math.log/1 on complex, real part

Percentage Accurate: 52.8% → 99.7%

Time: 6.2s

Alternatives: 9

Speedup: 1.2×

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: 52.8% 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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $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[(N[(re$95$m / im$95$m), $MachinePrecision] / im$95$m), $MachinePrecision]), $MachinePrecision] * re$95$m + N[Log[im$95$m], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.3%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-/l* N/A

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

   12. *-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-log.f64 24.6

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

5. Applied rewrites 24.6%

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

6. Final simplification 24.6%

   \[\leadsto \mathsf{fma}\left(0.5 \cdot \frac{\frac{re}{im}}{im}, re, \log im\right) \]
7. Add Preprocessing

Alternative 2: 99.3% accurate, 1.2× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
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

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

Fortran

im_m = abs(im)
re_m = abs(re)
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

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $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 51.3%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 26.8

      \[\leadsto \color{blue}{\log im} \]

5. Applied rewrites 26.8%

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

Alternative 3: 3.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \frac{1}{\frac{\frac{im\_m}{re\_m}}{re\_m}} \cdot \frac{0.5}{im\_m} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore (re_m im_m)
 :precision binary64
 (* (/ 1.0 (/ (/ im_m re_m) re_m)) (/ 0.5 im_m)))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double code(double re_m, double im_m) {
	return (1.0 / ((im_m / re_m) / re_m)) * (0.5 / im_m);
}

Fortran

im_m = abs(im)
re_m = abs(re)
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 = (1.0d0 / ((im_m / re_m) / re_m)) * (0.5d0 / im_m)
end function

Java

im_m = Math.abs(im);
re_m = Math.abs(re);
assert re_m < im_m;
public static double code(double re_m, double im_m) {
	return (1.0 / ((im_m / re_m) / re_m)) * (0.5 / im_m);
}

Python

im_m = math.fabs(im)
re_m = math.fabs(re)
[re_m, im_m] = sort([re_m, im_m])
def code(re_m, im_m):
	return (1.0 / ((im_m / re_m) / re_m)) * (0.5 / im_m)

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function code(re_m, im_m)
	return Float64(Float64(1.0 / Float64(Float64(im_m / re_m) / re_m)) * Float64(0.5 / im_m))
end

MATLAB

im_m = abs(im);
re_m = abs(re);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = code(re_m, im_m)
	tmp = (1.0 / ((im_m / re_m) / re_m)) * (0.5 / im_m);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $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[(1.0 / N[(N[(im$95$m / re$95$m), $MachinePrecision] / re$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.5 / im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.3%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-/l* N/A

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

   12. *-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-log.f64 24.6

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

5. Applied rewrites 24.6%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.3%

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

10. Applied rewrites 3.3%

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

12. Applied rewrites 3.3%

    \[\leadsto \frac{0.5}{im} \cdot \frac{1}{\frac{\frac{im}{re}}{\color{blue}{re}}} \]

13. Final simplification 3.3%

    \[\leadsto \frac{1}{\frac{\frac{im}{re}}{re}} \cdot \frac{0.5}{im} \]
14. Add Preprocessing

Alternative 4: 3.5% accurate, 3.2× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
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) re_m) im_m)))

C

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

Fortran

im_m = abs(im)
re_m = abs(re)
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) / re_m) * im_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $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[(0.5 / N[(N[(N[(im$95$m / re$95$m), $MachinePrecision] / re$95$m), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.3%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-/l* N/A

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

   12. *-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-log.f64 24.6

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

5. Applied rewrites 24.6%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.3%

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

10. Applied rewrites 3.3%

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

12. Applied rewrites 3.3%

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

13. Final simplification 3.3%

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

Alternative 5: 3.5% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

im_m = abs(im)
re_m = abs(re)
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 = ((re_m / im_m) * re_m) * (0.5d0 / im_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $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[(N[(re$95$m / im$95$m), $MachinePrecision] * re$95$m), $MachinePrecision] * N[(0.5 / im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.3%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-/l* N/A

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

   12. *-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-log.f64 24.6

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

5. Applied rewrites 24.6%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.3%

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

10. Applied rewrites 3.3%

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

11. Final simplification 3.3%

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

Alternative 6: 3.5% accurate, 3.8× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
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) (/ re_m im_m)))

C

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

Fortran

im_m = abs(im)
re_m = abs(re)
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) * (re_m / im_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $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[(N[(0.5 / im$95$m), $MachinePrecision] * re$95$m), $MachinePrecision] * N[(re$95$m / im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.3%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-/l* N/A

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

   12. *-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-log.f64 24.6

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

5. Applied rewrites 24.6%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.3%

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

Alternative 7: 3.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

im_m = abs(im)
re_m = abs(re)
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 = -re_m * (((-0.5d0) * re_m) / (im_m * im_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $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[((-re$95$m) * N[(N[(-0.5 * re$95$m), $MachinePrecision] / N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.3%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-/l* N/A

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

   12. *-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-log.f64 24.6

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

5. Applied rewrites 24.6%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.3%

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

10. Applied rewrites 3.3%

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

12. Applied rewrites 3.0%

    \[\leadsto \frac{-0.5 \cdot re}{im \cdot im} \cdot \left(-re\right) \]

13. Final simplification 3.0%

    \[\leadsto \left(-re\right) \cdot \frac{-0.5 \cdot re}{im \cdot im} \]
14. Add Preprocessing

Alternative 8: 3.1% accurate, 4.3× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
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

im_m = fabs(im);
re_m = fabs(re);
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

im_m = abs(im)
re_m = abs(re)
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

im_m = Math.abs(im);
re_m = Math.abs(re);
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

im_m = math.fabs(im)
re_m = math.fabs(re)
[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

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

MATLAB

im_m = abs(im);
re_m = abs(re);
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

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $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[(N[(-0.5 / N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * re$95$m), $MachinePrecision] * (-re$95$m)), $MachinePrecision]

Derivation

1. Initial program 51.3%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-/l* N/A

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

   12. *-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-log.f64 24.6

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

5. Applied rewrites 24.6%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.3%

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

10. Applied rewrites 3.3%

    \[\leadsto \frac{re \cdot \frac{0.5}{im}}{-im} \cdot \left(-re\right) \]

11. Taylor expanded in re around 0

    \[\leadsto \left(\frac{-1}{2} \cdot \frac{re}{{im}^{2}}\right) \cdot \left(-re\right) \]
12. Step-by-step derivation

13. Applied rewrites 3.0%

    \[\leadsto \left(\frac{-0.5}{im \cdot im} \cdot re\right) \cdot \left(-re\right) \]
14. Add Preprocessing

Alternative 9: 2.9% accurate, 4.6× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
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 re_m) re_m) (* im_m im_m)))

C

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

Fortran

im_m = abs(im)
re_m = abs(re)
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 * re_m) * re_m) / (im_m * im_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $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[(N[(0.5 * re$95$m), $MachinePrecision] * re$95$m), $MachinePrecision] / N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.3%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-/l* N/A

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

   12. *-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-log.f64 24.6

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

5. Applied rewrites 24.6%

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

6. Taylor expanded in re around inf

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

8. Applied rewrites 3.3%

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

10. Applied rewrites 2.8%

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

11. Final simplification 2.8%

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

Reproduce

herbie shell --seed 2024295 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))