math.log/1 on complex, real part

Percentage Accurate: 52.2% → 99.7%

Time: 6.6s

Alternatives: 5

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \log im\_m + 0.5 \cdot \left(re\_m \cdot \frac{\frac{re\_m}{im\_m}}{im\_m}\right) \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (+ (log im_m) (* 0.5 (* re_m (/ (/ re_m im_m) im_m)))))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return log(im_m) + (0.5 * (re_m * ((re_m / im_m) / im_m)));
}

Fortran

re_m = abs(re)
im_m = abs(im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function code(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = log(im_m) + (0.5d0 * (re_m * ((re_m / im_m) / im_m)))
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return Math.log(im_m) + (0.5 * (re_m * ((re_m / im_m) / im_m)));
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return math.log(im_m) + (0.5 * (re_m * ((re_m / im_m) / im_m)))

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(log(im_m) + Float64(0.5 * Float64(re_m * Float64(Float64(re_m / im_m) / im_m))))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = log(im_m) + (0.5 * (re_m * ((re_m / im_m) / im_m)));
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[Log[im$95$m], $MachinePrecision] + N[(0.5 * N[(re$95$m * N[(N[(re$95$m / im$95$m), $MachinePrecision] / im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.3%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Step-by-step derivation

   1. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in re around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\log im, \color{blue}{\left(\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}\right)}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{{re}^{2}}{{im}^{2}}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{{re}^{2}}{{im}^{2}}\right)}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{re \cdot re}{{\color{blue}{im}}^{2}}\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot \color{blue}{\frac{re}{{im}^{2}}}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{re}{{im}^{2}}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(\frac{re}{im \cdot \color{blue}{im}}\right)\right)\right)\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(\frac{\frac{re}{im}}{\color{blue}{im}}\right)\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\left(\frac{re}{im}\right), \color{blue}{im}\right)\right)\right)\right) \]

   10. /-lowering-/.f64 27.6%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{/.f64}\left(re, im\right), im\right)\right)\right)\right) \]

7. Simplified 27.6%

   \[\leadsto \color{blue}{\log im + 0.5 \cdot \left(re \cdot \frac{\frac{re}{im}}{im}\right)} \]
8. Add Preprocessing

Alternative 2: 99.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \log im\_m \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m) :precision binary64 (log im_m))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return log(im_m);
}

Fortran

re_m = abs(re)
im_m = abs(im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function code(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = log(im_m)
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return Math.log(im_m);
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return math.log(im_m)

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return log(im_m)
end

MATLAB

re_m = abs(re);
im_m = abs(im);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = log(im_m);
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[Log[im$95$m], $MachinePrecision]

Derivation

1. Initial program 47.3%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Step-by-step derivation

   1. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\log im} \]
6. Step-by-step derivation

   1. log-lowering-log.f64 29.2%

      \[\leadsto \mathsf{log.f64}\left(im\right) \]

7. Simplified 29.2%

   \[\leadsto \color{blue}{\log im} \]
8. Add Preprocessing

Alternative 3: 3.5% accurate, 23.0× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{\frac{0.5}{im\_m}}{\frac{\frac{im\_m}{re\_m}}{re\_m}} \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (/ (/ 0.5 im_m) (/ (/ im_m re_m) re_m)))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return (0.5 / im_m) / ((im_m / re_m) / re_m);
}

Fortran

re_m = abs(re)
im_m = abs(im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function code(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = (0.5d0 / im_m) / ((im_m / re_m) / re_m)
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return (0.5 / im_m) / ((im_m / re_m) / re_m);
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return (0.5 / im_m) / ((im_m / re_m) / re_m)

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(Float64(0.5 / im_m) / Float64(Float64(im_m / re_m) / re_m))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = (0.5 / im_m) / ((im_m / re_m) / re_m);
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[(0.5 / im$95$m), $MachinePrecision] / N[(N[(im$95$m / re$95$m), $MachinePrecision] / re$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.3%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Step-by-step derivation

   1. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in re around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\log im, \color{blue}{\left(\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}\right)}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{{re}^{2}}{{im}^{2}}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{{re}^{2}}{{im}^{2}}\right)}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{re \cdot re}{{\color{blue}{im}}^{2}}\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot \color{blue}{\frac{re}{{im}^{2}}}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{re}{{im}^{2}}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(\frac{re}{im \cdot \color{blue}{im}}\right)\right)\right)\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(\frac{\frac{re}{im}}{\color{blue}{im}}\right)\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\left(\frac{re}{im}\right), \color{blue}{im}\right)\right)\right)\right) \]

   10. /-lowering-/.f64 27.6%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{/.f64}\left(re, im\right), im\right)\right)\right)\right) \]

7. Simplified 27.6%

   \[\leadsto \color{blue}{\log im + 0.5 \cdot \left(re \cdot \frac{\frac{re}{im}}{im}\right)} \]

8. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {re}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), \left({\color{blue}{im}}^{2}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), \left({im}^{2}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \left({im}^{2}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \left(im \cdot \color{blue}{im}\right)\right) \]

   7. *-lowering-*.f64 2.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right) \]

10. Simplified 2.8%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \left(re \cdot re\right)}{im \cdot im}} \]
11. Step-by-step derivation

    1. *-lft-identity N/A

       \[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{2} \cdot \left(re \cdot re\right)}{im \cdot im}} \]

    2. associate-*r/ N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{2} \cdot \left(re \cdot re\right)\right)}{\color{blue}{im \cdot im}} \]

    3. frac-times N/A

       \[\leadsto \frac{1}{im} \cdot \color{blue}{\frac{\frac{1}{2} \cdot \left(re \cdot re\right)}{im}} \]

    4. frac-2neg N/A

       \[\leadsto \frac{1}{im} \cdot \frac{\mathsf{neg}\left(\frac{1}{2} \cdot \left(re \cdot re\right)\right)}{\color{blue}{\mathsf{neg}\left(im\right)}} \]

    5. frac-2neg N/A

       \[\leadsto \frac{1}{im} \cdot \frac{\frac{1}{2} \cdot \left(re \cdot re\right)}{\color{blue}{im}} \]

    6. remove-double-neg N/A

       \[\leadsto \frac{1}{im} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot \left(re \cdot re\right)}{im}\right)\right)\right)\right) \]

    7. associate-*r* N/A

       \[\leadsto \frac{1}{im} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{2} \cdot re\right) \cdot re}{im}\right)\right)\right)\right) \]

    8. associate-/l* N/A

       \[\leadsto \frac{1}{im} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot re\right) \cdot \frac{re}{im}\right)\right)\right)\right) \]

    9. distribute-rgt-neg-in N/A

       \[\leadsto \frac{1}{im} \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot re\right) \cdot \left(\mathsf{neg}\left(\frac{re}{im}\right)\right)\right)\right) \]

    10. distribute-lft-neg-in N/A

        \[\leadsto \frac{1}{im} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot re\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{re}{im}\right)\right)}\right) \]

    11. associate-*r* N/A

        \[\leadsto \left(\frac{1}{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot re\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{re}{im}\right)\right)} \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot re\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{re}{im}\right)\right)}\right) \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{im}\right), \left(\mathsf{neg}\left(\frac{1}{2} \cdot re\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{re}{im}}\right)\right)\right) \]

    14. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, im\right), \left(\mathsf{neg}\left(\frac{1}{2} \cdot re\right)\right)\right), \left(\mathsf{neg}\left(\frac{\color{blue}{re}}{im}\right)\right)\right) \]

    15. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, im\right), \left(\mathsf{neg}\left(re \cdot \frac{1}{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{re}{im}\right)\right)\right) \]

    16. distribute-rgt-neg-in N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, im\right), \left(re \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{re}{\color{blue}{im}}\right)\right)\right) \]

    17. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, im\right), \mathsf{*.f64}\left(re, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{re}{\color{blue}{im}}\right)\right)\right) \]

    18. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, im\right), \mathsf{*.f64}\left(re, \frac{-1}{2}\right)\right), \left(\mathsf{neg}\left(\frac{re}{im}\right)\right)\right) \]

    19. distribute-frac-neg2 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, im\right), \mathsf{*.f64}\left(re, \frac{-1}{2}\right)\right), \left(\frac{re}{\color{blue}{\mathsf{neg}\left(im\right)}}\right)\right) \]

    20. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, im\right), \mathsf{*.f64}\left(re, \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(re, \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right)\right) \]

    21. neg-sub0 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, im\right), \mathsf{*.f64}\left(re, \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(re, \left(0 - \color{blue}{im}\right)\right)\right) \]

    22. --lowering--.f64 3.3%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, im\right), \mathsf{*.f64}\left(re, \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(re, \mathsf{\_.f64}\left(0, \color{blue}{im}\right)\right)\right) \]

12. Applied egg-rr 3.3%

    \[\leadsto \color{blue}{\left(\frac{1}{im} \cdot \left(re \cdot -0.5\right)\right) \cdot \frac{re}{0 - im}} \]
13. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\left(\frac{1}{im} \cdot \left(re \cdot \frac{-1}{2}\right)\right) \cdot re}{\color{blue}{0 - im}} \]

    2. *-commutative N/A

       \[\leadsto \frac{\left(\left(re \cdot \frac{-1}{2}\right) \cdot \frac{1}{im}\right) \cdot re}{0 - im} \]

    3. un-div-inv N/A

       \[\leadsto \frac{\frac{re \cdot \frac{-1}{2}}{im} \cdot re}{0 - im} \]

    4. associate-/r/ N/A

       \[\leadsto \frac{\frac{re \cdot \frac{-1}{2}}{\frac{im}{re}}}{\color{blue}{0} - im} \]

    5. associate-/l/ N/A

       \[\leadsto \frac{re \cdot \frac{-1}{2}}{\color{blue}{\left(0 - im\right) \cdot \frac{im}{re}}} \]

    6. *-commutative N/A

       \[\leadsto \frac{\frac{-1}{2} \cdot re}{\color{blue}{\left(0 - im\right)} \cdot \frac{im}{re}} \]

    7. times-frac N/A

       \[\leadsto \frac{\frac{-1}{2}}{0 - im} \cdot \color{blue}{\frac{re}{\frac{im}{re}}} \]

    8. metadata-eval N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{0 - im} \cdot \frac{re}{\frac{im}{re}} \]

    9. sub0-neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\mathsf{neg}\left(im\right)} \cdot \frac{re}{\frac{im}{re}} \]

    10. frac-2neg N/A

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

    11. clear-num N/A

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

    12. un-div-inv N/A

        \[\leadsto \frac{\frac{\frac{1}{2}}{im}}{\color{blue}{\frac{\frac{im}{re}}{re}}} \]

    13. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{im}\right), \color{blue}{\left(\frac{\frac{im}{re}}{re}\right)}\right) \]

    14. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, im\right), \left(\frac{\color{blue}{\frac{im}{re}}}{re}\right)\right) \]

    15. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, im\right), \mathsf{/.f64}\left(\left(\frac{im}{re}\right), \color{blue}{re}\right)\right) \]

    16. /-lowering-/.f64 3.3%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, im\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(im, re\right), re\right)\right) \]

14. Applied egg-rr 3.3%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{im}}{\frac{\frac{im}{re}}{re}}} \]
15. Add Preprocessing

Alternative 4: 3.5% accurate, 23.0× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{0.5}{im\_m} \cdot \frac{re\_m}{\frac{im\_m}{re\_m}} \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (* (/ 0.5 im_m) (/ re_m (/ im_m re_m))))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return (0.5 / im_m) * (re_m / (im_m / re_m));
}

Fortran

re_m = abs(re)
im_m = abs(im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function code(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = (0.5d0 / im_m) * (re_m / (im_m / re_m))
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return (0.5 / im_m) * (re_m / (im_m / re_m));
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return (0.5 / im_m) * (re_m / (im_m / re_m))

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(Float64(0.5 / im_m) * Float64(re_m / Float64(im_m / re_m)))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = (0.5 / im_m) * (re_m / (im_m / re_m));
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[(0.5 / im$95$m), $MachinePrecision] * N[(re$95$m / N[(im$95$m / re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.3%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Step-by-step derivation

   1. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in re around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\log im, \color{blue}{\left(\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}\right)}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{{re}^{2}}{{im}^{2}}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{{re}^{2}}{{im}^{2}}\right)}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{re \cdot re}{{\color{blue}{im}}^{2}}\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot \color{blue}{\frac{re}{{im}^{2}}}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{re}{{im}^{2}}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(\frac{re}{im \cdot \color{blue}{im}}\right)\right)\right)\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(\frac{\frac{re}{im}}{\color{blue}{im}}\right)\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\left(\frac{re}{im}\right), \color{blue}{im}\right)\right)\right)\right) \]

   10. /-lowering-/.f64 27.6%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{/.f64}\left(re, im\right), im\right)\right)\right)\right) \]

7. Simplified 27.6%

   \[\leadsto \color{blue}{\log im + 0.5 \cdot \left(re \cdot \frac{\frac{re}{im}}{im}\right)} \]

8. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {re}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), \left({\color{blue}{im}}^{2}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), \left({im}^{2}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \left({im}^{2}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \left(im \cdot \color{blue}{im}\right)\right) \]

   7. *-lowering-*.f64 2.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right) \]

10. Simplified 2.8%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \left(re \cdot re\right)}{im \cdot im}} \]
11. Step-by-step derivation

    1. *-lft-identity N/A

       \[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{2} \cdot \left(re \cdot re\right)}{im \cdot im}} \]

    2. associate-*r/ N/A

       \[\leadsto \frac{1 \cdot \left(\frac{1}{2} \cdot \left(re \cdot re\right)\right)}{\color{blue}{im \cdot im}} \]

    3. frac-times N/A

       \[\leadsto \frac{1}{im} \cdot \color{blue}{\frac{\frac{1}{2} \cdot \left(re \cdot re\right)}{im}} \]

    4. frac-2neg N/A

       \[\leadsto \frac{1}{im} \cdot \frac{\mathsf{neg}\left(\frac{1}{2} \cdot \left(re \cdot re\right)\right)}{\color{blue}{\mathsf{neg}\left(im\right)}} \]

    5. frac-2neg N/A

       \[\leadsto \frac{1}{im} \cdot \frac{\frac{1}{2} \cdot \left(re \cdot re\right)}{\color{blue}{im}} \]

    6. remove-double-neg N/A

       \[\leadsto \frac{1}{im} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot \left(re \cdot re\right)}{im}\right)\right)\right)\right) \]

    7. associate-*r* N/A

       \[\leadsto \frac{1}{im} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{2} \cdot re\right) \cdot re}{im}\right)\right)\right)\right) \]

    8. associate-/l* N/A

       \[\leadsto \frac{1}{im} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot re\right) \cdot \frac{re}{im}\right)\right)\right)\right) \]

    9. distribute-rgt-neg-in N/A

       \[\leadsto \frac{1}{im} \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot re\right) \cdot \left(\mathsf{neg}\left(\frac{re}{im}\right)\right)\right)\right) \]

    10. distribute-lft-neg-in N/A

        \[\leadsto \frac{1}{im} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot re\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{re}{im}\right)\right)}\right) \]

    11. associate-*r* N/A

        \[\leadsto \left(\frac{1}{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot re\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{re}{im}\right)\right)} \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot re\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{re}{im}\right)\right)}\right) \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{im}\right), \left(\mathsf{neg}\left(\frac{1}{2} \cdot re\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{re}{im}}\right)\right)\right) \]

    14. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, im\right), \left(\mathsf{neg}\left(\frac{1}{2} \cdot re\right)\right)\right), \left(\mathsf{neg}\left(\frac{\color{blue}{re}}{im}\right)\right)\right) \]

    15. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, im\right), \left(\mathsf{neg}\left(re \cdot \frac{1}{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{re}{im}\right)\right)\right) \]

    16. distribute-rgt-neg-in N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, im\right), \left(re \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{re}{\color{blue}{im}}\right)\right)\right) \]

    17. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, im\right), \mathsf{*.f64}\left(re, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{re}{\color{blue}{im}}\right)\right)\right) \]

    18. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, im\right), \mathsf{*.f64}\left(re, \frac{-1}{2}\right)\right), \left(\mathsf{neg}\left(\frac{re}{im}\right)\right)\right) \]

    19. distribute-frac-neg2 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, im\right), \mathsf{*.f64}\left(re, \frac{-1}{2}\right)\right), \left(\frac{re}{\color{blue}{\mathsf{neg}\left(im\right)}}\right)\right) \]

    20. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, im\right), \mathsf{*.f64}\left(re, \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(re, \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right)\right) \]

    21. neg-sub0 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, im\right), \mathsf{*.f64}\left(re, \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(re, \left(0 - \color{blue}{im}\right)\right)\right) \]

    22. --lowering--.f64 3.3%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, im\right), \mathsf{*.f64}\left(re, \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(re, \mathsf{\_.f64}\left(0, \color{blue}{im}\right)\right)\right) \]

12. Applied egg-rr 3.3%

    \[\leadsto \color{blue}{\left(\frac{1}{im} \cdot \left(re \cdot -0.5\right)\right) \cdot \frac{re}{0 - im}} \]
13. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\left(\frac{1}{im} \cdot \left(re \cdot \frac{-1}{2}\right)\right) \cdot re}{\color{blue}{0 - im}} \]

    2. *-commutative N/A

       \[\leadsto \frac{\left(\left(re \cdot \frac{-1}{2}\right) \cdot \frac{1}{im}\right) \cdot re}{0 - im} \]

    3. un-div-inv N/A

       \[\leadsto \frac{\frac{re \cdot \frac{-1}{2}}{im} \cdot re}{0 - im} \]

    4. associate-/r/ N/A

       \[\leadsto \frac{\frac{re \cdot \frac{-1}{2}}{\frac{im}{re}}}{\color{blue}{0} - im} \]

    5. associate-/r* N/A

       \[\leadsto \frac{re \cdot \frac{-1}{2}}{\color{blue}{\frac{im}{re} \cdot \left(0 - im\right)}} \]

    6. times-frac N/A

       \[\leadsto \frac{re}{\frac{im}{re}} \cdot \color{blue}{\frac{\frac{-1}{2}}{0 - im}} \]

    7. metadata-eval N/A

       \[\leadsto \frac{re}{\frac{im}{re}} \cdot \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{0} - im} \]

    8. sub0-neg N/A

       \[\leadsto \frac{re}{\frac{im}{re}} \cdot \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\mathsf{neg}\left(im\right)} \]

    9. frac-2neg N/A

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

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{re}{\frac{im}{re}}\right), \color{blue}{\left(\frac{\frac{1}{2}}{im}\right)}\right) \]

    11. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(re, \left(\frac{im}{re}\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{im}\right)\right) \]

    12. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(re, \mathsf{/.f64}\left(im, re\right)\right), \left(\frac{\frac{1}{2}}{im}\right)\right) \]

    13. /-lowering-/.f64 3.3%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(re, \mathsf{/.f64}\left(im, re\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{im}\right)\right) \]

14. Applied egg-rr 3.3%

    \[\leadsto \color{blue}{\frac{re}{\frac{im}{re}} \cdot \frac{0.5}{im}} \]

15. Final simplification 3.3%

    \[\leadsto \frac{0.5}{im} \cdot \frac{re}{\frac{im}{re}} \]
16. Add Preprocessing

Alternative 5: 2.9% accurate, 23.0× speedup

Math

\[\begin{array}{l} re_m = \left|re\right| \\ im_m = \left|im\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{0.5}{im\_m \cdot im\_m} \cdot \left(re\_m \cdot re\_m\right) \end{array} \]

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (* (/ 0.5 (* im_m im_m)) (* re_m re_m)))

C

re_m = fabs(re);
im_m = fabs(im);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return (0.5 / (im_m * im_m)) * (re_m * re_m);
}

Fortran

re_m = abs(re)
im_m = abs(im)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function code(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    code = (0.5d0 / (im_m * im_m)) * (re_m * re_m)
end function

Java

re_m = Math.abs(re);
im_m = Math.abs(im);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return (0.5 / (im_m * im_m)) * (re_m * re_m);
}

Python

re_m = math.fabs(re)
im_m = math.fabs(im)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return (0.5 / (im_m * im_m)) * (re_m * re_m)

Julia

re_m = abs(re)
im_m = abs(im)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(Float64(0.5 / Float64(im_m * im_m)) * Float64(re_m * re_m))
end

MATLAB

re_m = abs(re);
im_m = abs(im);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = (0.5 / (im_m * im_m)) * (re_m * re_m);
end

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
code[re$95$m_, im$95$m_] := N[(N[(0.5 / N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(re$95$m * re$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.3%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
2. Step-by-step derivation

   1. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right)\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(re, im\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\log \left(\mathsf{hypot}\left(re, im\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in re around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\log im, \color{blue}{\left(\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}\right)}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{{re}^{2}}{{im}^{2}}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{{re}^{2}}{{im}^{2}}\right)}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{re \cdot re}{{\color{blue}{im}}^{2}}\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot \color{blue}{\frac{re}{{im}^{2}}}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{re}{{im}^{2}}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(\frac{re}{im \cdot \color{blue}{im}}\right)\right)\right)\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(\frac{\frac{re}{im}}{\color{blue}{im}}\right)\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\left(\frac{re}{im}\right), \color{blue}{im}\right)\right)\right)\right) \]

   10. /-lowering-/.f64 27.6%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(im\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{/.f64}\left(\mathsf{/.f64}\left(re, im\right), im\right)\right)\right)\right) \]

7. Simplified 27.6%

   \[\leadsto \color{blue}{\log im + 0.5 \cdot \left(re \cdot \frac{\frac{re}{im}}{im}\right)} \]

8. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2}}} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {re}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), \left({\color{blue}{im}}^{2}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), \left({im}^{2}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \left({im}^{2}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \left(im \cdot \color{blue}{im}\right)\right) \]

   7. *-lowering-*.f64 2.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right) \]

10. Simplified 2.8%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \left(re \cdot re\right)}{im \cdot im}} \]
11. Step-by-step derivation

    1. associate-*l/ N/A

       \[\leadsto \frac{\frac{1}{2}}{im \cdot im} \cdot \color{blue}{\left(re \cdot re\right)} \]

    2. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{im \cdot im}\right), \color{blue}{\left(re \cdot re\right)}\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(im \cdot im\right)\right), \left(\color{blue}{re} \cdot re\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), \left(re \cdot re\right)\right) \]

    5. *-lowering-*.f64 2.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right) \]

12. Applied egg-rr 2.7%

    \[\leadsto \color{blue}{\frac{0.5}{im \cdot im} \cdot \left(re \cdot re\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))