math.log10 on complex, imaginary part

Percentage Accurate: 98.7% → 99.6%

Time: 5.5s

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: 99.6% 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 98.6%

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

   1. clear-num 98.7%

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

   2. inv-pow 98.7%

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

   3. div-inv 98.6%

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

   4. unpow-prod-down 98.5%

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

   5. inv-pow 98.5%

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

3. Applied egg-rr 98.5%

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

   1. expm1-log1p-u 98.7%

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

   2. expm1-udef 97.7%

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

   3. metadata-eval 97.7%

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

   4. metadata-eval 97.7%

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

   5. neg-log 98.4%

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

   6. frac-2neg 98.4%

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

5. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-1}{\log 0.1}\right)} - 1\right)} \cdot {\left(\frac{1}{\tan^{-1}_* \frac{im}{re}}\right)}^{-1} \]
6. Step-by-step derivation

   1. expm1-def 99.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{\log 0.1}\right)\right)} \cdot {\left(\frac{1}{\tan^{-1}_* \frac{im}{re}}\right)}^{-1} \]

   2. expm1-log1p 99.8%

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

7. Simplified 99.8%

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

   1. associate-*l/ 99.8%

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

   2. metadata-eval 99.8%

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

   3. unpow-prod-down 99.8%

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

   4. div-inv 99.8%

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

   5. unpow-1 99.8%

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

   6. associate-/r* 99.8%

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

   7. *-commutative 99.8%

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

   8. associate-/r* 99.8%

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

   9. frac-2neg 99.8%

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

   10. metadata-eval 99.8%

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

   11. neg-log 97.9%

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

   12. metadata-eval 97.9%

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

9. Applied egg-rr 97.9%

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

    1. div-inv 97.9%

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

    2. frac-2neg 97.9%

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

    3. metadata-eval 97.9%

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

    4. neg-log 99.8%

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

    5. metadata-eval 99.8%

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

    6. metadata-eval 97.9%

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

    7. log1p-udef 97.9%

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

    8. log1p-udef 97.9%

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

    9. metadata-eval 99.8%

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

11. Applied egg-rr 99.8%

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

    1. associate-*r/ 99.8%

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

    2. metadata-eval 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

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 98.6%

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

   1. frac-2neg 98.6%

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

   2. div-inv 98.5%

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

   3. neg-log 99.8%

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

   4. metadata-eval 99.8%

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

3. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

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 98.6%

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

2. Final simplification 98.6%

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

Reproduce

herbie shell --seed 2023320 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))