math.log/2 on complex, imaginary part

Percentage Accurate: 50.3% → 99.5%

Time: 12.0s

Alternatives: 3

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: 50.3% 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 47.3%

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

   1. mul0-rgt 99.3%

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

   2. --rgt-identity 99.3%

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

   3. metadata-eval 99.3%

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

   4. +-rgt-identity 99.3%

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

   5. times-frac 99.5%

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

   6. *-inverses 99.5%

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

   7. *-rgt-identity 99.5%

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

3. Simplified 99.5%

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

Alternative 2: 13.4% accurate, 5.9× speedup

Math

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

FPCore

(FPCore (re im base) :precision binary64 (/ 1.0 (/ 1.0 (atan2 im re))))

C

double code(double re, double im, double base) {
	return 1.0 / (1.0 / 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 = 1.0d0 / (1.0d0 / atan2(im, re))
end function

Java

public static double code(double re, double im, double base) {
	return 1.0 / (1.0 / Math.atan2(im, re));
}

Python

def code(re, im, base):
	return 1.0 / (1.0 / math.atan2(im, re))

Julia

function code(re, im, base)
	return Float64(1.0 / Float64(1.0 / atan(im, re)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = 1.0 / (1.0 / atan2(im, re));
end

Wolfram

code[re_, im_, base_] := N[(1.0 / N[(1.0 / N[ArcTan[im / re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.3%

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

   1. mul0-rgt 99.3%

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

   2. --rgt-identity 99.3%

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

   3. metadata-eval 99.3%

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

   4. +-rgt-identity 99.3%

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

   5. times-frac 99.5%

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

   6. *-inverses 99.5%

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

   7. *-rgt-identity 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 98.6%

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

6. Applied egg-rr 98.6%

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

   1. clear-num 98.7%

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

   2. associate-/r/ 98.6%

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

8. Applied egg-rr 98.6%

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

10. Applied egg-rr 11.0%

    \[\leadsto \frac{1}{\color{blue}{\frac{\log base}{\log base \cdot {\tan^{-1}_* \frac{im}{re}}^{0.5}} \cdot \frac{\log base}{\log base \cdot {\tan^{-1}_* \frac{im}{re}}^{0.5}}}} \]
11. Step-by-step derivation

    1. associate-*r/ 11.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\log base}{\log base \cdot {\tan^{-1}_* \frac{im}{re}}^{0.5}} \cdot \log base}{\log base \cdot {\tan^{-1}_* \frac{im}{re}}^{0.5}}}} \]

    2. *-commutative 11.0%

       \[\leadsto \frac{1}{\frac{\frac{\log base}{\log base \cdot {\tan^{-1}_* \frac{im}{re}}^{0.5}} \cdot \log base}{\color{blue}{{\tan^{-1}_* \frac{im}{re}}^{0.5} \cdot \log base}}} \]

    3. times-frac 11.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\log base}{\log base \cdot {\tan^{-1}_* \frac{im}{re}}^{0.5}}}{{\tan^{-1}_* \frac{im}{re}}^{0.5}} \cdot \frac{\log base}{\log base}}} \]

    4. *-inverses 11.0%

       \[\leadsto \frac{1}{\frac{\frac{\log base}{\log base \cdot {\tan^{-1}_* \frac{im}{re}}^{0.5}}}{{\tan^{-1}_* \frac{im}{re}}^{0.5}} \cdot \color{blue}{1}} \]

    5. associate-/r/ 11.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\log base}{\log base \cdot {\tan^{-1}_* \frac{im}{re}}^{0.5}}}{\frac{{\tan^{-1}_* \frac{im}{re}}^{0.5}}{1}}}} \]

    6. associate-/r* 11.0%

       \[\leadsto \frac{1}{\frac{\color{blue}{\frac{\frac{\log base}{\log base}}{{\tan^{-1}_* \frac{im}{re}}^{0.5}}}}{\frac{{\tan^{-1}_* \frac{im}{re}}^{0.5}}{1}}} \]

    7. *-inverses 11.0%

       \[\leadsto \frac{1}{\frac{\frac{\color{blue}{1}}{{\tan^{-1}_* \frac{im}{re}}^{0.5}}}{\frac{{\tan^{-1}_* \frac{im}{re}}^{0.5}}{1}}} \]

    8. /-rgt-identity 11.0%

       \[\leadsto \frac{1}{\frac{\frac{1}{{\tan^{-1}_* \frac{im}{re}}^{0.5}}}{\color{blue}{{\tan^{-1}_* \frac{im}{re}}^{0.5}}}} \]

    9. associate-/l/ 11.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{{\tan^{-1}_* \frac{im}{re}}^{0.5} \cdot {\tan^{-1}_* \frac{im}{re}}^{0.5}}}} \]

    10. unpow1/2 11.0%

        \[\leadsto \frac{1}{\frac{1}{\color{blue}{\sqrt{\tan^{-1}_* \frac{im}{re}}} \cdot {\tan^{-1}_* \frac{im}{re}}^{0.5}}} \]

    11. unpow1/2 11.0%

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

    12. rem-square-sqrt 15.5%

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

12. Simplified 15.5%

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

Alternative 3: 13.3% 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 47.3%

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

   1. sub-neg 47.3%

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

   2. mul0-rgt 99.3%

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

   3. metadata-eval 99.3%

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

   4. +-rgt-identity 99.3%

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

   5. *-commutative 99.3%

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

   6. associate-/l* 99.2%

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

   7. metadata-eval 99.2%

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

   8. +-rgt-identity 99.2%

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

   9. pow2 99.2%

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

4. Applied egg-rr 99.2%

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

   1. *-commutative 99.2%

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

   2. associate-*l/ 99.3%

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

6. Applied egg-rr 99.3%

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

8. Applied egg-rr 15.4%

   \[\leadsto \color{blue}{\left(0 - \tan^{-1}_* \frac{im}{re} \cdot \log base\right) \cdot \frac{1}{0 - \log base}} \]
9. Step-by-step derivation

   1. sub0-neg 15.4%

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

   2. distribute-lft-neg-out 15.4%

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

   3. associate-/l* 15.4%

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

   4. *-rgt-identity 15.4%

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

   5. distribute-neg-frac2 15.4%

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

   6. sub0-neg 15.4%

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

   7. remove-double-neg 15.4%

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

   8. associate-/l* 15.4%

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

   9. *-inverses 15.4%

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

   10. *-rgt-identity 15.4%

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

10. Simplified 15.4%

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

Reproduce

herbie shell --seed 2024097 
(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))))