math.log/2 on complex, imaginary part

Percentage Accurate: 50.4% → 99.5%

Time: 7.2s

Alternatives: 2

Speedup: 3.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (/
  (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))

C

double code(double re, double im, double base) {
	return ((atan2(im, re) * log(base)) - (log(sqrt(((re * re) + (im * im)))) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

Fortran

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

Java

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

Python

def code(re, im, base):
	return ((math.atan2(im, re) * math.log(base)) - (math.log(math.sqrt(((re * re) + (im * im)))) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

Julia

function code(re, im, base)
	return Float64(Float64(Float64(atan(im, re) * log(base)) - Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((atan2(im, re) * log(base)) - (log(sqrt(((re * re) + (im * im)))) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[ArcTan[im / re], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] - N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 50.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (/
  (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))

C

double code(double re, double im, double base) {
	return ((atan2(im, re) * log(base)) - (log(sqrt(((re * re) + (im * im)))) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

Fortran

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

Java

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

Python

def code(re, im, base):
	return ((math.atan2(im, re) * math.log(base)) - (math.log(math.sqrt(((re * re) + (im * im)))) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

Julia

function code(re, im, base)
	return Float64(Float64(Float64(atan(im, re) * log(base)) - Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((atan2(im, re) * log(base)) - (log(sqrt(((re * re) + (im * im)))) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[ArcTan[im / re], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] - N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (re im base) :precision binary64 (/ (atan2 im re) (log base)))

C

double code(double re, double im, double base) {
	return atan2(im, re) / log(base);
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = atan2(im, re) / log(base)
end function

Java

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

Python

def code(re, im, base):
	return math.atan2(im, re) / math.log(base)

Julia

function code(re, im, base)
	return Float64(atan(im, re) / log(base))
end

MATLAB

function tmp = code(re, im, base)
	tmp = atan2(im, re) / log(base);
end

Wolfram

code[re_, im_, base_] := N[(N[ArcTan[im / re], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.1%

   \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

2. Taylor expanded in im around 0 99.5%

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

3. Final simplification 99.5%

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

Alternative 2: 13.8% accurate, 6.1× speedup

Math

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

FPCore

(FPCore (re im base) :precision binary64 (atan2 im re))

C

double code(double re, double im, double base) {
	return atan2(im, re);
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = atan2(im, re)
end function

Java

public static double code(double re, double im, double base) {
	return Math.atan2(im, re);
}

Python

def code(re, im, base):
	return math.atan2(im, re)

Julia

function code(re, im, base)
	return atan(im, re)
end

MATLAB

function tmp = code(re, im, base)
	tmp = atan2(im, re);
end

Wolfram

code[re_, im_, base_] := N[ArcTan[im / re], $MachinePrecision]

Derivation

1. Initial program 50.1%

   \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Step-by-step derivation

   1. *-un-lft-identity 50.1%

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\color{blue}{1 \cdot \left(\log base \cdot \log base + 0 \cdot 0\right)}} \]

   2. cancel-sign-sub-inv 50.1%

      \[\leadsto \frac{\color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \log base + \left(-\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot 0}}{1 \cdot \left(\log base \cdot \log base + 0 \cdot 0\right)} \]

   3. mul0-rgt 99.3%

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \log base + \color{blue}{0}}{1 \cdot \left(\log base \cdot \log base + 0 \cdot 0\right)} \]

   4. +-rgt-identity 99.3%

      \[\leadsto \frac{\color{blue}{\tan^{-1}_* \frac{im}{re} \cdot \log base}}{1 \cdot \left(\log base \cdot \log base + 0 \cdot 0\right)} \]

   5. *-commutative 99.3%

      \[\leadsto \frac{\color{blue}{\log base \cdot \tan^{-1}_* \frac{im}{re}}}{1 \cdot \left(\log base \cdot \log base + 0 \cdot 0\right)} \]

   6. times-frac 99.2%

      \[\leadsto \color{blue}{\frac{\log base}{1} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log base \cdot \log base + 0 \cdot 0}} \]

   7. metadata-eval 99.2%

      \[\leadsto \frac{\log base}{1} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log base \cdot \log base + \color{blue}{0}} \]

   8. +-rgt-identity 99.2%

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

   9. pow2 99.2%

      \[\leadsto \frac{\log base}{1} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{{\log base}^{2}}} \]

3. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{\log base}{1} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{{\log base}^{2}}} \]
4. Step-by-step derivation

   1. /-rgt-identity 99.2%

      \[\leadsto \color{blue}{\log base} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{{\log base}^{2}} \]

   2. *-commutative 99.2%

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{{\log base}^{2}} \cdot \log base} \]

   3. remove-double-div 99.1%

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{{\log base}^{2}} \cdot \color{blue}{\frac{1}{\frac{1}{\log base}}} \]

   4. frac-times 99.2%

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot 1}{{\log base}^{2} \cdot \frac{1}{\log base}}} \]

   5. *-commutative 99.2%

      \[\leadsto \frac{\color{blue}{1 \cdot \tan^{-1}_* \frac{im}{re}}}{{\log base}^{2} \cdot \frac{1}{\log base}} \]

   6. *-un-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{\tan^{-1}_* \frac{im}{re}}}{{\log base}^{2} \cdot \frac{1}{\log base}} \]

5. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{{\log base}^{2} \cdot \frac{1}{\log base}}} \]
6. Step-by-step derivation

   1. inv-pow 99.3%

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{{\log base}^{2} \cdot \color{blue}{{\log base}^{-1}}} \]

   2. pow-prod-up 99.5%

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{{\log base}^{\left(2 + -1\right)}}} \]

   3. metadata-eval 99.5%

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

   4. pow1 99.5%

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

   5. add-cube-cbrt 99.5%

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

   6. log-prod 99.4%

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

   7. pow2 99.4%

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log \color{blue}{\left({\left(\sqrt[3]{base}\right)}^{2}\right)} + \log \left(\sqrt[3]{base}\right)} \]

7. Applied egg-rr 99.4%

   \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\log \left({\left(\sqrt[3]{base}\right)}^{2}\right) + \log \left(\sqrt[3]{base}\right)}} \]
8. Step-by-step derivation

   1. log-pow 99.4%

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{2 \cdot \log \left(\sqrt[3]{base}\right)} + \log \left(\sqrt[3]{base}\right)} \]

   2. distribute-lft1-in 99.4%

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{base}\right)}} \]

   3. metadata-eval 99.4%

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

9. Simplified 99.4%

   \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{3 \cdot \log \left(\sqrt[3]{base}\right)}} \]
10. Step-by-step derivation

    1. frac-2neg 99.4%

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

    2. add-log-exp 99.4%

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

    3. neg-log 99.4%

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

    4. *-commutative 99.4%

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

    5. exp-to-pow 99.5%

       \[\leadsto \frac{-\tan^{-1}_* \frac{im}{re}}{\log \left(\frac{1}{\color{blue}{{\left(\sqrt[3]{base}\right)}^{3}}}\right)} \]

    6. pow3 99.5%

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

    7. add-cube-cbrt 99.5%

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

    8. neg-log 99.5%

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

    9. frac-2neg 99.5%

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

    10. add-sqr-sqrt 49.9%

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

    11. *-un-lft-identity 49.9%

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

    12. times-frac 49.9%

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

11. Applied egg-rr 9.8%

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

    1. associate-*r/ 9.8%

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

    2. *-commutative 9.8%

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

    3. associate-/l* 9.8%

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

    4. *-inverses 14.2%

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

    5. rem-square-sqrt 10.3%

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

    6. associate-*r/ 10.3%

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

    7. /-rgt-identity 10.3%

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

    8. rem-square-sqrt 14.2%

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

13. Simplified 14.2%

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

14. Final simplification 14.2%

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

Reproduce

herbie shell --seed 2023174 
(FPCore (re im base)
  :name "math.log/2 on complex, imaginary part"
  :precision binary64
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))