math.log/1 on complex, real part

Percentage Accurate: 51.7% → 99.7%

Time: 2.4s

Alternatives: 2

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: 51.7% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: re should be positive before calling this function
NOTE: im should be positive before calling this function
NOTE: re and im should be sorted in increasing order before calling this function.
code[re_, im_] := N[(N[Log[im], $MachinePrecision] + N[(N[(N[(0.5 * re), $MachinePrecision] / im), $MachinePrecision] * N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.0%

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

   1. hypot-def 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 19.2%

   \[\leadsto \color{blue}{\log im + 0.5 \cdot \frac{{re}^{2}}{{im}^{2}}} \]
5. Step-by-step derivation

   1. associate-*r/ 19.2%

      \[\leadsto \log im + \color{blue}{\frac{0.5 \cdot {re}^{2}}{{im}^{2}}} \]

   2. unpow2 19.2%

      \[\leadsto \log im + \frac{0.5 \cdot \color{blue}{\left(re \cdot re\right)}}{{im}^{2}} \]

   3. unpow2 19.2%

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

6. Simplified 19.2%

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

   1. associate-*r* 19.2%

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

   2. times-frac 22.0%

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

8. Applied egg-rr 22.0%

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

9. Final simplification 22.0%

   \[\leadsto \log im + \frac{0.5 \cdot re}{im} \cdot \frac{re}{im} \]

Alternative 2: 99.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} re = |re|\\ im = |im|\\ [re, im] = \mathsf{sort}([re, im])\\ \\ \log im \end{array} \]

FPCore

NOTE: re should be positive before calling this function
NOTE: im should be positive before calling this function
NOTE: re and im should be sorted in increasing order before calling this function.
(FPCore (re im) :precision binary64 (log im))

C

re = abs(re);
im = abs(im);
assert(re < im);
double code(double re, double im) {
	return log(im);
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: re should be positive before calling this function
NOTE: im should be positive before calling this function
NOTE: re and im should be sorted in increasing order before calling this function.
code[re_, im_] := N[Log[im], $MachinePrecision]

Derivation

1. Initial program 59.0%

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

   1. hypot-def 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 23.8%

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

5. Final simplification 23.8%

   \[\leadsto \log im \]

Reproduce

herbie shell --seed 2023199 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))