math.log/2 on complex, imaginary part

Percentage Accurate: 49.6% → 99.5%

Time: 7.4s

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: 49.6% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.5%

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

   1. mul0-rgt 99.4%

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

   2. --rgt-identity 99.4%

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

   3. metadata-eval 99.4%

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

   4. +-rgt-identity 99.4%

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

   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. Taylor expanded in base around inf 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 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 48.5%

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

   1. mul0-rgt 99.4%

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

   2. --rgt-identity 99.4%

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

   3. metadata-eval 99.4%

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

   4. +-rgt-identity 99.4%

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

   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. Final simplification 99.5%

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

Alternative 3: 13.7% 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 48.5%

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

   1. mul0-rgt 99.4%

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

   2. --rgt-identity 99.4%

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

   3. metadata-eval 99.4%

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

   4. +-rgt-identity 99.4%

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

   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. Step-by-step derivation

   1. div-inv 99.4%

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

5. Applied egg-rr 99.4%

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

   1. div-inv 99.5%

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

   2. clear-num 98.9%

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

7. Applied egg-rr 98.9%

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

   1. associate-/l* 99.5%

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

   2. add-sqr-sqrt 51.1%

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

   3. times-frac 51.0%

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

   4. metadata-eval 51.0%

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

   5. sqrt-div 51.2%

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

   6. add-exp-log 51.0%

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

   7. exp-neg 51.0%

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

   8. add-sqr-sqrt 0.0%

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

   9. sqrt-unprod 10.8%

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

   10. sqr-neg 10.8%

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

   11. sqrt-unprod 10.8%

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

   12. add-sqr-sqrt 10.8%

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

   13. add-exp-log 10.8%

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

   14. /-rgt-identity 10.8%

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

   15. /-rgt-identity 10.8%

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

9. Applied egg-rr 10.8%

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

    1. associate-*r/ 10.8%

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

    2. *-commutative 10.8%

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

    3. associate-/l* 10.8%

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

    4. *-inverses 14.1%

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

    5. /-rgt-identity 14.1%

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

11. Simplified 14.1%

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

12. Final simplification 14.1%

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

Reproduce

herbie shell --seed 2023189 
(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))))