by-step derivation

   1. associate-*r/ 98.7%

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot -1}{\mathsf{log1p}\left(-0.9\right)}} \]

   2. *-commutative 98.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot \tan^{-1}_* \frac{im}{re}}}{\mathsf{log1p}\left(-0.9\right)} \]

   3. neg-mul-1 98.7%

      \[\leadsto \frac{\color{blue}{-\tan^{-1}_* \frac{im}{re}}}{\mathsf{log1p}\left(-0.9\right)} \]

   4. log1p-undefine 98.7%

      \[\leadsto \frac{-\tan^{-1}_* \frac{im}{re}}{\color{blue}{\log \left(1 + -0.9\right)}} \]

   5. metadata-eval 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

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. Add Preprocessing

3. Final simplification 98.7%

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

Reproduce

herbie shell --seed 2024030 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))