math.log/1 on complex, real part

Percentage Accurate: 52.3% → 99.6%

Time: 6.6s

Alternatives: 7

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: 52.3% 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: 99.6% accurate, 1.0× speedup

Math

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

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
 (log (fma (/ re_m im_m) (* 0.5 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 log(fma((re_m / im_m), (0.5 * 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 log(fma(Float64(re_m / im_m), Float64(0.5 * 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[Log[N[(N[(re$95$m / im$95$m), $MachinePrecision] * N[(0.5 * re$95$m), $MachinePrecision] + im$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. lower-fma.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 29.4

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

5. Applied rewrites 29.4%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\frac{0.5}{im}, re \cdot re, im\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 30.7%

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

Alternative 2: 99.2% 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 57.9%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 30.4

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

5. Applied rewrites 30.4%

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

Alternative 3: 3.5% accurate, 2.5× speedup

Math

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

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) (/ 1.0 (/ (/ 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) * (1.0 / ((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) * (1.0d0 / ((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) * (1.0 / ((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) * (1.0 / ((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(0.5 / im_m) * Float64(1.0 / Float64(Float64(im_m / re_m) / 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) * (1.0 / ((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[(0.5 / im$95$m), $MachinePrecision] * N[(1.0 / N[(N[(im$95$m / re$95$m), $MachinePrecision] / re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.9%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. associate-/l* N/A

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

   13. *-rgt-identity N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. associate-/r* 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

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

   18. lower-/.f64 N/A

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

   19. lower-log.f64 29.1

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

5. Applied rewrites 29.1%

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

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%

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

10. Applied rewrites 3.3%

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

12. Applied rewrites 3.3%

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

13. Final simplification 3.3%

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

Alternative 4: 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 57.9%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. associate-/l* N/A

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

   13. *-rgt-identity N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. associate-/r* 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

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

   18. lower-/.f64 N/A

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

   19. lower-log.f64 29.1

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

5. Applied rewrites 29.1%

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

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%

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

10. Applied rewrites 3.3%

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

Alternative 5: 3.5% accurate, 3.8× speedup

Math

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

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 57.9%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. associate-/l* N/A

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

   13. *-rgt-identity N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. associate-/r* 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

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

   18. lower-/.f64 N/A

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

   19. lower-log.f64 29.1

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

5. Applied rewrites 29.1%

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

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%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 3.3%

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

Alternative 6: 3.1% accurate, 4.6× speedup

Math

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

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
 (* (/ re_m (* im_m im_m)) (* 0.5 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 (re_m / (im_m * im_m)) * (0.5 * 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 = (re_m / (im_m * im_m)) * (0.5d0 * 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 (re_m / (im_m * im_m)) * (0.5 * 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 (re_m / (im_m * im_m)) * (0.5 * 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(re_m / Float64(im_m * im_m)) * Float64(0.5 * 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 = (re_m / (im_m * im_m)) * (0.5 * 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[(re$95$m / N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.5 * re$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.9%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. associate-/l* N/A

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

   13. *-rgt-identity N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. associate-/r* 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

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

   18. lower-/.f64 N/A

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

   19. lower-log.f64 29.1

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

5. Applied rewrites 29.1%

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

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%

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

10. Applied rewrites 3.3%

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

12. Applied rewrites 3.1%

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

Alternative 7: 2.9% accurate, 4.6× speedup

Math

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

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(0.5 / Float64(im_m * im_m)) * Float64(re_m * 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[(0.5 / N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(re$95$m * re$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.9%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. associate-/l* N/A

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

   13. *-rgt-identity N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. associate-/r* 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

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

   18. lower-/.f64 N/A

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

   19. lower-log.f64 29.1

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

5. Applied rewrites 29.1%

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

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%

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

10. Applied rewrites 2.9%

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

11. Final simplification 2.9%

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

Reproduce

herbie shell --seed 2024249 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))