math.log10 on complex, imaginary part

Percentage Accurate: 98.7% → 99.5%

Time: 2.1s

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. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\log 10\right)}\right), \color{blue}{\left(\frac{1}{\mathsf{neg}\left(\tan^{-1}_* \frac{im}{re}\right)}\right)}\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\log 10\right)\right)\right), \left(\frac{\color{blue}{1}}{\mathsf{neg}\left(\tan^{-1}_* \frac{im}{re}\right)}\right)\right) \]

   7. neg-log N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \log \left(\frac{1}{10}\right)\right), \left(\frac{1}{\mathsf{neg}\left(\tan^{-1}_* \frac{im}{re}\right)}\right)\right) \]

   8. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{1}{10}\right)\right)\right), \left(\frac{1}{\mathsf{neg}\left(\tan^{-1}_* \frac{im}{re}\right)}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{log.f64}\left(\frac{1}{10}\right)\right), \left(\frac{1}{\mathsf{neg}\left(\tan^{-1}_* \frac{im}{re}\right)}\right)\right) \]

   10. frac-2neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{log.f64}\left(\frac{1}{10}\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\tan^{-1}_* \frac{im}{re}\right)\right)\right)}}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{log.f64}\left(\frac{1}{10}\right)\right), \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\tan^{-1}_* \frac{im}{re}\right)\right)}\right)}\right)\right) \]

   12. remove-double-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{log.f64}\left(\frac{1}{10}\right)\right), \left(\frac{-1}{\tan^{-1}_* \frac{im}{\color{blue}{re}}}\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{log.f64}\left(\frac{1}{10}\right)\right), \mathsf{/.f64}\left(-1, \color{blue}{\tan^{-1}_* \frac{im}{re}}\right)\right) \]

   14. atan2-lowering-atan2.f64 99.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{log.f64}\left(\frac{1}{10}\right)\right), \mathsf{/.f64}\left(-1, \mathsf{atan2.f64}\left(im, \color{blue}{re}\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\tan^{-1}_* \frac{im}{re}}{0 - \log 0.1} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (atan2 im re) (- 0.0 (log 0.1))))

C

double code(double re, double im) {
	return atan2(im, re) / (0.0 - log(0.1));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = atan2(im, re) / (0.0d0 - log(0.1d0))
end function

Java

public static double code(double re, double im) {
	return Math.atan2(im, re) / (0.0 - Math.log(0.1));
}

Python

def code(re, im):
	return math.atan2(im, re) / (0.0 - math.log(0.1))

Julia

function code(re, im)
	return Float64(atan(im, re) / Float64(0.0 - log(0.1)))
end

MATLAB

function tmp = code(re, im)
	tmp = atan2(im, re) / (0.0 - log(0.1));
end

Wolfram

code[re_, im_] := N[(N[ArcTan[im / re], $MachinePrecision] / N[(0.0 - N[Log[0.1], $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. remove-double-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{atan2.f64}\left(im, re\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log 10\right)\right)\right)\right)\right) \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{atan2.f64}\left(im, re\right), \mathsf{neg.f64}\left(\left(\mathsf{neg}\left(\log 10\right)\right)\right)\right) \]

   3. neg-log N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{atan2.f64}\left(im, re\right), \mathsf{neg.f64}\left(\log \left(\frac{1}{10}\right)\right)\right) \]

   4. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{atan2.f64}\left(im, re\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{1}{10}\right)\right)\right)\right) \]

   5. metadata-eval 99.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{atan2.f64}\left(im, re\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\frac{1}{10}\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

   \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{0 - \log 0.1} \]
6. 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. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))