math.log10 on complex, imaginary part

Percentage Accurate: 98.7% → 99.5%

Time: 7.1s

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

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

FPCore

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

C

double code(double re, double im) {
	return 1.0 / ((-1.0 / atan2(im, re)) * log(0.1));
}

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)) * log(0.1d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(1.0 / N[(N[(-1.0 / N[ArcTan[im / re], $MachinePrecision]), $MachinePrecision] * N[Log[0.1], $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. clear-num 98.6%

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

   2. inv-pow 98.6%

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

4. Applied egg-rr 98.6%

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

6. Applied egg-rr 98.9%

   \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{\frac{\mathsf{log1p}\left(9\right)}{\tan^{-1}_* \frac{im}{re}}}\right)}^{3}\right)}}^{-1} \]
7. Step-by-step derivation

   1. pow-pow 98.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\mathsf{log1p}\left(9\right)}{\tan^{-1}_* \frac{im}{re}}}\right)}^{\left(3 \cdot -1\right)}} \]

   2. pow1/3 52.9%

      \[\leadsto {\color{blue}{\left({\left(\frac{\mathsf{log1p}\left(9\right)}{\tan^{-1}_* \frac{im}{re}}\right)}^{0.3333333333333333}\right)}}^{\left(3 \cdot -1\right)} \]

   3. pow-pow 98.6%

      \[\leadsto \color{blue}{{\left(\frac{\mathsf{log1p}\left(9\right)}{\tan^{-1}_* \frac{im}{re}}\right)}^{\left(0.3333333333333333 \cdot \left(3 \cdot -1\right)\right)}} \]

   4. metadata-eval 98.6%

      \[\leadsto {\left(\frac{\mathsf{log1p}\left(9\right)}{\tan^{-1}_* \frac{im}{re}}\right)}^{\left(0.3333333333333333 \cdot \color{blue}{-3}\right)} \]

   5. metadata-eval 98.6%

      \[\leadsto {\left(\frac{\mathsf{log1p}\left(9\right)}{\tan^{-1}_* \frac{im}{re}}\right)}^{\color{blue}{-1}} \]

   6. inv-pow 98.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{log1p}\left(9\right)}{\tan^{-1}_* \frac{im}{re}}}} \]

   7. clear-num 98.6%

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\mathsf{log1p}\left(9\right)}} \]

   8. div-inv 98.6%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\mathsf{log1p}\left(9\right)}} \]

   9. remove-double-div 98.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\tan^{-1}_* \frac{im}{re}}}} \cdot \frac{1}{\mathsf{log1p}\left(9\right)} \]

   10. frac-2neg 98.6%

       \[\leadsto \color{blue}{\frac{-1}{-\frac{1}{\tan^{-1}_* \frac{im}{re}}}} \cdot \frac{1}{\mathsf{log1p}\left(9\right)} \]

   11. metadata-eval 98.6%

       \[\leadsto \frac{\color{blue}{-1}}{-\frac{1}{\tan^{-1}_* \frac{im}{re}}} \cdot \frac{1}{\mathsf{log1p}\left(9\right)} \]

   12. frac-2neg 98.6%

       \[\leadsto \frac{-1}{-\frac{1}{\tan^{-1}_* \frac{im}{re}}} \cdot \color{blue}{\frac{-1}{-\mathsf{log1p}\left(9\right)}} \]

   13. metadata-eval 98.6%

       \[\leadsto \frac{-1}{-\frac{1}{\tan^{-1}_* \frac{im}{re}}} \cdot \frac{\color{blue}{-1}}{-\mathsf{log1p}\left(9\right)} \]

   14. log1p-undefine 98.6%

       \[\leadsto \frac{-1}{-\frac{1}{\tan^{-1}_* \frac{im}{re}}} \cdot \frac{-1}{-\color{blue}{\log \left(1 + 9\right)}} \]

   15. neg-log 99.8%

       \[\leadsto \frac{-1}{-\frac{1}{\tan^{-1}_* \frac{im}{re}}} \cdot \frac{-1}{\color{blue}{\log \left(\frac{1}{1 + 9}\right)}} \]

   16. metadata-eval 99.8%

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

   17. metadata-eval 99.8%

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

   18. metadata-eval 98.6%

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

   19. log1p-undefine 98.6%

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

   20. frac-times 98.6%

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

   21. metadata-eval 98.6%

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

   22. distribute-neg-frac 98.6%

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

   23. metadata-eval 98.6%

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

   24. log1p-undefine 98.6%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

   \[\leadsto \frac{1}{\frac{-1}{\tan^{-1}_* \frac{im}{re}} \cdot \log 0.1} \]
10. 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(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. 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. metadata-eval 98.6%

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

   2. distribute-neg-frac 98.6%

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

   3. distribute-neg-frac2 98.6%

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

   4. associate-/l* 98.6%

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

   5. *-rgt-identity 98.6%

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

   6. log1p-undefine 98.6%

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

   7. metadata-eval 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

   \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{-\log 0.1} \]
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 98.6%

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

3. Final simplification 98.6%

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

Reproduce

herbie shell --seed 2024077 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))