math.log/1 on complex, real part

Percentage Accurate: 53.1% → 100.0%

Time: 3.9s

Alternatives: 3

Speedup: 0.6×

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: 53.1% 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: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{hypot}\left(re, im\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (log (hypot re im)))

C

double code(double re, double im) {
	return log(hypot(re, im));
}

Java

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

Python

def code(re, im):
	return math.log(math.hypot(re, im))

Julia

function code(re, im)
	return log(hypot(re, im))
end

MATLAB

function tmp = code(re, im)
	tmp = log(hypot(re, im));
end

Wolfram

code[re_, im_] := N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 57.5%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lower-hypot.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 27.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(\frac{re}{im} \cdot 0.5, re, im\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (log (fma (* (/ re im) 0.5) re im)))

C

double code(double re, double im) {
	return log(fma(((re / im) * 0.5), re, im));
}

Julia

function code(re, im)
	return log(fma(Float64(Float64(re / im) * 0.5), re, im))
end

Wolfram

code[re_, im_] := N[Log[N[(N[(N[(re / im), $MachinePrecision] * 0.5), $MachinePrecision] * re + im), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 57.5%

   \[\log \left(\sqrt{re \cdot re + im \cdot im}\right) \]
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. lower-fma.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 27.2

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

5. Applied rewrites 27.2%

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

7. Applied rewrites 28.9%

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

Alternative 3: 27.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \log im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (log im))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return log(im)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Log[im], $MachinePrecision]

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in re around 0

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

   1. lower-log.f64 28.9

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

5. Applied rewrites 28.9%

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

Reproduce

herbie shell --seed 2024322 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))