math.log/2 on complex, imaginary part

Percentage Accurate: 49.3% → 99.5%

Time: 8.7s

Alternatives: 4

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: 49.3% 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 47.0%

   \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
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. div-sub 99.3%

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

   3. div0 99.3%

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

   4. --rgt-identity 99.3%

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

   5. metadata-eval 99.3%

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

   6. +-rgt-identity 99.3%

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

   7. times-frac 99.6%

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

   8. *-inverses 99.6%

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

   9. *-rgt-identity 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 13.5% accurate, 203.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.1 \cdot 10^{+21}:\\ \;\;\;\;-0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im base) :precision binary64 (if (<= re 5.1e+21) -0.25 0.0))

C

double code(double re, double im, double base) {
	double tmp;
	if (re <= 5.1e+21) {
		tmp = -0.25;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    real(8) :: tmp
    if (re <= 5.1d+21) then
        tmp = -0.25d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im, double base) {
	double tmp;
	if (re <= 5.1e+21) {
		tmp = -0.25;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(re, im, base):
	tmp = 0
	if re <= 5.1e+21:
		tmp = -0.25
	else:
		tmp = 0.0
	return tmp

Julia

function code(re, im, base)
	tmp = 0.0
	if (re <= 5.1e+21)
		tmp = -0.25;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im, base)
	tmp = 0.0;
	if (re <= 5.1e+21)
		tmp = -0.25;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_, base_] := If[LessEqual[re, 5.1e+21], -0.25, 0.0]

Derivation

1. Split input into 2 regimes

2. if re < 5.1e21

   1. Initial program 52.8%

      \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
   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. div-sub 99.3%

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

      3. div0 99.3%

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

      4. --rgt-identity 99.3%

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

      5. metadata-eval 99.3%

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

      6. +-rgt-identity 99.3%

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

      7. times-frac 99.5%

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

      8. *-inverses 99.5%

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

      9. *-rgt-identity 99.5%

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

   3. Simplified 99.5%

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

   5. Applied egg-rr 9.0%

      \[\leadsto \color{blue}{-0.25} \]

   if 5.1e21 < re

   1. Initial program 28.9%

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

      1. mul0-rgt 99.5%

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

      2. div-sub 99.5%

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

      3. div0 99.5%

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

      4. --rgt-identity 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-rgt-identity 99.5%

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

      7. times-frac 99.7%

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

      8. *-inverses 99.7%

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

      9. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   5. Applied egg-rr 35.8%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.1 \cdot 10^{+21}:\\ \;\;\;\;-0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3: 7.1% accurate, 622.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \end{array} \]

FPCore

(FPCore (re im base) :precision binary64 -0.5)

C

double code(double re, double im, double base) {
	return -0.5;
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = -0.5d0
end function

Java

public static double code(double re, double im, double base) {
	return -0.5;
}

Python

def code(re, im, base):
	return -0.5

Julia

function code(re, im, base)
	return -0.5
end

MATLAB

function tmp = code(re, im, base)
	tmp = -0.5;
end

Wolfram

code[re_, im_, base_] := -0.5

Derivation

1. Initial program 47.0%

   \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
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. div-sub 99.3%

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

   3. div0 99.3%

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

   4. --rgt-identity 99.3%

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

   5. metadata-eval 99.3%

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

   6. +-rgt-identity 99.3%

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

   7. times-frac 99.6%

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

   8. *-inverses 99.6%

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

   9. *-rgt-identity 99.6%

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

3. Simplified 99.6%

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

5. Applied egg-rr 7.9%

   \[\leadsto \color{blue}{-0.5} \]

6. Final simplification 7.9%

   \[\leadsto -0.5 \]

Alternative 4: 7.3% accurate, 622.0× speedup

Math

\[\begin{array}{l} \\ -0.25 \end{array} \]

FPCore

(FPCore (re im base) :precision binary64 -0.25)

C

double code(double re, double im, double base) {
	return -0.25;
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = -0.25d0
end function

Java

public static double code(double re, double im, double base) {
	return -0.25;
}

Python

def code(re, im, base):
	return -0.25

Julia

function code(re, im, base)
	return -0.25
end

MATLAB

function tmp = code(re, im, base)
	tmp = -0.25;
end

Wolfram

code[re_, im_, base_] := -0.25

Derivation

1. Initial program 47.0%

   \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
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. div-sub 99.3%

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

   3. div0 99.3%

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

   4. --rgt-identity 99.3%

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

   5. metadata-eval 99.3%

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

   6. +-rgt-identity 99.3%

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

   7. times-frac 99.6%

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

   8. *-inverses 99.6%

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

   9. *-rgt-identity 99.6%

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

3. Simplified 99.6%

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

5. Applied egg-rr 8.0%

   \[\leadsto \color{blue}{-0.25} \]

6. Final simplification 8.0%

   \[\leadsto -0.25 \]

Reproduce

herbie shell --seed 2023297 
(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))))