math.log10 on complex, imaginary part

Percentage Accurate: 98.7% → 99.8%

Time: 6.4s

Alternatives: 2

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 98.7%

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

   1. log1p-expm1-u 98.8%

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

4. Applied egg-rr 98.8%

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

   1. log1p-expm1-u 98.7%

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

   2. div-inv 98.6%

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

   3. frac-2neg 98.6%

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

   4. metadata-eval 98.6%

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

   5. neg-log 99.7%

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

   6. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

   1. rem-square-sqrt 54.3%

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

   2. associate-*r* 54.3%

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

   3. *-lft-identity 54.3%

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

   4. associate-*l* 54.3%

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

   5. metadata-eval 54.3%

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

   6. associate-*r* 54.3%

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

   7. rem-square-sqrt 99.7%

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

   8. associate-*r/ 99.8%

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

   9. *-commutative 99.8%

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

   10. neg-mul-1 99.8%

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

   11. times-frac 99.8%

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

   12. neg-mul-1 99.8%

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

   13. neg-mul-1 99.8%

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

   14. remove-double-neg 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

   \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{-\log 0.1} \]
10. 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 98.7%

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

3. Final simplification 98.7%

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

Reproduce

herbie shell --seed 2024033 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))