math.log/1 on complex, real part

Percentage Accurate: 52.2% → 99.7%

Time: 3.8s

Alternatives: 2

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: 52.2% 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, 1.0× speedup

Math

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

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

Derivation

1. Initial program 50.4%

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

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. associate-*l* N/A

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

   10. /-rgt-identity N/A

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

   11. times-frac N/A

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

   12. *-lft-identity N/A

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

   13. *-rgt-identity N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 99.8

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

5. Applied rewrites 99.8%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(re, 0.5 \cdot \frac{re}{im}, im\right)\right)} \]
6. 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 52.2%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 99.3

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

5. Applied rewrites 99.3%

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

Reproduce

herbie shell --seed 2024230 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))