math.log10 on complex, real part

Percentage Accurate: 52.1% → 56.4%

Time: 1.4s

Alternatives: 6

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Initial Program: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Alternative 1: 56.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot re + im \cdot im\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{\log \left(\sqrt{t\_0}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\log 10\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* re re) (* im im))))
   (if (<= t_0 2e+302) (/ (log (sqrt t_0)) (log 10.0)) (log 10.0))))

C

double code(double re, double im) {
	double t_0 = (re * re) + (im * im);
	double tmp;
	if (t_0 <= 2e+302) {
		tmp = log(sqrt(t_0)) / log(10.0);
	} else {
		tmp = log(10.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (re * re) + (im * im)
    if (t_0 <= 2d+302) then
        tmp = log(sqrt(t_0)) / log(10.0d0)
    else
        tmp = log(10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * re) + (im * im);
	double tmp;
	if (t_0 <= 2e+302) {
		tmp = Math.log(Math.sqrt(t_0)) / Math.log(10.0);
	} else {
		tmp = Math.log(10.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * re) + (im * im)
	tmp = 0
	if t_0 <= 2e+302:
		tmp = math.log(math.sqrt(t_0)) / math.log(10.0)
	else:
		tmp = math.log(10.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * re) + Float64(im * im))
	tmp = 0.0
	if (t_0 <= 2e+302)
		tmp = Float64(log(sqrt(t_0)) / log(10.0));
	else
		tmp = log(10.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * re) + (im * im);
	tmp = 0.0;
	if (t_0 <= 2e+302)
		tmp = log(sqrt(t_0)) / log(10.0);
	else
		tmp = log(10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+302], N[(N[Log[N[Sqrt[t$95$0], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision], N[Log[10.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 re re) (*.f64 im im)) < 2.0000000000000002e302

   1. Initial program 91.8%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   2. Add Preprocessing

   if 2.0000000000000002e302 < (+.f64 (*.f64 re re) (*.f64 im im))

   1. Initial program 3.1%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{{re}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{re}^{2}}{{im}^{4} \cdot \log 10} + \frac{1}{2} \cdot \frac{1}{{im}^{2} \cdot \log 10}\right) + \frac{\log im}{\log 10}} \]

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\sqrt{re \cdot re + im \cdot im}} \]

   6. Taylor expanded in re around 0

      \[\leadsto im + \color{blue}{{re}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)} \]

   8. Applied rewrites 14.5%

      \[\leadsto \log 10 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 15.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot re + im \cdot im\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\log \left(\sqrt{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\log 10\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* re re) (* im im))))
   (if (<= t_0 2e+302) (log (sqrt t_0)) (log 10.0))))

C

double code(double re, double im) {
	double t_0 = (re * re) + (im * im);
	double tmp;
	if (t_0 <= 2e+302) {
		tmp = log(sqrt(t_0));
	} else {
		tmp = log(10.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (re * re) + (im * im)
    if (t_0 <= 2d+302) then
        tmp = log(sqrt(t_0))
    else
        tmp = log(10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * re) + (im * im);
	double tmp;
	if (t_0 <= 2e+302) {
		tmp = Math.log(Math.sqrt(t_0));
	} else {
		tmp = Math.log(10.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * re) + (im * im)
	tmp = 0
	if t_0 <= 2e+302:
		tmp = math.log(math.sqrt(t_0))
	else:
		tmp = math.log(10.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * re) + Float64(im * im))
	tmp = 0.0
	if (t_0 <= 2e+302)
		tmp = log(sqrt(t_0));
	else
		tmp = log(10.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * re) + (im * im);
	tmp = 0.0;
	if (t_0 <= 2e+302)
		tmp = log(sqrt(t_0));
	else
		tmp = log(10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+302], N[Log[N[Sqrt[t$95$0], $MachinePrecision]], $MachinePrecision], N[Log[10.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 re re) (*.f64 im im)) < 2.0000000000000002e302

   1. Initial program 91.8%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2} \cdot \log 10} + \frac{\log im}{\log 10}} \]

   5. Applied rewrites 17.3%

      \[\leadsto \color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \]

   if 2.0000000000000002e302 < (+.f64 (*.f64 re re) (*.f64 im im))

   1. Initial program 3.1%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{{re}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{re}^{2}}{{im}^{4} \cdot \log 10} + \frac{1}{2} \cdot \frac{1}{{im}^{2} \cdot \log 10}\right) + \frac{\log im}{\log 10}} \]

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\sqrt{re \cdot re + im \cdot im}} \]

   6. Taylor expanded in re around 0

      \[\leadsto im + \color{blue}{{re}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)} \]

   8. Applied rewrites 14.5%

      \[\leadsto \log 10 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 11.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \log 10 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (log 10.0))

C

double code(double re, double im) {
	return log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(10.0);
}

Python

def code(re, im):
	return math.log(10.0)

Julia

function code(re, im)
	return log(10.0)
end

MATLAB

function tmp = code(re, im)
	tmp = log(10.0);
end

Wolfram

code[re_, im_] := N[Log[10.0], $MachinePrecision]

Derivation

1. Initial program 56.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{{re}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{re}^{2}}{{im}^{4} \cdot \log 10} + \frac{1}{2} \cdot \frac{1}{{im}^{2} \cdot \log 10}\right) + \frac{\log im}{\log 10}} \]

5. Applied rewrites 4.2%

   \[\leadsto \color{blue}{\sqrt{re \cdot re + im \cdot im}} \]

6. Taylor expanded in re around 0

   \[\leadsto im + \color{blue}{{re}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{re}^{2}}{{im}^{3}} + \frac{1}{2} \cdot \frac{1}{im}\right)} \]

8. Applied rewrites 11.5%

   \[\leadsto \log 10 \]
9. Add Preprocessing

Alternative 4: 4.1% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{re \cdot re + im \cdot im} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))

C

double code(double re, double im) {
	return sqrt(((re * re) + (im * im)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt(((re * re) + (im * im)))
end function

Java

public static double code(double re, double im) {
	return Math.sqrt(((re * re) + (im * im)));
}

Python

def code(re, im):
	return math.sqrt(((re * re) + (im * im)))

Julia

function code(re, im)
	return sqrt(Float64(Float64(re * re) + Float64(im * im)))
end

MATLAB

function tmp = code(re, im)
	tmp = sqrt(((re * re) + (im * im)));
end

Wolfram

code[re_, im_] := N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 56.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{{re}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{re}^{2}}{{im}^{4} \cdot \log 10} + \frac{1}{2} \cdot \frac{1}{{im}^{2} \cdot \log 10}\right) + \frac{\log im}{\log 10}} \]

5. Applied rewrites 4.2%

   \[\leadsto \color{blue}{\sqrt{re \cdot re + im \cdot im}} \]
6. Add Preprocessing

Alternative 5: 3.9% accurate, 39.2× speedup

Math

\[\begin{array}{l} \\ re \cdot re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* re re))

C

double code(double re, double im) {
	return re * re;
}

Fortran

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

Java

public static double code(double re, double im) {
	return re * re;
}

Python

def code(re, im):
	return re * re

Julia

function code(re, im)
	return Float64(re * re)
end

MATLAB

function tmp = code(re, im)
	tmp = re * re;
end

Wolfram

code[re_, im_] := N[(re * re), $MachinePrecision]

Derivation

1. Initial program 56.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2} \cdot \log 10} + \frac{\log im}{\log 10}} \]

5. Applied rewrites 11.6%

   \[\leadsto \color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \log im + \color{blue}{\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}} \]

8. Applied rewrites 4.0%

   \[\leadsto re \cdot \color{blue}{re} \]
9. Add Preprocessing

Alternative 6: 3.9% accurate, 39.2× speedup

Math

\[\begin{array}{l} \\ im \cdot im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* im im))

C

double code(double re, double im) {
	return im * im;
}

Fortran

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

Java

public static double code(double re, double im) {
	return im * im;
}

Python

def code(re, im):
	return im * im

Julia

function code(re, im)
	return Float64(im * im)
end

MATLAB

function tmp = code(re, im)
	tmp = im * im;
end

Wolfram

code[re_, im_] := N[(im * im), $MachinePrecision]

Derivation

1. Initial program 56.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2} \cdot \log 10} + \frac{\log im}{\log 10}} \]

5. Applied rewrites 11.6%

   \[\leadsto \color{blue}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \log im + \color{blue}{{re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{6} \cdot \frac{{re}^{2}}{{im}^{6}} - \frac{1}{4} \cdot \frac{1}{{im}^{4}}\right) + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)} \]

8. Applied rewrites 3.8%

   \[\leadsto im \cdot \color{blue}{im} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024313 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))